A Course in Model Theory1

Katrin Tent and Martin Ziegler

December 20, 2012

1
v3.0-77-gd59fd6c, Wed Dec 19 13:06:41 2012 +0100
Contents

1 The basics 1
1.1 Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2 Elementary extensions and compactness 17

2.1 Elementary substructures . . . . . . . . . . . . . . . . . . . . . . 17
2.2 The Compactness Theorem . . . . . . . . . . . . . . . . . . . . . 19
2.3 The Löwenheim–Skolem Theorem . . . . . . . . . . . . . . . . . . 24

3 Quantifier elimination 27
3.1 Preservation theorems . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Quantifier elimination . . . . . . . . . . . . . . . . . . . . . . . . 31
3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4 Countable models 47
4.1 The omitting types theorem . . . . . . . . . . . . . . . . . . . . . 47
4.2 The space of types . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.3 ℵ0 -categorical theories . . . . . . . . . . . . . . . . . . . . . . . . 51
4.4 The amalgamation method . . . . . . . . . . . . . . . . . . . . . 55
4.5 Prime models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5 ℵ1 -categorical theories 63
5.1 Indiscernibles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.2 ω-stable theories . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.3 Prime extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.4 Lachlan’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.5 Vaughtian pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.6 Algebraic formulas . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.7 Strongly minimal sets . . . . . . . . . . . . . . . . . . . . . . . . 81
5.8 The Baldwin–Lachlan Theorem . . . . . . . . . . . . . . . . . . . 86

vii
CONTENTS viii

6 Morley rank 87
6.1 Saturated models and the monster . . . . . . . . . . . . . . . . . 87
6.2 Morley rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.3 Countable models of ℵ1 -categorical theories . . . . . . . . . . . . 98
6.4 Computation of Morley rank . . . . . . . . . . . . . . . . . . . . 102

7 Simple theories 107

7.1 Dividing and forking . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.2 Simplicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
7.3 The independence theorem . . . . . . . . . . . . . . . . . . . . . 116
7.4 Lascar strong types . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.5 Example: pseudo-finite fields . . . . . . . . . . . . . . . . . . . . 124

8 Stable theories 127

8.1 Heirs and coheirs . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
8.2 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
8.3 Definable types . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
8.4 Elimination of imaginaries and T eq . . . . . . . . . . . . . . . . . 137
8.5 Properties of forking in stable theories . . . . . . . . . . . . . . . 143
8.6 SU-rank and the stability spectrum . . . . . . . . . . . . . . . . . 149

9 Prime extensions 154

9.1 Indiscernibles in stable theories . . . . . . . . . . . . . . . . . . . 154
9.2 Totally transcendental theories . . . . . . . . . . . . . . . . . . . 156
9.3 Countable stable theories . . . . . . . . . . . . . . . . . . . . . . 159

10 The fine structure of ℵ1 -categorical theories 161

10.1 Internal types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
10.2 Analysable types . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
10.3 Locally modular strongly minimal sets . . . . . . . . . . . . . . . 168
10.4 Hrushovski’s examples . . . . . . . . . . . . . . . . . . . . . . . . 171

A Set theory 181

A.1 Sets and classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
A.2 Ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
A.3 Cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

B Fields 187
B.1 Ordered fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
B.2 Differential fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
B.3 Separable and regular field extensions . . . . . . . . . . . . . . . 194
B.4 Pseudo-finite fields and profinite groups . . . . . . . . . . . . . . 197

C Combinatorics 202
C.1 Pregeometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
C.2 The Erdős–Makkai Theorem . . . . . . . . . . . . . . . . . . . . . 207
C.3 The Erdős–Rado Theorem . . . . . . . . . . . . . . . . . . . . . . 207
CONTENTS ix

D Solutions to exercises 209

Preface

This book aims to be an introduction to model theory which can be used with-
out any background in logic. We start from scratch, introducing first-order
logic, structures, languages etc. but move on fairly quickly to the fundamental
results in model theory and stability theory. We also decided to cover simple
theories and Hrushovski constructions, which over the last decade have devel-
oped into an important subject. We try to give the necessary background in
algebra, combinatorics and set theory either in the course of the text or in the
corresponding section of the appendices. The exercises form an integral part
of the book. Some of them are used later on, others complement the text or
present aspects of the theory that we felt should not be completely ignored. For
the most important exercises (and the more difficult ones) we include (hints for)
solutions at the end of the book. Those exercises which will be used in the text
have their solution marked with an asterisk.
The book falls into four parts. The first three chapters introduce the basics
as would be contained in a course giving a general introduction to model theory.
This first part ends with Chapter 4 which introduces and explores the notion of
a type, the topology on the space of types and a way to make sure that a certain
type will not be realised in a model to be constructed. The chapter ends with
Fraı̈ssé’s amalgamation method, a simple but powerful tool for constructing
models.
Chapter 5 is devoted to Morley’s famous theorem that a theory with a unique
model in some uncountable cardinality has a unique model in every uncountable
cardinality. To prove this theorem, we describe the analysis of uncountably
categorical theories due to Baldwin and Lachlan in terms of strongly minimal
sets. These are in some sense the easiest examples of stable theories and serve
as an introduction to the topic. This chapter forms a unit with Chapter 6 in
which the Morley rank is studied in a bit more detail.
For the route to more general stable theories we decided to go via simplicity.
The notion of a simple theory was introduced by Shelah in [56]. Such theories
allow for a notion of independence which is presented in Chapter 7. Funda-
mental examples such as pseudo-finite fields make simple theories an important
generalisation of the stable ones. We specialise this notion of independence in
Chapter 8 to characterise forking in stable theories.
In Chapters 9 and 10 we go back to more classical topics of stability theory
such as existence and uniqueness of prime extensions and their analysis in the

x
CONTENTS xi

uncountably categorical case due to Hrushovski. We end the exposition by

explaining a variant of Hrushovski’s construction of a strongly minimal set.
Model theory does not exist independently of set theory or other areas of
mathematics. Many proofs require a knowledge of certain principles of infinite
combinatorics which we were hesitant to assume as universally known. Similarly,
to study theories of fields we felt it necessary to explain a certain amount of
algebra. In the three appendices we try to give enough background about set
theory and algebra to be able to follow the exposition in the text.
Other books, some general introductions, others emphasising particular as-
pects of the theory, that we recommend for further reading include those by
Pillay [44] and [42], by Marker [39], Buechler [12], Hodges [24], Poizat [45] [46],
Casanovas [14], Wagner [60] and of course Shelah [54]. We refer the reader to
these books also for their excellent accounts of the historical background on the
material we present.
We would like to thank Manuel Bleichner, Juan-Diego Caycedo, Philipp
Doebler, Heinz-Dieter Ebbinghaus, Antongiulio Fornasiero, Nina Frohn, Za-
niar Ghadernezhad, John Goodrick, Guntram Hainke, Immanuel Halupczok,
Franziska Jahnke, Leander Jehl, Itay Kaplan, Magnus Kollmann, Alexander
Kraut, Moritz Müller, Alexandra Omar Aziz, Amador Martin Pizarro, Sebas-
tian Rombach, Lars Scheele and Nina Schwarze for carefully reading earlier
versions of the manuscript and Bijan Afshordel for suggesting Exercises 1.1.2
and 5.4.1. We also thank Andreas Baudisch for trying out the book in a semi-
nar and Bernhard Herwig, who translated early parts of the lecture notes from
which parts of this book evolved.

Freiburg, Münster February 2011

Udpate

This is an early version of a second edition. Up to now a list of minor errors

were corrected. We thank A. Berarducci, A. Blumensath and M. Junker for
bugreports.

Freiburg, Münster December 2012

Chapter 1

The basics

1.1 Structures
In this section we start at the very beginning, by introducing the prerequisites
for the objects of study. We deal with first-order logic and its structures. To
this end we first introduce the languages. These will be chosen in different ways
for the different mathematical structures that one wants to study.
Definition 1.1.1. A language L is a set of constants, function symbols and
relation symbols1 .
Function symbols and relation symbols have an arity ≥ 1. One can think
of constants as 0-ary function symbols2 . This allows us to omit the constant
symbol case in many proofs.
The language per se has no inherent meaning. However, the choice of lan-
guage will reflect the nature of the intended objects. Here are some standard
examples:
L∅ =∅ The empty language.
LAbG = {0, +, −} The language of abelian groups.
LRing = LAbG ∪ {1, ·} The language of rings.
LGroup = {e, ◦, −1 } The language of groups.
LOrder = {<} The language of orders.
LORing = LRing ∪ LOrder The language of ordered rings.
LNumbers = {0, S, +, ·, <} The language of the natural numbers.
LSet = {} The language of set theory.
The symbols are
constants: 0, 1, e
unary function symbols: −, −1 , S
binary function symbols: +, ·, ◦
binary relation symbols: <, .
1 We also use predicate for relation symbol.
2 By an unfortunate convention 0-ary relation symbols are not considered. See p. 32.

1
CHAPTER 1. THE BASICS 2

The languages obtain their meaning only when interpreted in an appropriate

structure:


Definition 1.1.2. Let L be a language. An L-structure is a pair A = A, (Z A )Z∈L ,
where
A is a non-empty set, the domain or universe of A,
ZA ∈ A if Z is a constant,
Z A : An −→ A if Z is an n-ary function symbol, and
Z A ⊆ An if Z is an n-ary relation symbol.

We call Z A the A-interpretation of Z.

The requirement on A to be not empty is merely a (sometimes annoying)
convention. The cardinality of a structure is the cardinality of its universe. We
write |A| or |A| for the cardinality of A.
Definition 1.1.3. Let A and B be L-structures. A map h : A → B is called a
homomorphism if for all a1 , . . . , an ∈ A
h(cA ) = cB
h(f A (a1 , . . . , an )) = f B (h(a1 ), . . . , h(an ))
RA (a1 , . . . , an ) ⇒ RB (h(a1 ), . . . , h(an ))
for all constants c, n-ary function symbols f and relation symbols R from L.
We denote this by
h : A → B.
If in addition h is injective and
RA (a1 , . . . , an ) ⇔ RB (h(a1 ), . . . , h(an ))
for all a1 , . . . , an ∈ A, then h is called an (isomorphic) embedding. An isomor-
phism is a surjective embedding. We denote isomorphisms by
∼
h : A → B.
If there is an isomorphism between A and B, the two structures are called
isomorphic and we write
A∼= B.
It is easy to see that being isomorphic is an equivalence relation between struc-
tures and that bijections can be used to transfer the structures between sets.
∼
Definition 1.1.4. An automorphism of A is an isomorphism A → A. The set
of automorphisms Aut(A) forms a group under composition.
Definition 1.1.5. We call A a substructure of B if A ⊆ B and if the inclusion
map is an embedding from A to B. We denote this by
A ⊆ B.
We say B is an extension of A if A is a substructure of B.
CHAPTER 1. THE BASICS 3

Remark 1.1.6. If B is an L-structure and A a non-empty subset of B, then

A is the universe of a (uniquely determined) substructure A if and only if A
contains all cB and A is closed under all operations f B . In particular, if L does
not contain any constants or function symbols, then any non-empty subset of
an L-structure is again an L-structure. Also, if h : A → B, then h(A) is the
universe of a substructure of B.
It is also clear that for any family Ai of substructures of B, the intersection
of the Ai is either empty or a substructure of B. Therefore, if S is any non-
empty subset of B, then there exists a smallest substructure A = hSiB which
contains S. We call A the substructure generated by S. If S is finite, then A is
said to be finitely generated.
If L contains a constant, then the intersection of all substructures of B is
not empty as it contains the B-interpretation of this constant. Thus B has a
smallest substructure h∅iB . If L has no constants, we set h∅iB = ∅
Lemma 1.1.7. If A is generated by S, then every homomorphism h : A → B
is determined by its values on S.

Proof. If h0 : A → B is another homomorphism, then C = {b | h(b) = h0 (b)} is

either empty or a substructure. If h and h0 coincide on S, then S is a subset of
C, and therefore C = A.
∼
Lemma 1.1.8. Let h : A → A0 be an isomorphism and B an extension of
∼
A. Then there exists an extension B0 of A0 and an isomorphism g : B → B0
extending h.
Proof. First extend the bijection h : A → A0 to a bijection g : B → B 0 and use
g to define an L-structure on B 0 .
Definition 1.1.9. Let (I, ≤) be a directed partial order. This means that for
all i, j ∈ I there exists a k ∈ I such that i ≤ k and j ≤ k. A family (Ai )i∈I of
L-structures is called directed if

i ≤ j ⇒ Ai ⊆ Aj .

If I is linearly ordered, we call (Ai )i∈I a chain. If, for example, a structure
A1 is isomorphic to a substructure A0 of itself,
∼
h0 : A0 → A1 ,

then Lemma 1.1.8 gives an extension

∼
h1 : A1 → A2 .

Continuing in this way, we obtain a chain A0 ⊆ A1 ⊆ A2 ⊆ · · · and an increasing

∼
sequence hi : Ai → Ai+1 of isomorphisms.
CHAPTER 1. THE BASICS 4

Lemma
S 1.1.10. Let (Ai )i∈I be a directed family of L-structures. Then A =
i∈I A i is the universe of a (uniquely determined) L-structure
[
A= Ai ,
i∈I

which is an extension of all Ai .

Proof. Let R be an n-ary relation symbol and a1 , . . . , an ∈ A. As I is directed,
there exists k ∈ I such that all ai are in Ak . We define (and this is the only
possibility)
RA (a1 , . . . , an ) ⇔ RAk (a1 , . . . , an ).
Constants and function symbols are treated similarly.
A subset K of L is called a sublanguage. An L-structure becomes a K-
structure, the reduct, by simply forgetting the interpretations of the symbols
from L \ K:
A  K = A, (Z A )Z∈K .

Conversely we call A an expansion of A  K. Here are some examples:

Let A be an L-structure.
a) Let R be an n-ary relation on A. We introduce a new relation symbol R and
we denote by
(A, R)
the expansion of A to an L ∪ {R}-structure in which R is interpreted by R.
b) For given elements a1 , . . . , an we may introduce new constants a1 , . . . , an and
consider the L ∪ {a1 , . . . , an }-structure

(A, a1 , . . . , an ) .

c) Let B be a subset of A. By considering every element of B as a new constant,

we obtain the new language

L(B) = L ∪ B

and the L(B)-structure

AB = (A, b)b∈B .
Note that Aut(AB ) is the group of automorphisms of A fixing B elementwise.
We denote this group by Aut(A/B).
Similarly, if C is a set of new constants, we write L(C) for the language
L ∪ C.
CHAPTER 1. THE BASICS 5

Many-sorted structures
Without much effort, the concepts introduced here can be extended to many-
sorted languages and structures, which we shall need to consider later on.
Let S be a set, which we call the set of sorts. An S-sorted language L is given
by a set of constants for each sort in S, and typed function and relation symbols
which carry the information about their arity and the sorts of their domain and
range. More precisely, for any tuple (s1 , . . . , sn ) and (s1 , . . . , sn , t) there is a set
of relation symbols and function

symbols, respectively. An S-sorted structure
is a pair A = A, (Z A )Z∈L , where

A is a family (As )s∈S of non-empty sets.

Z A ∈ As if Z is a constant of sort s ∈ S,
Z A : As1 × · · · × Asn −→ At if Z is a function symbol of type (s1 , . . . , sn , t),
Z A ⊆ As1 × · · · × Asn if Z is a relation symbol of type (s1 , . . . , sn ).

It should be clear how to define homomorphisms between many-sorted structures

A and B: they are given by maps taking As to Bs for s ∈ S and behaving as
before with respect to constants, function and relation symbols.
Example. Consider the two-sorted language LPerm for permutation groups with
a sort x for the set and a sort g for the group. The constants and function
symbols for LPerm are those of LGroup restricted to the sort g and an additional
function symbol ϕ of type (x, g, x). Thus, an LPerm -structure (X, G) is given by
a set X and an LGroup -structure G together with a function X × G −→ X.
Exercise 1.1.1 (Direct products). Let A1 , A2 be L-structures. Define an L-
structure A1 × A2 with universe A1 × A2 such that the natural epimorphisms
πi : A1 × A2 −→ Ai for i = 1, 2 satisfy the following universal property: given
any L-structure D and homomorphisms ϕi : D −→ Ai , i = 1, 2 there is a unique
homomorphism ψ : D −→ A1 × A2 such that πi ◦ ψ = ϕi , i = 1, 2, i.e., this is
the product in the category of L-structures with homomorphisms.
Exercise 1.1.2. Let f : A → A be an embedding. Then there is an extension
A ⊆ B and an extension of f to an automorphism g of B. We can find B as the
union of the chain A ⊆ g −1 (A) ⊆ g −2 (A) ⊆ · · · . The pair (B, g) is uniquely
determined by that property.

1.2 Language
Starting from the inventory of the languages defined in Section 1.1 we now
describe the grammar which allows us to build well-formed terms and formulas
which will again be interpreted in the according structures.
Definition 1.2.1. An L-term is a word (sequence of symbols) built from con-
stants, the function symbols of L and the variables v0 , v1 , . . . according to the
following rules:
CHAPTER 1. THE BASICS 6

1. Every variable vi and every constant c is an L-term.

2. If f is an n-ary function symbol and t1 , . . . , tn are L-terms, then f t1 . . . tn
is also an L-term.
The number of occurrences of function symbols in a term is called its com-
plexity. This will be used in induction arguments.
We often write f (t1 , . . . , tn ) instead of f t1 . . . tn for better readability and
use the usual conventions for some particular function symbols. For example

(x + y) · (z + w)

stands for
· + xy + zw
−1 −1
and (x ◦ y) for ◦ xy.
Let A be an L-structure and ~b = (b0 , b1 , . . .) a sequence of elements which
we consider as assignments to the variables v0 , v1 , . . .. If we replace in t each
variable vi by ai , the term t determines an element tA [~b] of A in an obvious way:

Definition 1.2.2. For an L-term t, an L-structure A and an assignment ~b we

define the interpretation tA [~b] by

viA [~b] = bi
cA [~b] = cA
f t1 . . . tA ~ f A (tA ~ A~
n [b] = 1 [b], . . . , tn [b]).

This (recursive) definition is possible because every term has a unique decom-
position into its constituents: if f t1 . . . tn = f t01 . . . t0n , then t1 = t01 , . . . , tn = t0n .
This as well as the following lemma are easy to prove using induction on the
complexity of the terms involved.
Lemma 1.2.3. The interpretation tA [~b] depends on bi only if vi occurs in t.
If the variables x1 , . . . , xn are pairwise distinct3 and if no other variables
occur in t, we write
t = t(x1 , . . . , xn ).
According to the previous lemma, if ~b is an assignment for the variables which
assigns ai to xi , we can define

tA [a1 , . . . , an ] = tA [~b].

If t1 , . . . , tn are terms, we can substitute t1 , . . . , tn for the variables x1 , . . . , xn .

The resulting term is denoted by

t(t1 , . . . , tn ).

One easily proves:

3 Remember that xi ∈ {v0 , v1 , . . .}.
CHAPTER 1. THE BASICS 7

Lemma 1.2.4 (Substitution Lemma).

h i
t(t1 , . . . , tn )A [~b] = tA tA
1 [~b], . . . , tA [~b] .
n

If we expand A to the L(A)-structure AA , we get as a special case

t(a1 , . . . , an )AA = tA [a1 , . . . , an ].

Lemma 1.2.5. Let h : A → B be a homomorphism and t(x1 , . . . , xn ) a term.

Then we have for all a1 , . . . , an from A

tB [h(a1 ), . . . , h(an )] = h tA [a1 , . . . , an ] .

Proof. Induction on the complexity of t.

Lemma 1.2.6. Let S be a subset of the L-structure A. Then

hSiA = tA [s1 , . . . , sn ] | t(x1 , . . . , xn ) L-term, s1 , . . . , sn ∈ S .



Proof. We may assume that S is not empty or that L contains a constant since
otherwise both sides of the equation are empty. It follows from Lemma 1.2.5
that the universe of a substructure is closed under interpretations of terms
tA [−, . . . , −]. Thus the right hand side is contained in hSiA . For the converse
we have to show that the right hand side is closed under the operations f A . The
assertion now follows using Remark 1.1.6.
A constant term is a term without variables. As a special case of Lemma 1.2.6
we thus have
h∅iA = tA | t constant L-term .


The previous lemma implies:

Corollary 1.2.7. |hSiA | ≤ max(|S|, |L|, ℵ0 ).
Proof. There are at most max(|L|, ℵ0 ) many L-terms and for every term t at
most max(|S|, ℵ0 ) many assignments of elements of S to the variables of t.
We still need to define L-formulas. These are sequences of symbols which are
built from the symbols of L, the parentheses “(” and “)” as auxiliary symbols
and the following logical symbols:
variables v0 , v1 , . . .
.
equality symbol =
negation symbol ¬
conjunction symbol ∧
existential quantifier ∃
Definition 1.2.8. L-formulas are
.
1. t1 = t2 where t1 , t2 are L-terms,
CHAPTER 1. THE BASICS 8

2. Rt1 . . . tn where R is an n-ary relation symbol from L and t1 , . . . , tn

are L-terms,
3. ¬ψ where ψ is an L-formula,
4. (ψ1 ∧ ψ2 ) where ψ1 and ψ2 are L-formulas,
5. ∃x ψ where ψ is an L-formula and x a variable.
.
Formulas of the form t1 = t2 or Rt1 . . . tn are called atomic.
As with terms, we define the complexity of a formula as the number of
occurrences of ¬, ∃ and ∧. This allows us to do induction on (the complexity
of) formulas.
We use the following abbreviations:
(ψ1 ∨ ψ2 ) = ¬(¬ψ1 ∧ ¬ψ2 )
(ψ1 → ψ2 ) = ¬(ψ1 ∧ ¬ψ2 )
(ψ1 ↔ ψ2 ) = ((ψ1 → ψ2 ) ∧ (ψ2 → ψ1 ))
∀x ψ = ¬∃x¬ψ
for disjunction, implication, equivalence and universal quantifier.
Sometimes we write t1 Rt2 for Rt1 t2 , ∃x1 . . . xn for ∃x1 . . . ∃xn and ∀x1 . . . xn
for ∀x1 . . . ∀xn . To improve readability we might use superfluous parentheses.
On the other hand we might omit parentheses with an implicit understanding
of the binding strength of logical symbols: ¬, ∃, and ∀ bind more strongly
than ∧ which in turn binds more strongly than ∨. Finally → and ↔ have the
least binding strength. For example ¬ψ1 ∧ ψ2 → ψ3 is understood to mean
((¬ψ1 ∧ ψ2 ) → ψ3 ).
Given an L-structure A and an L-formula ϕ(x1 , . . . xn ) it should now be
clear what it means for ϕ to hold for ~b. Here is the formal definition.
Definition 1.2.9. Let A be an L-structure. For L-formulas ϕ and all assign-
ments ~b we define the relation
A |= ϕ[~b]
recursively over ϕ:
.
A |= t1 = t2 [~b] ⇔ tA ~ A~
1 [b] = t2 [b]
 
A |= Rt1 . . . tn [~b] ⇔ R A tA ~ A~
1 [b], . . . , tn [b]

A |= ¬ψ [~b] ⇔ A 6|= ψ [~b]

A |= (ψ1 ∧ ψ2 ) [~b] ⇔ A |= ψ1 [~b] and A |= ψ2 [~b]
h ai
A |= ∃xψ [~b] ⇔ there exists a ∈ A such that A |= ψ ~b .
x
Here we use the notation
~b a = (b0 , . . . , bi−1 , a, bi+1,... ) if x = vi .
x
If A |= ϕ[~b] holds we say ϕ holds in A for ~b or ~b satisfies ϕ (in A).
CHAPTER 1. THE BASICS 9

For this definition to work one has to check that every formula has a unique
decomposition 4 into subformulas: if Rt1 . . . tn = Rt01 . . . t0n , then t1 = t01 , . . . , tn =
t0n ; and (ψ1 ∧ ψ2 ) = (ψ10 ∧ ψ20 ) implies ψ1 = ψ10 and ψ2 = ψ20 .
It should be clear that our abbreviations have the intended meaning, e.g.,

A |= (ψ1 → ψ2 )[~b] if and only if (A |= ψ1 [~b] implies A |= ψ2 [~b]).

Whether ϕ holds in A for ~b depends only on the free variables of ϕ:

Definition 1.2.10. The variable x occurs freely in the formula ϕ if it occurs at
a place which is not within the scope of a quantifier ∃x. Otherwise its occurrence
is called bound. Here is the formal definition (recursive in ϕ):
.
x free in t1 = t2 ⇔ x occurs in t1 or in t2 .
x free in Rt1 . . . tn ⇔ x occurs in one of the ti .
x free in ¬ψ ⇔ x free in ψ.
x free in (ψ1 ∧ ψ2 ) ⇔ x free in ψ1 or x free in ψ2 .
x free in ∃y ψ ⇔ x 6= y and x free in ψ.

For example the variable v0 does not occur freely in ∀v0 (∃v1 R(v0 , v1 ) ∧
P (v1 )); v1 occurs both freely and bound5 .

Lemma 1.2.11. Suppose ~b and ~c agree on all variables which are free in ϕ.
Then
A |= ϕ[~b] ⇔ A |= ϕ[~c].
Proof. By induction on the complexity of ϕ.
If we write a formula in the form ϕ(x1 , . . . , xn ), we mean:
• the xi are pairwise distinct,
• all free variables in ϕ are among {x1 , . . . , xn }.
If furthermore a1 , . . . , an are elements of the structure A, we define

A |= ϕ[a1 , . . . , an ]

by A |= ϕ[~b], where ~b is an assignment satisfying ~b(xi ) = ai . Because of

Lemma 1.2.11 this is well defined.
Thus ϕ(x1 , . . . , xn ) defines an n-ary relation

ϕ(A) = {a | A |= ϕ[a]}

on A, the realisation set of ϕ. Such realisation sets are called 0-definable

subsets of An , or 0-definable relations.
4 It is precisely because of this uniqueness that we introduced brackets when defining for-

mulas.
5 However, we usually make sure that no variable occurs both freely and bound. This can

be done by renaming the bound occurrence with an unused variable.

CHAPTER 1. THE BASICS 10

Let B be a subset of A. A B-definable subset of A is a set of the form ϕ(A)

for an L(B)-formula ϕ(x). We also say that ϕ (and ϕ(A)) are defined over B and
that the set ϕ(A) is defined by ϕ. Often we don’t explicitly specify a parameter
set B and just talk about definable subsets. A 0-definable set is definable over
the empty set. We call two formulas equivalent if in every structure they define
the same set.
Definition 1.2.12. A formula ϕ without free variables is called a sentence. We
write A |= ϕ if A |= ϕ[~b] for some (all) ~b.
In that case A is called a model of ϕ. We also say ϕ holds in A. If Σ is a set
of sentences, then A is a model of Σ if all sentences of Σ hold in A. We denote
this by
A |= Σ.
Let ϕ = ϕ(x1 , . . . , xn ) and let t1 , . . . , tn be terms. The formula
ϕ(t1 , . . . , tn )
is the formula obtained by first renaming all bound variables by variables which
do not occur in the ti and then replacing every free occurrence of xi by ti .
Lemma 1.2.13 (Substitution lemma).

A |= ϕ(t1 , . . . , tn )[~b] ⇐⇒ A |= ϕ tA ~ A~
 
1 [b], . . . , tn [b] .

Proof. The proof is an easy induction on ϕ.

Also note this (trivial) special case:
AA |= ϕ(a1 , . . . , an ) ⇐⇒ A |= ϕ[a1 , . . . , an ].
Henceforth we often suppress the assignment (and the subscript) and simply
write
A |= ϕ(a1 , . . . , an ).
Atomic formulas and their negations are called basic. Formulas without
quantifiers (or: quantifier-free formulas) are Boolean combinations of basic for-
V successivelyWapplying ¬ and
mulas, i.e., they can be built from basic formulas by
∧. The conjunction of formulas πiVis denotedWby i<m πi and i<m πi denotes
their disjunction. By convention i<1 πi = i<1 πi = π0 . It is convenient to
allow the empty conjunction and the empty disjunction. For that we introduce
two new atomic formulas: the formula >, which is always true, and the formula
⊥, which is always false. We define
^
πi = >
i<0
_
πi =⊥
i<0

A formula is in negation normal form if it is built from basic formulas using

∧, ∨, ∃, ∀.
CHAPTER 1. THE BASICS 11

Lemma 1.2.14. Every formula can be transformed into an equivalent formula

which is in negation normal form.
Proof. Let ∼ denote equivalence of formulas. We consider formulas which are
built using ∧, ∨, ∃, ∀ and ¬ and move the negation symbols in front of atomic
formulas using

¬(ϕ ∧ ψ) ∼ (¬ϕ ∨ ¬ψ)

¬(ϕ ∨ ψ) ∼ (¬ϕ ∧ ¬ψ)
¬∃xϕ ∼ ∀x¬ϕ
¬∀xϕ ∼ ∃x¬ϕ
¬¬ϕ ∼ ϕ.

Definition 1.2.15. A formula in negation normal form which does not contain
any existential quantifier is called universal. Formulas in negation normal form
without universal quantifiers are called existential.

Clearly an isomorphism h : A → B preserves the validity of every formula:

A |= ϕ[a1 , . . . , an ] ⇐⇒ B |= ϕ[h(a1 ), . . . , h(an )].

Embeddings preserve the validity of existential formulas:

Lemma 1.2.16. Let h : A → B be an embedding. Then for all existential

formulas ϕ(x1 , . . . , xn ) and all a1 , . . . , an ∈ A we have

A |= ϕ[a1 , . . . , an ] =⇒ B |= ϕ[h(a1 ), . . . , h(an )].

For universal ϕ, the dual holds:

B |= ϕ[h(a1 ), . . . , h(an )] =⇒ A |= ϕ[a1 , . . . , an ].

Proof. By an easy induction on ϕ: for basic formulas the assertion follows from
the definition of an embedding and Lemma 1.2.5. The inductive step is trivial
for the cases ∧ and ∨. Let finally ϕ(x) be ∃y ψ(x, y). If A |= ϕ[a], there exists
an a ∈ A such that A |= ψ[a, a]. By induction we have B |= ψ[h(a), h(a)]. Thus
B |= ϕ[h(a)].
Let A be an L-structure. The atomic diagram of A is
 
Diag(A) = ϕ basic L(A)-sentence  AA |= ϕ ,

the set of all basic sentences with parameters from A which hold in A.
Lemma 1.2.17. The models of Diag(A) are precisely those structures
(B, h(a))a∈A for embeddings h : A → B.
CHAPTER 1. THE BASICS 12

Proof. The structures (B, h(a))a∈A are models of the atomic diagram by Lemma 1.2.16.
For the converse note that a map h is an embedding if and only if it preserves
the validity of all formulas of the form
.
(¬) x1 = x2
.
c = x1
.
f (x1 , . . . , xn ) = x0
(¬) R(x1 , . . . , xn ).

Many-sorted languages
In a many-sorted language with sorts in S, terms and formulas are built with
respect to the sorts. For each sort s ∈ S we have variables v0s , v1s , . . . from which
we build the following terms of sort s.
Every variable vis is an L-term of sort s.
Every constant c of sort s is an L-term of sort s.
If f is a function symbol of type (s1 , . . . , sn , s) and ti is an L-term of sort
si for i = 1, . . . , n, then f t1 . . . tn is an L-term of sort s.
We construct L-formulas as before with the following adjustments:
.
t1 = t2 where t1 , t2 are L-terms of the same sort,
Rt1 . . . tn where R is a relation symbol from L of type (s1 , . . . , sn )
and ti is an L-term of sort si ,
∃x ψ where ψ is an L-formula and x a variable (of some sort s).
It should be clear how to extend the definitions of this section to the many-
sorted situation and that the results presented here continue to hold without
change. In what follows we will not deal separately with many-sorted languages
until we meet them again in Section 8.4.
Exercise 1.2.1. Let L be a language and P be a new n-ary relation symbol. Let
ϕ = ϕ(P ) be an L(P ) = L ∪ {P }-sentence and π(x1 , . . . , xn ) an L-formula. Now
replace every occurrence of P in ϕ by π. More precisely, every subformula of the
form P t1 . . . tn is replaced by π(t1 . . . tn ). We denote the resulting L-formula by
ϕ(π). Show that

A |= ϕ(π) if and only if (A, π(A)) |= ϕ(P ).

Exercise 1.2.2. Every quantifier-free formula is equivalent to a formula of the

form ^ _
πij
i<m j<mi
CHAPTER 1. THE BASICS 13

and to a formula of the form _ ^

πij
i<m j<mi

where the πij are basic formulas. The first form is called the conjunctive normal
form; the second, the disjunctive normal form.
Exercise 1.2.3. Every formula is equivalent to a formula in prenex normal
form:
Q1 x1 . . . Qn xn ϕ.
The Qi are quantifiers (∃ or ∀) and ϕ is quantifier-free.
Exercise 1.2.4 (Ultraproducts and Los’s Theorem). A filter on a set I is a
non-empty set F ⊆ P(I) which does not contain the empty set and is closed
under intersections and supersets, i.e., for A, B ∈ F, we have A ∩ B ∈ F and if
A ∈ F and A ⊆ C ⊆ I we have C ∈ F. A filter F is called an ultrafilter if for
every A ∈ P we have A ∈ F or I \ A ∈ F. (By Zorn’s Lemma, any filter can be
extended to an ultrafilter.)
For a family (Ai | i ∈ I) of L-structures and F an ultrafilter on I we define
the ultraproduct Πi∈I Ai /F as follows. On the Cartesian product Πi∈I Ai , the
ultrafilter F defines an equivalence relation ∼F by

(ai )i∈I ∼F (bi )i∈I ⇔ {i ∈ I | ai = bi } ∈ F.

On the set of equivalence classes (ai )F we define an L-structure Πi∈I Ai /F.

• For constants c ∈ L, put cΠF Ai = (cAi )F .
• For n-ary function symbols f ∈ L put

f ΠF Ai ((a1i )F , . . . , (ani )F )) = (f Ai (a1i , . . . , ani ))F .

• For n-ary relation symbols R ∈ L put

RΠF Ai ((a1i )F , . . . , (ani )F )) ⇔ {i ∈ I | RAi (a1i , . . . , ani )} ∈ F.

1. Show that the ultraproduct Πi∈I Ai /F is well-defined.

2. Prove Los’s Theorem: for any L-formula ϕ we have

Πi∈I Ai /F |= ϕ((a1i )F , . . . , (ani )F ) ⇔ {i ∈ I | Ai |= ϕ(a1i , . . . , ani )} ∈ F.

1.3 Theories
Having defined a language, we can now take a closer look at which formulas
hold in a given structure. Conversely, we can start with a set of sentences and
consider those structures in which they hold. In this way, these sentences serve
as a set of axioms for a theory.
CHAPTER 1. THE BASICS 14

Definition 1.3.1. An L-theory T is a set of L-sentences.

A theory which has a model is a consistent theory. More generally, we call
a set Σ of L-formulas consistent if there is an L-structure A and an assignment
~b such that A |= ϕ[~b] for all ϕ ∈ Σ. We say that Σ is consistent with T if T ∪ Σ
is consistent.
Lemma 1.3.2. Let T be an L-theory and L0 be an extension of L. Then T is
consistent as an L-theory if and only if T is consistent as a L0 -theory.
Proof. This follows from the (trivial) fact, that every L-structure is expandable
to an L0 -structure.
Example. To keep algebraic expressions readable we will write 0 and 1 for
the symbols 0 and 1 in the following examples. We will omit the dot for the
multiplication and brackets if they are implied by the order of operations rule.
AbG, the theory of abelian groups, has the axioms:
.
• ∀x, y, z (x + y) + z = x + (y + z)
.
• ∀x 0 + x = x
.
• ∀x (−x) + x = 0
.
• ∀x, y x + y = y + x .
Ring, the theory of commutative rings:
• AbG
.
• ∀x, y, z (xy)z = x(yz)
.
• ∀x 1x = x
.
• ∀x, y xy = yx
.
• ∀x, y, z x(y + z) = xy + xz .
Field, the theory of fields:
• Ring
.
• ¬0 = 1
. .
• ∀x (¬x = 0 → ∃y xy = 1) .
Definition 1.3.3. If a sentence ϕ holds in all models of T , we say that ϕ follows
from T (or that T proves ϕ) and write6

T ` ϕ.

By Lemma 1.3.2 this relation is independent of the language. Sentences ϕ

which follow from the empty theory ∅ are called valid . We denote this by ` ϕ.
The most important properties of ` are:
6 Note that sometimes this relation is denoted by T |= ϕ to distinguish this notion from

the more syntactic notion of logical inference, see [57, section 2.6].
CHAPTER 1. THE BASICS 15

Lemma 1.3.4. 1. If T ` ϕ and T ` (ϕ → ψ), then T ` ψ.

2. If T ` ϕ(c1 , . . . , cn ) and the constants c1 , . . . , cn occur neither in T nor in
ϕ(x1 , . . . , xn ), then T ` ∀x1 . . . xn ϕ(x1 , . . . , xn ).
Proof. We prove 2. Let L0 = L \ {c1 , . . . , cn }. If the L0 -structure A is a model
of T and a1 , . . . , an are arbitrary elements, then (A, a1 , . . . , an ) |= ϕ(c1 , . . . , cn ).
That means A |= ∀x1 . . . xn ϕ(x1 , . . . , xn ). Thus T ` ∀x1 . . . xn
ϕ(x1 , . . . , xn ).
We generalise this relation to theories S: we write T ` S if all models of T
are models of S. S and T are called equivalent, S ≡ T , if S and T have the
same models.
Definition 1.3.5. A consistent L-theory T is called complete if for all L-
sentences ϕ
T ` ϕ or T ` ¬ϕ.
This notion clearly depends on L. If T is complete and L0 is an extension of
L, then T will in general not be complete as an L0 -theory.
Definition 1.3.6. For a complete theory T we define

|T | = max(|L|, ℵ0 ) :

|T | is exactly the number of L–formulas. This will be explained in the proof of

Corollary 2.1.3.
The typical (and, as we will see below, only) example of a complete theory
is the theory of a structure A

Th(A) = {ϕ | A |= ϕ}.

Lemma 1.3.7. A consistent theory is complete if and only if it is maximal

consistent, i.e., if it is equivalent to every consistent extension.
Proof. We call ϕ independent from T if neither ϕ nor ¬ϕ follows from T . So ϕ
is independent from T exactly when T ∪ {ϕ} is a proper (i.e., not equivalent)
consistent extension of T . From this the lemma follows directly.
Definition 1.3.8. Two L-structures A and B are called elementary equivalent,

A ≡ B,

if they have the same theory; that is, if for all L-sentences ϕ

A |= ϕ ⇐⇒ B |= ϕ.

Isomorphic structures are always elementarily equivalent. The converse

holds only for finite structures, see Exercise 1.3.3 and Theorem 2.3.1.
Lemma 1.3.9. Let T be a consistent theory. Then the following are equivalent:
CHAPTER 1. THE BASICS 16

a) T is complete.
b) All models of T are elementarily equivalent.
c) There exists a structure A with T ≡ Th(A).
Proof. a) ⇒ c): Let A be a model of T . If ϕ holds in A, then T 6` ¬ϕ and thus
T ` ϕ. So T ≡ Th(A) holds.
c) ⇒ b): If B |= T , then B |= Th(A) and therefore B ≡ A. Note that ≡ is
an equivalence relation.
b) ⇒ a): Let A be a model of T . If ϕ holds in A, then ϕ holds in all models
of T , i.e., T ` ϕ. Otherwise, ¬ϕ holds in A and we have T ` ¬ϕ.
From now on, when we fix a complete theory it is generally assumed to have
an infinite model. In many cases, the results will still be true for the complete
theory of a finite model, often for trivial reasons since in this case the model is
unique up to isomorphism (see Exercise 1.3.3).
A class of L-structures forms an elementary class if it is the class of models
of some L-theory T . By the previous examples, the class of all abelian groups
(commutative rings, fields, respectively) is an elementary class as is the subclass
of elementary abelian p-groups for some prime p. However, the class of all finite
abelian p-groups does not form an elementary class since by Los’s Theorem
elementary classes are closed under ultraproducts (see Exercises 1.3.4, 2.1.2 and
1.2.4).
Exercise 1.3.1. Write down the axioms for the theory DLO of dense linear
orders without endpoint in the language LOrder of orders and the axioms for the
theory ACF of algebraically closed fields in LRing . Is ACF complete?
Exercise 1.3.2. 1. For a prime number p, let Zp∞ denote the p-Prüfer
group, i.e., the group of all pk -th roots of unity for all k ∈ N. Show
that the groups Zm n
p∞ and Zp∞ are not elementarily equivalent for m 6= n.

2. Show that Zn 6≡ Zm in the language of groups if n 6= m.

Exercise 1.3.3. Show that if A is a finite L-structure and B is elementarily
equivalent to A, then they are isomorphic. (Show this first for finite L.)
Exercise 1.3.4. Use ultraproducts to show that the class of all finite groups (all
torsion groups, all nilpotent groups, respectively) does not form an elementary
class.
Exercise 1.3.5. Two structures A and B are partially isomorphic if there is a
non-empty set I of isomorphisms between substructures of A and B with the
back-and-forth property:
1. For every f ∈ I and a ∈ A there is an extension of f in I with a in its
domain.
2. For every f ∈ I and b ∈ B there is an extension of f in I with b in its
image.
Show that partially isomorphic structures are elementarily equivalent.
Chapter 2

Elementary extensions and

compactness

2.1 Elementary substructures

As in other fields of mathematics, we need to compare structures and consider
maps from one structure to another. For this to make sense we consider a fixed
language L. Maps and extensions are then required to respect this language.
Let A and B be two L-structures. A map h : A → B is called elementary
if it preserves the validity of arbitrary formulas ϕ(x1 , . . . , xn )1 . More precisely,
for all a1 , . . . , an ∈ A we have:

A |= ϕ[a1 , . . . , an ] ⇐⇒ B |= ϕ[h(a1 ), . . . , h(an )].

In particular, h preserves quantifier-free formulas and is therefore an embedding.

Hence h is also called elementary embedding. We write
≺
h : A −→ B.

The following lemma is clear.

Lemma 2.1.1. The models of Th(AA ) are exactly the structures of the form
≺
(B, h(a))a∈A for elementary embeddings h : A −→ B.
We call Th(AA ) the elementary diagram of A.
A substructure A of B is called elementary if the inclusion map is elementary,
i.e., if
A |= ϕ[a1 , . . . , an ] ⇐⇒ B |= ϕ[a1 , . . . , an ]
for all a1 , . . . , an ∈ A. In this case we write

A≺B
1 This only means that formulas which hold in A also hold in B. But taking negations, the
converse follows.

17
CHAPTER 2. ELEMENTARY EXTENSIONS AND COMPACTNESS 18

and B is called an elementary extension of A.

Theorem 2.1.2 (Tarski’s Test). Let B be an L-structure and A a subset of
B. Then A is the universe of an elementary substructure if and only if every
L(A)-formula ϕ(x) which is satisfiable in B can be satisfied by an element of A.

Proof. If A ≺ B and B |= ∃xϕ(x), we also have A |= ∃xϕ(x) and there exists

a ∈ A such that A |= ϕ(a). Thus B |= ϕ(a).
Conversely, suppose that the condition of Tarski’s test is satisfied. First
.
we show that A is the universe of a substructure A. The L(A)-formula x = x
is satisfiable in B, so A is not empty. If f ∈ L is an n-ary function symbol
(n ≥> 0; 278; 0c0) and a1 , . . . , an is from A, we consider the formula
.
ϕ(x) = f (a1 , . . . , an ) = x.

Since ϕ(x) is always satisfied by an element of A, it follows that A is closed

under f B .
Now we show, by induction on ψ, that

A |= ψ ⇐⇒ B |= ψ

for all L(A)-sentences ψ. This is clear for atomic sentences. The induction steps
for ψ = ¬ϕ and ψ = (ϕ1 ∧ ϕ2 ) are trivial.
It remains to consider the case ψ = ∃xϕ(x). If ψ holds in A, there exists
a ∈ A such that A |= ϕ(a). The induction hypothesis yields B |= ϕ(a), thus
B |= ψ. For the converse suppose ψ holds in B. Then ϕ(x) is satisfiable in B
and by Tarski’s test we find a ∈ A such that B |= ϕ(a). By induction A |= ϕ(a)
and A |= ψ holds.
We use Tarski’s Test to construct small elementary substructures.
Corollary 2.1.3. Suppose S is a subset of the L-structure B. Then B has an
elementary substructure A containing S and of cardinality at most

max(|S|, |L|, ℵ0 ).

Proof. We construct A as the union of an ascending sequence S0 ⊆ S1 ⊆ · · ·

of subsets of B. We start with S0 = S. If Si is already defined, we choose
an element aϕ ∈ B for every L(Si )-formula ϕ(x) which is satisfiable in B and
define Si+1 to be Si together with these aϕ . It is clear that A is the universe of
an elementary substructure. It remains to prove the bound on the cardinality
of A.
An L-formula is a finite sequence of symbols from L, auxiliary symbols and
logical symbols. These are |L| + ℵ0 = max(|L|, ℵ0 ) many symbols and therefore
there are exactly max(|L|, ℵ0 ) many L-formulas (see Corollary A.3.4).
Let κ = max(|S|, |L|, ℵ0 ). There are κ many L(S)-formulas: therefore |S1 | ≤
κ. Inductively it follows for every i that |Si | ≤ κ. Finally we have |A| ≤
κ · ℵ0 = κ.
CHAPTER 2. ELEMENTARY EXTENSIONS AND COMPACTNESS 19

A directed family (Ai )i∈I of structures is elementary if Ai ≺ Aj for all i ≤ j.

The following lemma is mainly applied to elementary chains, hence its name.
Theorem 2.1.4 (Tarski’s Chain Lemma). The union of an elementary directed
family is an elementary extension of all its members.
S
Proof. Let A = i∈I (Ai )i∈I . We prove by induction on ϕ(x) that for all i and
a ∈ Ai
Ai |= ϕ(a) ⇐⇒ A |= ϕ(a).
If ϕ is atomic, nothing is to be proved. If ϕ is a negation or a conjunction, the
claim follows directly from the induction hypothesis.
If ϕ(x) = ∃yψ(x, y), then ϕ(a) holds in A exactly if there exists b ∈ A with
A |= ψ(a, b). As the family is directed, there always exists a j ≥ i with b ∈ Aj .
By the induction hypothesis we have A |= ψ(a, b) ⇐⇒ Aj |= ψ(a, b). Thus ϕ(a)
holds in A exactly if it holds in an Aj (j ≥ i). Now the claim follows from
Ai ≺ Aj .
Exercise 2.1.1. Let A be an S L-structure and (Ai )i∈I a chain of elementary
substructures of A. Show that i∈I Ai is an elementary substructure of A.
Exercise 2.1.2. Consider a class C of L-structures. Prove:

1. Let Th(C) = {ϕ | A |= ϕ for all A ∈ C} be the theory of C. Then M

is a model of Th(C) if and only if M is elementarily equivalent to an
ultraproduct of elements of C.
2. Show that C is an elementary class if and only if C is closed under ultra-
products and elementary equivalence.

3. Assume that C is a class of finite structures containing only finitely many

structures of size n for each n ∈ ω. Then the infinite models of Th(C) are
exactly the models of

Tha (C) = {ϕ | A |= ϕ for all but finitely many A ∈ C}.

2.2 The Compactness Theorem

In this section we prove the Compactness Theorem, one of the fundamental
results in first-order logic. It states that a first-order theory has a model if
every finite part of it does. Its name is motivated by the results in Section 4.2
which associate to each theory a certain compact topological space.
We call a theory T finitely satisfiable if every finite subset of T is consistent.

Theorem 2.2.1 (Compactness Theorem). Finitely satisfiable theories are

consistent.
CHAPTER 2. ELEMENTARY EXTENSIONS AND COMPACTNESS 20

Let L be a language and C a set of new constants. An L(C)-theory T 0 is

called a Henkin theory if for every L(C)-formula ϕ(x) there is a constant c ∈ C
such that
∃xϕ(x) → ϕ(c) ∈ T 0 .
The elements of C are called Henkin constants of T 0 .
Let us call an L-theory T finitely complete if it is finitely satisfiable and if
every L-sentence ϕ satisfies ϕ ∈ T or ¬ϕ ∈ T . This terminology is only prelim-
inary (and not standard): by the Compactness Theorem a theory is equivalent
to a finitely complete one if and only if it is complete.
The Compactness Theorem follows from the following two lemmas.
Lemma 2.2.2. Every finitely satisfiable L-theory T can be extended to a finitely
complete Henkin theory T ∗ .
Note that conversely the lemma follows directly from the Compactness The-
orem. Choose a model A of T . Then Th(AA ) is a finitely complete Henkin
theory with A as a set of Henkin constants.
Proof. We define an increasing sequence ∅ = C0 ⊆ C1 ⊆ · · · of new constants
by assigning to every L(Ci )-formula ϕ(x) a constant cϕ(x) and
 
Ci+1 = cϕ(x)  ϕ(x) L(Ci )-formula .

Let C be the union of the Ci and T H the set of all Henkin axioms

∃xϕ(x) → ϕ(cϕ(x) )

for L(C)-formulas ϕ(x). It is easy to see that one can expand every L-structure
to a model of T H . Hence T ∪ T H is a finitely satisfiable Henkin theory. Using
the fact that the union of a chain of finitely satisfiable theories is also finitely
satisfiable, we can apply Zorn’s Lemma and get a maximal finitely satisfiable
L(C)-theory T ∗ which contains T ∪ T H . As in Lemma 1.3.7 we show that T ∗
is finitely complete: if neither ϕ nor ¬ϕ belongs to T ∗ , neither T ∗ ∪ {ϕ} nor
T ∗ ∪ {¬ϕ} would be finitely satisfiable. Hence there would be a finite subset
∆ of T ∗ which would be consistent neither with ϕ nor with ¬ϕ. Then ∆ itself
would be inconsistent and T ∗ would not be finitely satisfiable. This proves the
lemma.
Lemma 2.2.3. Every finitely complete Henkin theory T ∗ has a model A (unique
up to isomorphism) consisting of constants; i.e.,

(A, ac )c∈C |= T ∗

with A = {ac | c ∈ C}.

Proof. Let us first note that since T ∗ is finitely complete, every sentence which
follows from a finite subset of T ∗ belongs to T ∗ . Otherwise the negation of that
sentence would belong to T ∗ , but would also be inconsistent together with a
finite part of T ∗ .
CHAPTER 2. ELEMENTARY EXTENSIONS AND COMPACTNESS 21

We define for c, d ∈ C
.
c ' d ⇐⇒ c = d ∈ T ∗.
. . . . .
As c = c is valid, and d = c follows from c = d, and c = e follows from c = d and
.
d = e, we have that ' is an equivalence relation. We denote the equivalence
class of c by ac and set
A = {ac | c ∈ C}.
We expand A to an L-structure A by defining

RA (ac1 , . . . , acn ) ⇐⇒ R(c1 , . . . , cn ) ∈ T ∗ (2.1)

.
A
f (ac1 , . . . , acn ) = ac0 ⇐⇒ f (c1 , . . . , cn ) = c0 ∈ T ∗ (2.2)

for relation symbols R and function symbols f (n ≥ 0-ary) from L.

We have to show that this is well-defined. For (2.1) we have to show that

ac1 = ad1 , . . . , acn = adn , R(c1 , . . . , cn ) ∈ T ∗

implies R(d1 , . . . , dn ) ∈ T ∗ . But clearly R(d1 , . . . , dn ) holds in any structure

satisfying
. .
c1 = d1 , . . . , cn = dn , R(c1 , . . . , cn ).
Similarly for (2.2) we first notice that ac0 = ad0 follows from
. .
ac1 = ad1 , . . . , acn = adn , f (c1 , . . . , cn ) = c0 ∈ T ∗ , f (d1 , . . . , dn ) = d0 ∈ T ∗ .

For (2.2) we also have to show that for all c1 , . . . , cn there exists c0 with
.
f (c1 , . . . , cn ) = c0 ∈ T ∗ . As T ∗ is a Henkin theory, there exists c0 with
. .
∃xf (c1 , . . . , cn ) = x → f (c1 , . . . , cn ) = c0 ∈ T ∗ .
. .
Now the valid sentence ∃xf (c1 , . . . , cn ) = x belongs to T ∗ , so f (c1 , . . . , cn ) = c0
belongs to T ∗ . This shows that everything is well defined.
Let A∗ be the L(C)-structure (A, ac )c∈C . We show by induction on the
complexity of ϕ that for every L(C)-sentence ϕ

A∗ |= ϕ ⇐⇒ ϕ ∈ T ∗ .

There are four cases:

.
a) ϕ is atomic. If ϕ has the form c = d or R(c1 , . . . , cn ), the statement fol-
∗
lows from the construction of A . Other atomic sentences contain function
symbols f or constants (which we consider in this proof as 0-ary function
symbols) from L. We inductively reduce the number of such occurrences
and apply the previous case. Suppose ϕ contains a function symbol from L.
Then ϕ can be written as

ϕ = ψ(f (c1 , . . . , cn ))
CHAPTER 2. ELEMENTARY EXTENSIONS AND COMPACTNESS 22

for a function symbol f ∈ L and an L(C)-formula ψ(x). Choose c0 satisfying

. .
f (c1 , . . . , cn ) = c0 ∈ T ∗ . By construction f (c1 , . . . , cn ) = c0 holds in A∗ .
Thus A |= ϕ ⇐⇒ A |= ψ(c0 ) and ϕ ∈ T ⇐⇒ ψ(c0 ) ∈ T ∗ . From the
∗ ∗ ∗

induction hypothesis on the number of occurrences we have A∗ |= ψ(c0 ) ⇐⇒

ψ(c0 ) ∈ T ∗ . This suffices.

b) ϕ = ¬ψ. As T ∗ is finitely complete, ϕ ∈ T ∗ ⇐⇒ ψ 6∈ T ∗ holds and by the

induction hypothesis we have

A∗ |= ϕ ⇐⇒ A∗ 6|= ψ ⇐⇒ ψ 6∈ T ∗ ⇐⇒ ϕ ∈ T ∗ .

c) ϕ = (ψ1 ∧ ψ2 ). As T ∗ contains all sentences which follow from a finite subset

of T ∗ , ϕ belongs to T ∗ if and only if ψ1 and ψ2 belong to T ∗ . Thus

A∗ |= ϕ ⇐⇒ A∗ |= ψi (i = 1, 2) ⇐⇒ ψi ∈ T ∗ (i = 1, 2) ⇐⇒ ϕ ∈ T ∗ .

d) ϕ = ∃xψ(x). We have

A∗ |= ϕ ⇔ A∗ |= ψ(c) for some c ∈ C ⇔

ψ(c) ∈ T ∗ for some c ∈ C ⇔ ϕ ∈ T ∗ .

The second equivalence is the induction hypothesis and for the third we argue
as follows: if ϕ ∈ T ∗ , we choose c satisfying ϕ → ψ(c) ∈ T ∗ . As ϕ ∈ T ∗ we
also have ψ(c) ∈ T ∗ .

Corollary 2.2.4. We have T ` ϕ if and only if ∆ ` ϕ for a finite subset ∆ of

T.

Proof. The formula ϕ follows from T if and only if T ∪ {¬ϕ} is inconsistent.

Corollary 2.2.5. A set of formulas Σ(x1 , . . . , xn ) is consistent with T if and
only if every finite subset of Σ is consistent with T .
Proof. Introduce new constants c1 , . . . , cn . Then Σ is consistent with T if and
only if T ∪Σ(c1 , . . . , cn ) is consistent. Now apply the Compactness Theorem.

Definition 2.2.6. Let A be an L-structure and B ⊆ A. Then a ∈ A realises a

set of L(B)-formulas Σ(x) (containing at most the free variable x), if a satisfies
all formulas from Σ(x). We write

A |= Σ(a).

We call Σ(x) finitely satisfiable in A if every finite subset of Σ is realised in A.

Lemma 2.2.7. The set Σ(x) is finitely satisfiable in A if and only if there is
an elementary extension of A in which Σ(x) is realised.
CHAPTER 2. ELEMENTARY EXTENSIONS AND COMPACTNESS 23

Proof. By Lemma 2.1.1, Σ is realised in an elementary extension of A if and only

if Σ is consistent with Th(AA ). So the lemma follows from the easy observation
that a finite set of L(A)-formulas is consistent with Th(AA ) if and only if it is
realised in A.
Definition 2.2.8. Let A be an L-structure and B a subset of A. A set p(x) of
L(B)-formulas is a type over B if p(x) is maximal finitely satisfiable in A. We
call B the domain of p. Let

S(B) = SA (B)

denote the set of types over B.

Every element a of A determines a type

tp(a/B) = tpA (a/B) = {ϕ(x) | A |= ϕ(a), ϕ an L(B)-formula}.

So an element a realises the type p ∈ S(B) exactly if p = tp(a/B). Note that if

A0 is an elementary extension of A, then
0 0
SA (B) = SA (B) and tpA (a/B) = tpA (a/B).

We will use the notation tp(a) for tp(a/∅).

Similarly, maximal finitely satisfiable sets of formulas in x1 , . . . , xn are called
n-types and
Sn (B) = SA n (B)

denotes the set of n-types over B. For an n-tuple a from A, there is an obvious
definition of tpA (a/B) ∈ SA
n (B). Very much in the same way, we can define the
type tp(C/B) of an arbitrary set C over B. This will be convenient in later
chapters. In order to do this properly we allow free variables xc indexed by
c ∈ C and define

tp(C/B) = {ϕ(xc1 , . . . , xcn ) | A |= ϕ(c1 , . . . , cn ), ϕ an L(B)-formula}.

Many theorems which we will formulate for 1-types will hold, with the same
proofs, for n-types and often for types with infinitely many variables.
Corollary 2.2.9. Every structure A has an elementary extension B in which
all types over A are realised.

Proof. We choose for every p ∈ S(A) a new constant cp . We have to find a

model of [
Th(AA ) ∪ p(cp ).
p∈S(A)

It is easy to see that this theory is finitely satisfiable using that every p is finitely
satisfiable in A. The Compactness Theorem now shows that the model exists.
CHAPTER 2. ELEMENTARY EXTENSIONS AND COMPACTNESS 24

We give a second proof which only uses Lemma 2.2.7. Let (pα )α<λ be an
enumeration of S(A), where λ is an ordinal number (see Section A.2). Using
Theorem A.2.2, we construct an elementary chain

A = A0 ≺ A1 ≺ · · · ≺ Aβ ≺ · · · (β ≤ λ)

such that each pα is realised in Aα+1 .

Let us suppose that the elementary chain (Aα0 )α0 <β is already constructed.
If β is a limit ordinal, we let Aβ = α<β Aα .2 The longer chain (Aα0 )α0 ≤β is
S
elementary because of Lemma 2.1.4. If β = α + 1 we first note that pα is also
finitely satisfiable in Aα . Therefore we can realise pα in a suitable elementary
extension Aβ  Aα . Then B = Aλ is the model we were looking for.
Exercise 2.2.1. Prove the Compactness Theorem using ultraproducts (see Ex-
ercise 1.2.4).

Exercise 2.2.2. A class C of L-structure is finitely axiomatisable if it is the

class of models of a finite theory. Show that C is finitely axiomatisable if and
only if both C and its complement form an elementary class.
Exercise 2.2.3. Show that the class of connected graphs is not an elementary
class. A graph (V, R) is a set V with a symmetric, irreflexive binary relation. It
is connected if for any x, y ∈ V there is a sequence of elements x0 = x, . . . , xk = y
such that (xi−1 , xi ) ∈ R for i = 1, . . . , n.
Exercise 2.2.4. Let A = (R, 0, <, f A ), where f is a unary function symbol.
Call an element x ∈ A∗  A infinitesimal if − n1 < x < n1 for all all positive
natural numbers n. Show that if f A (0) = 0, then f A is continuous in 0 if and
∗
only if for any elementary extension A∗ of A the map f A takes infinitesimal
elements to infinitesimal elements.
Exercise 2.2.5. Let T be an LRing -theory containing Field. Show that:
1. If T has models of arbitrary large characteristic, then it has a model of
characteristic 0.

2. The theory of fields of characteristic 0 is not finitely axiomatisable.

2.3 The Löwenheim–Skolem Theorem

One of the consequences of the Compactness Theorem is the fact that a first-
order theory cannot pin down the size of an infinite structure. This is the
content of the following theorem.
Theorem 2.3.1 (Löwenheim–Skolem). Let B be an L-structure, S a subset of
B and κ an infinite cardinal.
2 We
S
call a chain (Aα ) indexed by ordinal numbers continuous if Aβ = α<β Aα for all
limit ordinals β.
CHAPTER 2. ELEMENTARY EXTENSIONS AND COMPACTNESS 25

1. If
max(|S|, |L|) ≤ κ ≤ |B|,
then B has an elementary substructure of cardinality κ containing S.
2. If B is infinite and
max(|B|, |L|) ≤ κ,
then B has an elementary extension of cardinality κ.
Proof. 1: Choose a set S ⊆ S 0 ⊆ B of cardinality κ and apply Corollary 2.1.3.
2: We first construct an elementary extension B0 of cardinality at least κ.
Choose a set C of new constants of cardinality κ. As B is infinite, the theory
.
Th(BB ) ∪ {¬ c = d | c, d ∈ C, c 6= d}

is finitely satisfiable (even in B: just interpret the finitely many new constants
in a finite subset by elements of B). By Lemma 2.1.1, any model (B0B , bc )c∈C
is an elementary extension of B with κ many different elements (bc ).
Finally we apply the first part of the theorem to B0 and S = B.
Note that in Theorem 2.3.1(1) the assumption κ ≥ max(|S|, |L|) is certainly
necessary in general.
Corollary 2.3.2. A theory which has an infinite model has a model in every
cardinality κ ≥ max(|L|, ℵ0 ).

Thus, no theory with an infinite model can describe this model up to iso-
morphism. So the best we can hope for is a unique model for a given cardinality.
Definition 2.3.3 (preliminary, see 2.3.5). Let κ be an infinite cardinal. A
theory T is called κ-categorical if all models of T of cardinality κ are isomorphic.

Theorem 2.3.4 (Vaught’s Test). A κ-categorical theory T is complete if the

following conditions are satisfied:
a) T is consistent,
b) T has no finite model,

c) |L| ≤ κ.
Proof. We have to show that all models A and B of T are elementarily equiv-
alent. As A and B are infinite, Th(A) and Th(B) have models A0 and B0 of
cardinality κ. By assumption A0 and B0 are isomorphic, and it follows that

A ≡ A0 ≡ B0 ≡ B.
CHAPTER 2. ELEMENTARY EXTENSIONS AND COMPACTNESS 26

Examples. 1. (Theorem of Cantor, see Exercise 1.3.1) The theory DLO of

dense linear orders without endpoints is ℵ0 -categorical and by Vaught’s
test complete. To see this let A and B be countable dense linear orders
without endpoints, and let A = {ai | i ∈ ω}, B = {bi | i ∈ ω}. We induc-
tively define sequences (ci )i<ω , (di )i<ω exhausting A and B, respectively,
and such that the assignment ci 7→ di is the required isomorphism. As-
sume that (ci )i<m , (di )i<m have been defined so that ci 7→ di , i < m is an
order isomorphism. If m = 2k, let cm = aj where aj is the element with
minimal index in {ai | i ∈ ω} not occurring in (ci )i<m . Since B is a dense
linear order without endpoints, there is some element dm ∈ {bi | i ∈ ω}
such that (ci )i≤m and (di )i≤m are order isomorphic. If m = 2k + 1 we
interchange the roles of A and B.
2. For any prime p or p = 0, the theory ACFp of algebraically closed fields
of characteristic p is κ-categorical for any κ > ℵ0 . Any two algebraically
closed fields of the same characteristic and of cardinality κ > ℵ0 have
transcendence bases (over the algebraic closure of the prime field) of car-
dinality κ, see Corollary A.3.4 and Section C.1. Any bijection between
these transcendence bases induces an isomorphism of the fields. It follows
that ACFp is complete.
Considering Theorem 2.3.4 we strengthen our definition.
Definition 2.3.5. Let κ be an infinite cardinal. A theory T is called κ-
categorical if it is complete, |T | ≤ κ and, up to isomorphism, has exactly one
model of cardinality κ.
Exercise 2.3.1. 1. Two functions f, g : N → N are almost disjoint if f (n) 6=
g(n) for almost all n. Show that there are 2ℵ0 -many almost disjoint func-
tions from N to N. (Hint: For every real r choose a sequence of rational
numbers which converges to r.)

2. Let F be the set of all functions N → N. Show that (N, yf )f ∈F has no

countable proper elementary extension.
3. Let Q be the ordered field of rational numbers. For every real r introduce
two predicates Pr , Rr for {q ∈ Q | q < r} and {q ∈ Q | r ≤ q}. Show that
(Q, Pr , Qr )r∈R has no countable proper elementary extension.
Exercise 2.3.2. 1. The theory of K-vector spaces Mod(K) (see p. 38) is
κ-categorical for κ > |K|.
2. Is ACFp ℵ0 -categorical?

Exercise 2.3.3. Show that an ∀∃-sentence, which holds in all finite fields, is
true in all algebraically closed fields.
Chapter 3

Quantifier elimination

3.1 Preservation theorems

In general, it can be difficult to tell which sentences belong to a given theory or
which extensions are consistent. Therefore it is helpful to know that in certain
theories one can restrict attention to sentences of a certain class, e.g., quantifier-
free or, say, existential formulas. We consider a fixed language L and first prove
some separation results.
Lemma 3.1.1 (Separation Lemma). Let T1 and T2 be two theories. Assume H
is a set of sentences which is closed under ∧ and ∨ and contains > and ⊥ (true
and false). Then the following are equivalent:
a) There is a sentence ϕ ∈ H which separates T1 from T2 . This means

T1 ` ϕ and T2 ` ¬ϕ.

b) All models A1 of T1 can be separated from all models A2 of T2 by a

sentence ϕ ∈ H. This means

A1 |= ϕ and A2 |= ¬ϕ.

Proof. a) ⇒ b): If ϕ separates T1 from T2 , it separates all models of T1 from all

models of T2 .
b) ⇒ a): For any model A1 of T1 let HA1 be the set of all sentences from H
which are true in A1 . b) implies that HA1 and T2 cannot have a common model.
By the Compactness Theorem there is a finite conjunction ϕA1 of sentences from
HA1 inconsistent with T2 . Clearly,

T1 ∪ {¬ϕA1 | A1 |= T1 }

is inconsistent since any model A1 of T1 satisfies ϕA1 . Again by compactness T1

implies a disjunction ϕ of finitely many of the ϕA1 . This formula ϕ is in H and
separates T1 from T2 .

27
CHAPTER 3. QUANTIFIER ELIMINATION 28

For structures A, B and a map f : A → B preserving all formulas from a set

of formulas ∆, we use the notation

f : A →∆ B.

We also write
A ⇒∆ B
to express that all sentences from ∆ true in A are also true in B.

Lemma 3.1.2. Let T be a theory, A a structure and ∆ a set of formulas, closed

under existential quantification, conjunction and substitution of variables. Then
the following are equivalent:
a) All sentences ϕ ∈ ∆ which are true in A are consistent with T .

b) There is a model B |= T and a map f : A →∆ B.

Proof. b) ⇒ a): Assume f : A →∆ B |= T . If ϕ ∈ ∆ is true in A it is also true
in B and therefore is consistent with T .
a) ⇒ b): Consider Th∆ (AA ), the set of all sentences δ(a), (δ(x) ∈ ∆), which
are true in AA . The models (B, f (a)a∈A ) of this theory correspond to maps
f : A →∆ B. This means that we have to find a model of T ∪ Th∆ (AA ). To
show finite satisfiability it is enough to show that T ∪ D is consistent for every
finite subset D of Th∆ (AA ). Let δ(a) be the conjunction of the elements of D.
Then A is a model of ϕ = ∃x δ(x), so by assumption T has a model B which is
also a model of ϕ. This means that there is a tuple b such that (B, b) |= δ(a).
Note that Lemma 3.1.2 applied to T = Th(B) shows that A ⇒∆ B if and only
if there exists a map f and a structure B0 ≡ B such that f : A →∆ B0 .
Theorem 3.1.3. Let T1 and T2 be two theories. Then the following are equiv-
alent:
a) There is a universal sentence which separates T1 from T2 .

b) No model of T2 is a substructure of a model of T1 .

Proof. a) ⇒ b): Let ϕ be a universal sentence which separates T1 from T2 . Let
A1 be a model of T1 and A2 a substructure of A1 . Since A1 is a model of ϕ,
then by Lemma 1.2.16 A2 is also model of ϕ. Therefore A2 cannot be a model
of T2 .
b) ⇒ a): If T1 and T2 cannot be separated by a universal sentence, then they
have models A1 and A2 which cannot be separated by a universal sentence. This
can be denoted by
A2 ⇒∃ A1 .
Now Lemma 3.1.2 implies that A2 has an extension A01 ≡ A1 . Then A01 is again
a model of T1 contradicting b).
CHAPTER 3. QUANTIFIER ELIMINATION 29

Definition 3.1.4. For any L-theory T , the formulas ϕ(x), ψ(x) are said to be
equivalent modulo T (or relative to T ) if T ` ∀x(ϕ(x) ↔ ψ(x̄)).
Corollary 3.1.5. Let T be a theory.
1. Consider a formula ϕ(x1 , . . . , xn ). The following are equivalent:

a) ϕ(x1 , . . . , xn ) is, modulo T , equivalent to a universal formula.

b) If A ⊆ B are models of T and a1 , . . . , an ∈ A, then B |= ϕ(a1 , . . . , an )
implies A |= ϕ(a1 , . . . , an ).
2. We say that a theory which consists of universal sentences is universal.
Then T is equivalent to a universal theory if and only if all substructures
of models of T are again models of T .
Proof. 1): Assume b). We extend L by an n-tuple c of new constants c1 , . . . , cn
and consider the theory

T1 = T ∪ {ϕ(c)} and T2 = T ∪ {¬ϕ(c)}.

Then b) says that substructures of models of T1 cannot be models of T2 . By

Theorem 3.1.3, T1 and T2 can be separated by a universal L(c)-sentence ψ(c).
By Lemma 1.3.4(2), T1 ` ψ(c) implies

T ` ∀x (ϕ(x) → ψ(x))

and from T2 ` ¬ψ(c) we see

T ` ∀x (¬ϕ(x) → ¬ψ(x)).

2): It is clear that substructures of models of a universal theory are models

again. Now suppose that a theory T has this property. Let ϕ be an axiom of T .
If A is a substructure of B, it is not possible for B to be a model of T and for
A to be a model of ¬ϕ at the same time. By 3.1.3 there is a universal sentence
ψ with T ` ψ and ¬ϕ ` ¬ψ. Hence all axioms of T follow from

T∀ = {ψ | T ` ψ, ψ universal}.

An ∀∃-formula is of the form

∀x1 . . . xn ψ

where ψ is existential (see p. 11). The following is clear.

Lemma 3.1.6. Suppose ϕ is an ∀∃-sentence, (Ai )i∈I is a directed family of

models of ϕ and B the union of the Ai . Then B is also a model of ϕ.
CHAPTER 3. QUANTIFIER ELIMINATION 30

Proof. Write
ϕ = ∀x ψ(x),
where ψ is existential. For any a ∈ B there is an Ai containing a. Since Ai |= ϕ,
clearly ψ(a) holds in Ai . As ψ(a) is existential it must also hold in B.
Definition 3.1.7. We call a theory T inductive if the union of any directed
family of models of T is again a model.
Theorem 3.1.8. Let T1 and T2 be two theories. Then the following are equiv-
alent:
a) There is an ∀∃-sentence which separates T1 from T2 .
b) No model of T2 is the union of a chain (or of a directed family) of models
of T1 .

Proof. a) ⇒ b): Assume ϕ is a ∀∃-sentence which separates T1 from T2 , (Ai )i∈I

is a directed family of models of T1 and B the union of the Ai . Since the Ai are
models of ϕ, by Lemma 3.1.6 B is also a model of ϕ. Since B |= ϕ, B cannot
be a model of T2 .
b) ⇒ a): If a) is not true, T1 and T2 have models which cannot be separated
by an ∀∃-sentence. Since ∃∀-formulas are equivalent to negated ∀∃-formulas,
we have
B0 ⇒∃∀ A.
By Lemma 3.1.2 there is a map

f : B0 → ∀ A 0

with A0 ≡ A. We can assume that B0 ⊆ A0 and f is the inclusion map. Then

A0B ⇒∃ B0B .

Applying Lemma 3.1.2 again, we obtain an extension B1B of A0B with B1B ≡ B0B ,
i.e., B0 ≺ B1 .
B0 ⊆ A0 ⊆ B1
≺

The same procedure applied to A and B1 gives us two extensions A1 ⊆ B2 with

A1 ≡ A and B1 ≺ B2 . This results in an infinite chain

B0 ⊆ A0 ⊆ B1 ⊆ A1 ⊆ B2 ⊆ . . .
≺ ≺ ≺

with Ai ≡ A and Bi ≺ Bi+1 . Let B be the union of the Ai . Since B is also the
union of the elementary chain of the Bi , it is an elementary extension of B0
and hence a model of T2 . But the Ai are models of T1 , so b) does not hold.
CHAPTER 3. QUANTIFIER ELIMINATION 31

Corollary 3.1.9. Let T be a theory.

1. For each sentence ϕ the following are equivalent:
a) ϕ is, modulo T , equivalent to an ∀∃-sentence.
b) If
A0 ⊆ A1 ⊆ · · ·
and their union B are models of T , then ϕ holds in B if it is true in
all the Ai .

2. T is inductive if and only if it can be axiomatised by ∀∃-sentences.

Proof. 1): Theorem 3.1.8 shows that ∀∃-formulas are preserved by unions of
chains. Hence a) ⇒ b). For the converse consider the theories

T1 = T ∪ {ϕ} and T2 = T ∪ {¬ϕ}.

Part b) says that the union of a chain of models of T1 cannot be a model of T2 .

By Theorem 3.1.8 we can separate T1 and T2 by an ∀∃-sentence ψ. Now T1 ` ψ
implies T ` ϕ → ψ and T2 ` ¬ψ implies T ` ¬ϕ → ¬ψ.
2): Clearly ∀∃-axiomatised theories are inductive. For the converse assume
that T is inductive and ϕ an axiom of T . If B is a union of models of T , it
cannot be a model of ¬ϕ. By Theorem 3.1.8 there is an ∀∃-sentence ψ with
T ` ψ and ¬ϕ ` ¬ψ. Hence all axioms of T follow from

T∀∃ = {ψ | T ` ψ, ψ ∀∃-formula}.

Exercise 3.1.1. Let X be a topological space, Y1 and Y2 quasi-compact1 sub-

sets, and H a set of clopen subsets. Then the following are equivalent:
a) There is a positive Boolean combination B of elements from H such that
Y1 ⊆ B and Y2 ∩ B = ∅.

b) For all y1 ∈ Y1 and y2 ∈ Y2 there is an H ∈ H such that y1 ∈ H and y2 6∈ H.

(This, in fact, is a generalisation of the Separation Lemma 3.1.1.)

3.2 Quantifier elimination

Having quantifier elimination in a reasonable language is a property which makes
a theory ‘tame’. In this section we will collect some criteria and extensions of
this concept. They will be applied in Section 3.3 to show that a number of
interesting theories have quantifier elimination in the appropriate language.
1 That is, compact but not necessarily Hausdorff.
CHAPTER 3. QUANTIFIER ELIMINATION 32

Definition 3.2.1. A theory T has quantifier elimination if every L-formula

ϕ(x1 , . . . , xn ) in theory is equivalent modulo T to some quantifier-free formula
ρ(x1 , . . . , xn ).
For n = 0 this means that modulo T every sentence is equivalent to a
quantifier-free sentence. If L has no constants, > and ⊥ are the only quantifier-
free sentences. Then T is either inconsistent or complete.
Note that it is easy to transform any theory T into a theory with quantifier
elimination if one is willing to expand the language: just enlarge L by adding an
n-place relation symbol Rϕ for every L-formula ϕ(x1 , . . . , xn ) and T by adding
all axioms
∀x1 , . . . , xn (Rϕ (x1 , . . . , xn ) ↔ ϕ(x1 , . . . , xn )).
The resulting theory, the Morleyisation T m of T , has quantifier elimination.2
Many other properties of a theory are not affected by Morleyisation. So T is
complete if and only if T m is; similarly for κ-categoricity and other properties
we will define in later chapters.
A prime structure of T is a structure which embeds into all models of T .
The following is clear.
Lemma 3.2.2. A consistent theory T with quantifier elimination which pos-
sesses a prime structure is complete.
Definition 3.2.3. A simple existential formula has the form

ϕ = ∃y ρ

for a quantifier-free formula ρ. If ρ is a conjunction of basic formulas, ϕ is called

primitive existential.
Lemma 3.2.4. The theory T has quantifier elimination if and only if every
primitive existential formula is, modulo T , equivalent to a quantifier-free for-
mula.
W
Proof. We can write every simple existential formula in the form ∃y i<n ρi for
ρi which are conjunctions of basic formulas. This shows that every simple ex-
istential formula
W is equivalent to a disjunction of primitive existential formulas,
namely to i<n (∃y ρi ). We can therefore assume that every simple existential
formula is, modulo T , equivalent to a quantifier-free formula.
We are now able to eliminate the quantifiers in arbitrary formulas in prenex
normal form (see Exercise 1.2.3)

Q1 x1 . . . Qn xn ρ.

If Qn = ∃, we choose a quantifier-free formula ρ0 which, modulo T , is equivalent

to ∃xn ρ. Then we proceed with the formula Q1 x1 . . . Qn−1 xn−1 ρ0 . If Qn = ∀,
we find a quantifier-free ρ1 which is, modulo T , equivalent to ∃xn ¬ρ and proceed
with Q1 x1 . . . Qn−1 xn−1 ¬ρ1 .
2 If T has no constants, is consistent and incomplete, one has to allow 0-ary relation symbols.
CHAPTER 3. QUANTIFIER ELIMINATION 33

The following theorem gives useful criteria for quantifier elimination.

Theorem 3.2.5. For a theory T the following are equivalent:
a) T has quantifier elimination.
b) For all models M1 and M2 of T with a common substructure A we have

M1A ≡ M2A .

c) For all models M1 and M2 of T with a common finitely generated sub-

structure A and for all primitive existential formulas ϕ(x1 , . . . , xn ) and
parameters a1 , . . . , an from A we have

M1 |= ϕ(a1 , . . . , an ) ⇒ M2 |= ϕ(a1 , . . . , an ).

If L has no constants, A is allowed to be the empty “structure”.

Proof. a) ⇒ b): Let ϕ(a) be an L(A)-sentence which holds in M1 . Choose a
quantifier-free ρ(x) which is, modulo T , equivalent to ϕ(x). Then

M1 |= ϕ(a) ⇒ M1 |= ρ(a)
⇒ A |= ρ(a) ⇒
M2 |= ρ(a) ⇒ M2 |= ϕ(a).

b) ⇒ c): Clear.
c) ⇒ a): Let ϕ(x) be a primitive existential formula. In order to show
that ϕ(x) is equivalent, modulo T , to a quantifier-free formula ρ(x) we extend
L by an n-tuple c of new constants c1 , . . . , cn . We have to show that we can
separate T ∪ {ϕ(c)} and T ∪ {¬ϕ(c)} by a quantifier free sentence ρ(c). We
apply the Separation Lemma. Let M1 and M2 be two models of T with two
distinguished n-tuples a1 and a2 . Suppose that (M1 , a1 ) and (M2 , a2 ) satisfy
the same quantifier-free L(c)-sentences. We have to show that

M1 |= ϕ(a1 ) ⇒ M2 |= ϕ(a2 ). (3.1)

i
Consider the substructures Ai = hai iM , generated by ai . If we can show that
there is an isomorphism
f : A1 → A2
taking a1 to a2 , we may assume that A1 = A2 = A and a1 = a2 = a. Then (3.1)
follows directly from c).
1
Every element of A1 has the form tM [a1 ] for an L-term t(x), (see 1.2.6).
The isomorphism f to be constructed must satisfy
1 2
f tM [a1 ] = tM [a2 ].

We now define f by this equation, and we have to check that f is well defined
and injective. Assume
1 1
sM [a1 ] = tM [a1 ].
CHAPTER 3. QUANTIFIER ELIMINATION 34

.
Then s(c) = t(c) holds in (M1 , a1 ), and by our assumption also in (M2 , a2 ),
which means 2 2
sM [a2 ] = tM [a2 ].
This shows that f is well defined. Swapping the two sides yields injectivity.
That f is surjective is clear. It remains to show that f commutes with the
interpretation of the relation symbols. Now
 1 1 M1 1
M1 |= R tM

1 [a ], . . . , tm [a ] ,

is equivalent to (M1 , a1 ) |= R(t1 (c), . . . , tm (c)), which is equivalent to

(M2 , a2 ) |= R(t1 (c), . . . , tm (c)), which in turn is equivalent to
 2 2 M2 2
M2 |= R tM

1 [a ], . . . , tm [a ] .

Note that part b) of Theorem 3.2.5 is saying that T is substructure complete;

i.e., for any model M |= T and substructure A ⊆ M the theory T ∪ Diag(A) is
complete.
Definition 3.2.6. We call T model complete if for all models M1 and M2 of T

M1 ⊆ M2 ⇒ M1 ≺ M2 .

Note that T is model complete if and only if for any M |= T the theory
T ∪ Diag(M) is complete.
Clearly, by 3.2.5(b) applied to A = M1 all theories with quantifier elimina-
tion are model complete.
Lemma 3.2.7 (Robinson’s Test). Let T be a theory. Then the following are
equivalent:
a) T is model complete.
b) For all models M1 ⊆ M2 of T and all existential sentences ϕ from L(M 1 )

M2 |= ϕ ⇒ M1 |= ϕ.

c) Each formula is, modulo T , equivalent to a universal formula.

Proof. a) ⇒ b) is trivial.
a) ⇔ c) follows from 3.1.5(1).
b) implies that every existential formula is, modulo T , equivalent to a uni-
versal formula. As in the proof of 3.2.4 this implies c).

If M1 ⊆ M2 satisfies b), we call M1 existentially closed in M2 . We denote

this by
M 1 ≺1 M 2 .
CHAPTER 3. QUANTIFIER ELIMINATION 35

Definition 3.2.8. Let T be a theory. A theory T ∗ is a model companion of T

if the following three conditions are satisfied.
a) Each model of T can be extended to a model of T ∗ .
b) Each model of T ∗ can be extended to a model of T .
c) T ∗ is model complete.
Theorem 3.2.9. A theory T has, up to equivalence, at most one model com-
panion T ∗ .
Proof. If T + is another model companion of T , every model of T + is contained
in a model of T ∗ and conversely. Let A0 be a model of T + . Then A0 can be
embedded in a model B0 of T ∗ . In turn B0 is contained in a model A1 of T + .
In this way we find two elementary chains, (Ai ) and (Bi ), which have a common
union C. Then A0 ≺ C and B0 ≺ C implies A0 ≡ B0 . Thus A0 is a model of T ∗ .
Interchanging T ∗ and T + yields that every model of T ∗ is a model of T + .

Digression: existentially closed structures and the Kaiser

hull
Let T be an L-theory. It follows from 3.1.2 that the models of T ∀ are the
substructures of models of T . The conditions a) and b) in the definition of
“model companion” can therefore be expressed as
T ∀ = T ∀∗ .
Hence the model companion of a theory T depends only on T ∀ .
Definition 3.2.10. An L-structure A is called T -existentially closed (or T -ec),
if
a) A can be embedded in a model of T .
b) A is existentially closed in every extension which is a model of T .
A structure A is T -ec exactly if it is T ∀ -ec. This is clear for condition a)
since every model B of T ∀ can be embedded in a model M of T . For b) this
follows from the fact that A ⊆ B ⊆ M and A ≺1 M implies A ≺1 B.
Lemma 3.2.11. Every model of a theory T can be embedded in a T -ec structure.
Proof. Let A be a model of T ∀ . We choose an enumeration (ϕα )α<κ of all
existential L(A)-sentences and construct an ascending chain (Aα )α≤κ of models
of T ∀ . We begin with A0 = A. Let Aα be constructed. If ϕα holds in an
extension of Aα which is a model of T , we let Aα+1 be such a model. Otherwise
we set Aα+1 = Aα . For limit ordinals λ we define Aλ to be the union of all Aα ,
(α < λ). Note that Aλ is again a model of T ∀ .
The structure A1 = Aκ has the following property: every existential L(A)-
sentence which holds in an extension of A1 that is a model of T holds in A1 .
Now, in the same manner, we construct A2 from A1 , etc. The union M of the
chain A0 ⊆ A1 ⊆ A2 ⊆ · · · is the desired T -ec structure.
CHAPTER 3. QUANTIFIER ELIMINATION 36

The structure M constructed in the proof can be very big. On the other
hand, it is easily seen that every elementary substructure N of a T -ec structure
M is again T -ec. To this end let N ⊆ A be a model of T . Since MN ⇒∃
AN , there is an embedding of M in an elementary extension B of A which is
the identity on N . Since M is existentially closed in B, it follows that N is
existentially closed in B and therefore also in A.
B
7 So
≺ S≺1
 S
A M
o
S 7

≺1S ≺
S 
N

Lemma 3.2.12 ([32]). Let T be a theory. Then there is a biggest inductive

theory T KH with T ∀ = T ∀KH . We call T KH the Kaiser hull of T .

Proof. Let T 1 and T 2 be two inductive theories with T ∀1 = T ∀2 = T ∀ . We have

to show that (T 1 ∪ T 2 ) ∀ = T ∀ . Let M be a model of T . As in the proof of 3.2.9
we extend M by a chain A0 ⊆ B0 ⊆ A1 ⊆ B1 ⊆ · · · of models of T 1 and T 2 .
The union of this chain is a model of T 1 ∪ T 2 .

Lemma 3.2.13. The Kaiser hull T KH is the ∀∃-part of the theory of all T -ec
structures.
Proof. Let T ∗ be the ∀∃-part of the theory of all T -ec structures. Since T -ec
structures are models of T ∀ , we have T ∀ ⊆ T ∀∗ . It follows from 3.2.11 that
T ∀∗ ⊆ T ∀ . Hence T ∗ is contained in the Kaiser hull.
It remains to show that every T -ec structure M is a model of the Kaiser
hull. Choose a model N of T KH which contains M. Then M ≺1 N. This
implies N ⇒∀∃ M and therefore M |= T KH .
The previous lemma implies immediately that T -ec structures are models of
T∀∃ .

Theorem 3.2.14. For any theory T the following are equivalent:

a) T has a model companion T ∗ .
b) All models of T KH are T -ec.

c) The T -ec structures form an elementary class.

If T ∗ exists, we have

T ∗ = T KH = theory of all T -ec structures.

CHAPTER 3. QUANTIFIER ELIMINATION 37

Proof. a) ⇒ b): Let T ∗ be the model companion of T . As a model complete

theory, T ∗ is inductive. So T ∗ is contained in the Kaiser hull and it suffices to
show that every model M of T ∗ is T -ec. Let A be a model of T which extends
M. A can be embedded in a model N of T ∗ . Now M ≺ N implies M ≺1 A.
b) ⇒ c): By the last lemma all T -ec structures are models of T KH . Thus b)
implies that T -ec structures are exactly the models of T KH .
c) ⇒ a): Assume that the T -ec structures are exactly the models of the
theory T + . By 3.2.11 we have T ∀ = T ∀+ . Criterion 3.2.7 implies that T + is
model complete. So T + is the model companion of T .
The last assertion of the theorem follows easily from the proof.
Exercise 3.2.1. Let L be the language containing a unary function f and
a binary relation symbol R and consider the L-theory T = {∀x∀y(R(x, y) →
(R(x, f (y))}. Show the following
/ {a, f M (a), (f M )2 (a),
1. For any T -ec structure M and a, b ∈ M with b ∈
. . .} we have M |= ∃z(R(z, a) ∧ ¬R(z, b)).
2. Let M be a model of T and a an element of M such that {a, f M (a),
(f M )2 (a), . . .} is infinite. Then in an elementary extension M0 there is an
element b with M0 |= ∀z(R(z, a) → R(z, b)).
3. The class of T -ec structures is not elementary, so T does not have a model
companion.

Exercise 3.2.2. Prove:

1. If T is inductive and has infinite models, then it has existentially closed
models in every cardinality κ ≥ |T |.
2. If T has an infinite model which is not existentially closed, it has such a
model in every cardinality κ ≥ |T |.

3. Lindström’s Theorem: Every inductive κ-categorical theory is model com-

plete.
End of digression.
Exercise 3.2.3. A theory T with quantifier elimination is axiomatisable by
sentences of the form
∀x1 . . . xn ψ
where ψ is a primitive existential formula.

3.3 Examples
In this section we present a number of theories with quantifier elimination,
or at least elimination down to some well-understood formulas, among them
the theories of vector spaces and of algebraically, differentially, and real closed
CHAPTER 3. QUANTIFIER ELIMINATION 38

fields. Since such theories are comparatively easy to understand, they form a
core inventory of the working model theorist. One notable omission is the theory
of valued fields. Their model theory can be found in [15] and in [48] and [23].

3.3.1 Infinite sets

The models of the theory Infset of infinite sets are all infinite sets without
additional structure. The language L∅ is empty, the axioms are (for n = 1, 2, . . .)
V .
• ∃x0 . . . xn−1 i<j<n ¬xi = xj
Theorem 3.3.1. The theory Infset of infinite sets has quantifier elimination
and is complete.
Proof. Clear.

3.3.2 Dense linear orderings

Theorem 3.3.2. DLO has quantifier elimination.
Proof. Let A be a finite common substructure of the two models O1 and O2 .
We choose an ascending enumeration A = {a1 , . . . , an }. Let ∃y ρ(y) be a simple
existential L(A)-sentence, which is true in O1 and assume O1 |= ρ(b1 ). We
want to extend the order preserving map ai 7→ ai to an order preserving map
A ∪ {b1 } → O2 . For this we have to find an image b2 of b1 . There are four cases:
i) b1 ∈ A. We set b2 = b1 .
ii) b1 lies between ai and ai+1 . We choose b2 in O2 with the same property.
iii) b1 is smaller than all elements of A. We choose a b2 ∈ O2 of the same kind.
iv) b1 is bigger that all ai . Choose b2 in the same manner.
This defines an isomorphism A ∪ {b1 } → A ∪ {b2 }, which shows that O2 |=
ρ(b2 ).
Since LOrder has no constants, we have another proof that DLO is complete.

3.3.3 Modules
Let R be a (possibly non-commutative) ring with 1. An R-module

M = (M, 0, +, −, r)r∈R

is an abelian group (M, 0, +, −) together with operations r : M → M for every

ring element r ∈ R which satisfies certain axioms. We formulate the axioms in
the language LMod (R) = LAbG ∪ {r | r ∈ R}. The theory Mod(R) of R-modules
consists of
• AbG
CHAPTER 3. QUANTIFIER ELIMINATION 39

.
• ∀x, y r(x + y) = rx + ry
.
• ∀x (r + s)x = rx + sx
.
• ∀x (rs)x = r(sx)
.
• ∀x 1x = x
for all r, s ∈ R. Then Infset ∪ Mod(R) is the theory of all infinite R-modules.
We start with the case where the ring is a field K. Of course, a K-module
is just a vector space over K.
Theorem 3.3.3. Let K be a field. Then the theory of all infinite K-vector
spaces has quantifier elimination and is complete.
Proof. Let A be a common finitely generated substructure (i.e., a subspace) of
the two infinite K-vector spaces V1 and V2 . Let ∃y ρ(y) be a simple existential
L(A)-sentence which holds in V1 . Choose a b1 from V1 which satisfies ρ(y). If
b1 belongs to A, we are finished since then V2 |= ρ(b1 ). If not, we choose a
b2 ∈ V2 \ A. Possibly we have to replace V2 by an elementary extension. The
vector spaces A + Kb1 and A + Kb2 are isomorphic by an isomorphism which
maps b1 to b2 and fixes A elementwise. Hence V2 |= ρ(b2 ).
The theory is complete since a quantifier-free sentence is true in a vector
space if and only if it is true in the zero-vector space.
For arbitrary rings R, we only get a relative elimination result down to
positive primitive formulas.
Definition 3.3.4. An equation is an LMod (R)-formula γ(x) of the form

r1 x1 + r2 x2 + · · · + rm xm = 0.

A positive primitive formula (pp-formula) is of the form

∃y(γ1 ∧ · · · ∧ γn )

where the γi (xy) are equations.

Theorem 3.3.5. For every ring R and any R-module M , every LMod (R)-
formula is equivalent (modulo the theory of M ) to a Boolean combination of
positive primitive formulas.
Remark 3.3.6. 1. We assume the class of positive primitive formulas to be
closed under ∧.
2. A pp-formula ϕ(x1 , . . . , xn ) defines a subgroup ϕ(M n ) of M n :

M |= ϕ(0) and M |= ϕ(x) ∧ ϕ(y) → ϕ(x − y).

Lemma 3.3.7. Let ϕ(x, y) be a pp-formula and a ∈ M . Then ϕ(M, a) is empty

or a coset of ϕ(M, 0).
CHAPTER 3. QUANTIFIER ELIMINATION 40

Proof. M |= ϕ(x, a) → (ϕ(y, 0) ↔ ϕ(x + y, a)).

Corollary 3.3.8. Let a, b ∈ M , ϕ(x, y) a pp-formula. Then (in M ) ϕ(x, a)
and ϕ(x, b) are equivalent or contradictory.
For the proof of Theorem 3.3.5 we need two further lemmas.

Lemma 3.3.9 (B. H. Sn Neumann). Let Hi denote subgroups of some abelian

group. If H0 + a0 ⊆ i=1 Hi + ai and H0 /(H0 ∩ Hi ) is infinite for i > k, then
Sk
H0 + a0 ⊆ i=1 Hi + ai .
For a proof see Exercise 6.1.16. The following is an easy calculation.
Sk
Lemma 3.3.10. Let Ai , i ≤ k, be any sets. If A0 is finite, then A0 ⊆ i=1 Ai
if and only if X  \ 
(−1)|∆| A0 ∩ Ai  = 0.

∆⊆{1,...,k} i∈∆

Proof of Theorem 3.3.5. Fix M . It is enough to show that if ψ(x, y) is in M

equivalent to a Boolean combination of pp-formulas, then so is ∀xψ. Since pp-
formulas are closed under conjunction, ψ is M -equivalent to a conjunction of
formulas ϕ0 (x, y) → ϕ1 (x, y) ∨ · · · ∨ ϕn (x, y) where the ϕi (x, y) are pp-formulas.
We may assume that ψ itself is of this form. Let Hi = ϕi (M, 0), so the
ϕi (M, y) are empty or cosets of Hi . (Think of y as being fixed in M .) Let
H0 /(H0 ∩ Hi ) be finite for i = 1, . . . , k and infinite for i = k + 1, . . . , n, k ≥ 0.
By Neumann’s Lemma we have

M |= ∀xψ ↔ ∀x (ϕ0 (x, y) → ϕ1 (x, y) ∨ · · · ∨ ϕk (x, y)) .

T Lemma 3.3.10 to the sets Ai = ϕi (M, y)/(H0 ∩ · · · ∩ Hk ): so

We apply
ϕ(M, y) ∩ i∈∆ ϕi (M, y) is empty or consists of N∆ cosets of H0 ∩ · · · ∩ Hk
where  \ 
N∆ = H0 ∩ Hi /(H0 ∩ · · · ∩ Hk ).
 
i∈∆

Whence X
M |= ∀xψ ↔ (−1)|∆| N∆ = 0.
∆∈N

where
n  ^
o
N = ∆ ⊆ {1, . . . , k}  ∃x ϕ0 (x, y) ∧ ϕi (x, y) .

i∈∆
CHAPTER 3. QUANTIFIER ELIMINATION 41

3.3.4 Algebraically closed fields

As the next group of examples we consider fields.
Theorem 3.3.11 (Tarski). The theory ACF of algebraically closed fields has
quantifier elimination.
Proof. Let K1 and K2 be two algebraically closed fields and R a common sub-
ring. Let ∃y ρ(y) be a simple existential sentence with parameters in R which
holds in K1 . We have to show that ∃y ρ(y) is also true in K2 .
Let F1 and F2 be the quotient fields of R in K1 and K2 , respectively, and
let f : F1 → F2 be an isomorphism which is the identity on R (see e.g., [35],
Ch. II.4). Then f extends to an isomorphism g : G1 → G2 between the relative
algebraic closures Gi of Fi in Ki , (i = 1, 2) (see e.g., [35], Ch. V.2). Choose an
element b1 of K1 which satisfies ρ(y).

K1 K2

G1 (b1 )
g
G1 -G2

f
F1 - F2

id
R -R

There are two cases:

Case 1: b1 ∈ G1 . Then b2 = g(b1 ) satisfies the formula ρ(y) in K2 .
Case 2: b1 6∈ G1 . Then b1 is transcendental over G1 and the field extension
G1 (b1 ) is isomorphic to the rational function field G1 (X). If K2 is a proper
extension of G2 , we choose any element from K2 \ G2 for b2 . Then g extends
to an isomorphism between G1 (b1 ) and G2 (b2 ) which maps b1 to b2 . Hence
b2 satisfies ρ(y) in K2 . In case that K2 = G2 we take a proper elementary
extension K20 of K2 . (Such a K20 exists by 2.3.1)(2) since K2 is infinite.) Then
(for the same reason) ∃y ρ(y) holds in K20 and therefore in K2 .
Corollary 3.3.12. ACF is model complete.
Obviously, ACF is not complete: for prime numbers p let
.
ACFp = ACF ∪ {p · 1 = 0}

be the theory of algebraically closed fields of characteristic p and

.
ACF0 = ACF ∪ {¬ n · 1 = 0 | n = 1, 2, . . .}
CHAPTER 3. QUANTIFIER ELIMINATION 42

the theory of algebraically closed fields of characteristic 0. We use here the

notation n · 1 = 1 + · · · + 1.
| {z }
n− times

Corollary 3.3.13. The theories ACFp and ACF0 are complete.

Proof. This follows from Theorem 3.2.2 since the prime fields are prime struc-
tures for these theories.
Corollary 3.3.14 (Hilbert’s Nullstellensatz). Let K be a field. Then any proper
ideal I in K[X1 , . . . , Xn ] has a zero in the algebraic closure acl(K).
Proof. As a proper ideal, I is contained in a maximal ideal P . Then L =
K[X1 , . . . , Xn ]/P is an extension field of K in which the cosets of the Xi are a
zero of I. If I is generated by f0 , . . . , fk−1 and
^ .
ϕ = ∃x1 . . . xn fi (x1 , . . . , xn ) = 0,
i<k

then ϕ holds in L and therefore in acl(L). We can assume that acl(K) lies in
acl(L). Since acl(K) ≺ acl(L), we have that ϕ holds in acl(K).

3.3.5 Real closed fields

The theory of real closed fields, RCF, will be discussed in Section B.1. It is
axiomatised in the language LORing of ordered rings.

Theorem 3.3.15 (Tarski–Seidenberg). RCF has quantifier elimination and is

complete.
Proof. Let (K1 , <) and (K2 , <) be two real closed field with a common subring
R. Consider an LORing (R)-sentence ∃y ρ(y) (for a quantifier-free ρ) which holds
in (K1 , <). We have to show ∃y ρ(y) also holds in (K2 , <).
We build first the quotient fields F1 and F2 of R in K1 and K2 . By B.1.1 there
is an isomorphism f : (F1 , <) → (F2 , <) which fixes R. The relative algebraic
closure Gi of Fi in Ki is a real closure of (Fi , <), (i = 1, 2). By B.1.5 f extends
to an isomorphism g : (G1 , <) → (G2 , <).
Let b1 be an element of K1 which satisfies ρ(y). There are two cases:
Case 1: b1 ∈ G1 : Then b2 = g(b1 ) satisfies ρ(y) in K2 .
Case 2: b1 6∈ G1 : Then b1 is transcendental over G1 and the field extension
G1 (b1 ) is isomorphic to the rational function field G1 (X). Let G`1 be the set
of all elements of G1 which are smaller than b1 , and Gr1 the set of all elements
of G1 which are larger than b1 . Then all elements of G`2 = g(G`1 ) are smaller
than all elements of Gr2 = g(Gr1 ). Since fields are densely ordered, we find in an
elementary extension (K20 , <) of (K2 , <) an element b2 which lies between the
elements of G`2 and the elements of Gr2 . Since b2 is not in G2 , it is transcendental
over G2 . Hence g extends to an isomorphism h : G1 (b1 ) → G2 (b2 ) which maps
b1 to b2 .
CHAPTER 3. QUANTIFIER ELIMINATION 43

In order to show that h is order preserving it suffices to show that h is

order preserving on G1 [b1 ] (Lemma B.1.1). Let p(b1 ) be an element of G1 [b1 ].
Corollary B.1.8 gives us a decomposition
Y Y
(X − cj )2 + dj


p(X) =  (X − ai )
i<m j<n

with positive dj . The sign of p(b1 ) depends only on the signs of the factors
, b1 − a0 , . . . , b1 − am−1 . The sign of h(p(b1 )) depends in the same way on the
signs of g(), b2 − g(a0 ), . . . , b2 − g(am−1 ). But b2 was chosen in such a way that

b1 < ai ⇔ b2 < g(ai ).

Hence p(b1 ) is positive if and only if h(p(b1 )) is positive.

Finally we have

(K1 , <) |= ρ(b1 ) ⇒ (G1 (b1 ), <) |= ρ(b1 ) ⇒ (G2 (b2 ), <) |= ρ(b2 ) ⇒
⇒ (K20 , <) |= ∃y ρ(y) ⇒ (K2 , <) |= ∃y ρ(y),

which proves quantifier elimination.

RCF is complete since the ordered field of the rationals is a prime structure.

Corollary 3.3.16 (Hilbert’s 17th Problem). Let (K, <) be a real closed
field. A polynomial f ∈ K[X1 , . . . , Xn ] is a sum of squares

f = g12 + · · · + gk2

of rational functions gi ∈ K(X1 , . . . , Xn ) if and only if

f (a1 , . . . , an ) ≥ 0

for all a1 , . . . , an ∈ K.
Proof. Clearly a sum of squares cannot have negative values. For the converse
assume that f is not a sum of squares. Then, by Corollary B.1.3, K(X1 , . . . , Xn )
has an ordering in which f is negative. Since in K the positive elements are
squares, this ordering, which we denote also by <,
extends the ordering of K.
Let (L, <) be the real closure of K(X1 , . . . , Xn ), < . In (L, <) the sentence

∃ x1 , . . . , xn f (x1 , . . . , xn ) < 0

is true. Hence it is also true in (K, <).

3.3.6 Separably closed fields

A field is separably closed if every non-constant separable polynomial has a
zero, or equivalently, if it has no proper separable algebraic extension. Clearly,
separably closed fields which are also perfect are algebraically closed.
CHAPTER 3. QUANTIFIER ELIMINATION 44

For any field K of characteristic p > 0, K p = {ap | a ∈ K} is a subfield of

K. If the degree [K : K p ] is finite, it has the form pe and e is called the degree
of imperfection of K. If the degree [K : K p ] is infinite, then we say that K has
infinite degree of imperfection. See also page 204.
For any natural number e we denote by SCFp,e the theory of separably closed
fields with degree of imperfection e. By SCFp,∞ we denote the theory of sepa-
rably closed closed field of characteristic p with infinite degree of imperfection.
We will prove below that SCFp,e is complete. A proof of the completeness of
SCFp,∞ can be found in [19].
To study SCFp,e we consider an expansion of it: SCFp (c1 , · · · , ce ), the theory
of separably closed fields of characteristic p in the language L(c1 , . . . , ce ) of rings
with constants c1 , . . . , ce for a distinguished finite p-basis. We show
Proposition 3.3.17. SCFp (c1 , · · · , ce ) is model complete.
For the proof we need the following lemma.
Lemma 3.3.18. Let K and L be extensions of F . Assume that K/F is sepa-
rable and that L is separably closed. Then K embeds over F in an elementary
extension of L.
Proof. Since L is infinite, it has arbitrarily large elementary extensions. So we
may assume that the transcendence degree tr. deg(L/F ) of L over F is infinite.
Let K 0 be a finitely generated subfield of K over F . By compactness it suffices
to show that all such K 0 /F can be embedded into L/F . By Lemma B.3.12
K 0 /F has a transcendence basis x1 , . . . , xn so that K 0 /F (x1 , . . . , xn ) is separably
algebraic. F (x1 , . . . , xn )/F can be embedded into L/F . Since L is separably
closed this embedding extend to K 0 .
Proof of Proposition 3.3.17. Let (F, b1 , . . . , be ) ⊆ (K, b1 , . . . , be ) be an extension
of models of SCFp (c1 , · · · , ce ). Since F and K have the same p-basis, K is
separable over F by Remark B.3.9. By Lemma 3.3.18, K embeds over F into
an elementary extension of F , showing that F is existentially closed in K. Now
the claim follows by Robinson’s Test (Lemma 3.2.7).
Corollary 3.3.19 (Ershov). SCFp,e is complete.
Proof. Consider the polynomial ring R = Fp [x1 , . . . , xe ]. It is easy to see that
x1 , . . . , xe is a p-basis of R in the sense of Lemma B.3.11. The same lemma
implies that x1 , . . . , xe is a p-basis of F = Fp (x1 , . . . , xe )sep (where Lsep denotes
the separable algebraic closure of the field L). So (F, x1 , . . . , xe ) is a model of
SCFp (c1 , · · · , ce ). If (L, b1 , . . . , be ) is another model of SCFp (c1 , · · · , ce ), then by
Lemma B.3.10 the bi are algebraically independent over Fp , so we can embed
(F, x1 , . . . , xe ) into (L, b1 , . . . , be ). This embedding is elementary by Proposi-
tion 3.3.17, so L is elementarily equivalent to F .
Let K be a field with p-basis b1 , . . . , be . The λ-functions λν : K → K are
defined by X
x= λν (x)p bν ,
CHAPTER 3. QUANTIFIER ELIMINATION 45

where the ν are multi-indices (ν1 , . . . , νe ) with 0 ≤ νi < p and bν denotes

bν11 · · · bνee .
Theorem 3.3.20 (Delon). SCFp (c1 , · · · , ce ) has quantifier elimination in the
language L(c1 , . . . , ce , λν )ν∈pe .
It can be shown that SCFp (c1 , · · · , ce ) has quantifier elimination already in the
language L(λν )ν∈pe without naming a p-basis.
Proof. Let K = (K, b1 , . . . , be ) and L = (L, b1 , . . . , be ) each be models of SCFp (c1 , · · · , ce )
and let R be a common subring which contains the bi and is closed under the
λ-functions of K and L.3 Since R is closed under the λ-functions, the bi form a
p-basis of R in the sense of Lemma B.3.11. Let F be the separable closure of the
quotient field of R. By Lemma B.3.11 the bi also form a p-basis of F . So F is
an elementary subfield of K and of L by Proposition 3.3.17, and hence KR and
LR are elementarily equivalent. Now the claim follows from Theorem 3.2.5.

3.3.7 Differentially closed fields

We next consider fields with a derivation in the language of fields expanded
by a function symbol d. Differential fields are introduced and discussed in
Section B.2.
Definition 3.3.21. The theory of differentially closed fields, DCF0 , is the
theory of differential fields (K, d) in characteristic 0 satisfying the following
property:
For f ∈ K[x0 , . . . , xn ] \ K[x0 , . . . , xn−1 ] and g ∈ K[x0 , . . . , xn−1 ], g 6= 0, there
is some a ∈ K such that f (a, da, . . . , dn a) = 0 and g(a, da, . . . , dn−1 a) 6= 0.
Clearly, models of DCF0 are algebraically closed.
Theorem 3.3.22.
1. Any differential field can be extended to a model of DCF0 .
2. DCF0 is complete and has quantifier elimination.
Proof. Let (K, d) be a differential field and f and g as in the definition. We
may assume that f is irreducible. Then f determines a field extension F =
K(t0 , . . . , tn−1 , b) where the ti are algebraically independent over K and

f (t0 , . . . , tn−1 , b) = 0

(see Remark B.3.7). By Lemma B.2.2 and B.2.3 there is an extension of

the derivation to F with dti = ti+1 and dtn−1 = b. For a = t0 we have
f (a, da, . . . , dn a) = 0 and g(a, da, . . . , dn−1 a) 6= 0. Let K0 denote the differen-
tial field that we obtain from K by doing this for all pairs f, g as above with
coefficients from K. Inductively, we define differential fields Ki+1 satisfying the
3 Note that the λ-functions of K and L agree automatically on R.
CHAPTER 3. QUANTIFIER ELIMINATION 46

S
required condition for polynomials with coefficients in Ki . Their union i<ω Ki
is a model of DCF0 .
Since the rational numbers with trivial derivation are a prime structure for
DCF0 , by Lemma 3.2.2 it suffices for part 2 to prove quantifier elimination.
For this, let K be a differential field with two extensions F1 and F2 which are
models of DCF0 . Let a be an element of F1 and let K{a} = K(a, da, d2 a, . . .)
be the differential field generated by K and a. We have to show that K{a} can
be embedded over K into an elementary extension of F2 . We distinguish two
cases.
1. The derivatives a, da, d2 a, . . . are algebraically independent over K: since
F2 is a model of DCF0 , there is an element b in some elementary extension
such that g(b, db, . . . , dn−1 b) 6= 0 for all n and all g ∈ K[x0 , . . . , xn−1 ] \ 0.
The isomorphism from K{a} to K{b} defined by di a 7→ di b is the required
embedding.
2. Let dn a be algebraic over K(a, da, . . . , dn−1 a) and n minimal: choose
an irreducible f ∈ K[x0 , . . . , xn ] such that f (a, da, . . . , dn a) = 0 (see Re-
mark B.3.7). We may find some b with f (b, db, . . . , dn b) = 0 and g(b, ba, . . . ,
dn−1 b) 6= 0 for all g ∈ K[x0 , . . . , xn−1 ] \ 0 in an elementary extension of F2 .
The field isomorphism from K1 = K(a, . . . , dn a) to K2 = K(b, . . . , dn b) fixing
K and taking di a to di b takes the derivation of F1 restricted to K(a, . . . , dn−1 a)
to the derivation of F2 restricted to K(b, . . . , dn−1 b). The uniqueness part of
Lemma B.2.3 implies that K1 and K2 are closed under the respective deriva-
tions, and that K1 and K2 are isomorphic over K as differential fields.
Exercise 3.3.1. Let Graph be the theory of graphs. The theory RG of the
random graph is the extension of Graph by the following axiom scheme:
^
.
∀x0 . . . xm−1 y1 . . . yn−1 ¬xi = yj →
i6=j

^
^ .

∃z zRxi ∧ ¬zRyj ∧ ¬z = yj
i<m j<n

Show that RG has quantifier elimination and is complete. Show also that RG is
the model companion of Graph.
Exercise 3.3.2. In models of ACF, RCF and DCF0 , the model-theoretic alge-
braic closure of a set A coincides with the algebraic closure of the (differential)
field generated by A.
Exercise 3.3.3. Show that the following is true in any algebraically closed
field K: every injective polynomial map of a definable subset of K n in itself is
surjective.
In fact, more is true: use Exercise 6.1.14 to show that the previous statement
holds for every injective definable map.
Chapter 4

Countable models

4.1 The omitting types theorem

As we have seen in Corollary 2.2.9, it is not hard to realise a given type or in fact
any number of them. But as Sacks [50] pointed out, it needs a model theorist
to avoid realising a given type.
Definition 4.1.1. Let T be an L-theory and Σ(x) a set of L-formulas. A model
A of T not realising Σ(x) is said to omit Σ(x). A formula ϕ(x) isolates Σ(x) if
a) ϕ(x) is consistent with T .
b) T ` ∀x (ϕ(x) → σ(x)) for all σ(x) in Σ(x).
A set of formulas is often called a partial type. This explains the name of
the following theorem.
Theorem 4.1.2 (Omitting Types). If T is countable1 and consistent and if
Σ(x) is not isolated in T , then T has a model which omits Σ(x).
If Σ(x) is isolated by ϕ(x) and A is a model of T , then Σ(x) is realised in
A by all realisations of ϕ(x). Therefore the converse of the theorem is true for
complete theories T : if Σ(x) is isolated in T , then it is realised in every model
of T .
Proof. We choose a countable set C of new constants and extend T to a theory
T ∗ with the following properties:

a) T ∗ is a Henkin theory: for all L(C)-formulas ψ(x) there exists a constant

c ∈ C with ∃x ψ(x) → ψ(c) ∈ T ∗ .
b) For all c ∈ C there is a σ(x) ∈ Σ(x) with ¬σ(c) ∈ T ∗ .
1 An L-theory is countable if L is at most countable.

47
CHAPTER 4. COUNTABLE MODELS 48

We construct T ∗ inductively as the union of an ascending chain

T = T0 ⊆ T1 ⊆ · · ·

of consistent extensions of T by finitely many axioms from L(C), in each step

making an instance of a) or b) true.
Enumerate C = {ci | i < ω} and let {ψi (x) | i < ω} be an enumeration of
the L(C)-formulas.
Assume that T2i is already constructed. Choose some c ∈ C which does not
occur in T2i ∪ {ψi (x)} and set T2i+1 = T2i ∪ {∃x ψi (x) → ψi (c)}. Clearly T2i+1
is consistent.
Up to equivalence T2i+1 has the form T ∪ {δ(ci , c)} for an L-formula δ(x, y)
and a tuple c ∈ C which does not contain ci . Since ∃ȳ δ(x, y) does not isolate
Σ(x), for some σ ∈ Σ the formula ∃y δ(x, ȳ) ∧ ¬σ(x) is consistent with T . Thus,
T2i+2 = T2i+1 ∪ {¬σ(ci )} consistent.
Take a model (A0 , ac )c∈C of T ∗ . Since T ∗ is a Henkin theory, Tarski’s Test
2.1.2 shows that A = {ac | c ∈ C} is the universe of an elementary substructure
A (see Lemma 2.2.3). By property b), Σ(x) is omitted in A.

Corollary 4.1.3. Let T be countable and consistent and let

Σ0 (x1 , . . . , xn0 ), Σ1 (x1 , . . . , xn1 ), . . .

be a sequence of partial types. If all Σi are not isolated, then T has a model
which omits all Σi .
Proof. Generalise the proof of the Omitting Types Theorem.
Exercise 4.1.1. Prove Corollary 4.1.3.

4.2 The space of types

We now endow the set of types of a given theory with a topology. The Com-
pactness Theorem 2.2.1 then translates into the statement that this topology is
compact, whence its name.
Fix a theory T . An n-type is a maximal set of formulas p (x1 , . . . , xn ) con-
sistent with T . We denote by Sn (T ) the set of all n-types of T . We also write
S(T ) for S1 (T ).2
If B is a subset of an L-structure A, we recover SA n (B) (see p. 23) as
Sn (Th(AB )). In particular, if T is complete and A is any model of T , we
have SA (∅) = S(T ).
For any L-formula ϕ(x1 , . . . , xn ), let [ϕ] denote the set of all types contain-
ing ϕ.
Lemma 4.2.1.
2S
0 (T ) can be considered as the set of all complete extensions of T , up to equivalence.
CHAPTER 4. COUNTABLE MODELS 49

1. [ϕ] = [ψ] if and only if ϕ and ψ are equivalent modulo T .

2. The sets [ϕ] are closed under Boolean operations. In fact [ϕ]∩[ψ] = [ϕ∧ψ],
[ϕ] ∪ [ψ] = [ϕ ∨ ψ], Sn (T ) \ [ϕ] = [¬ϕ], Sn (T ) = [>] and ∅ = [⊥].
Proof. For the first part just notice that if ϕ and ψ are not equivalent modulo
T , then ϕ ∧ ¬ψ or ¬ϕ ∧ ψ is consistent with T and hence [ϕ] 6= [ψ]. The rest is
clear.
It follows that the collection of sets of the form [ϕ] is closed under finite
intersections and includes Sn (T ). So these sets form a basis of a topology on
Sn (T ).

Lemma 4.2.2. The space Sn (T ) is 0-dimensional and compact.

Proof. Being 0-dimensional means having a basis of clopen sets. Our basic open
sets are clopen since their complements are also basic open.
If p and q are two different types, there is a formula ϕ contained in p but
not in q. It follows that [ϕ] and [¬ϕ] are open sets which separate p and q. This
shows that Sn (T ) is Hausdorff.
To show compactness consider a family [ϕi ], (i ∈ I), with the finite intersec-
tion property. This means that all ϕi1 ∧ · · · ∧ ϕik are consistent with T . So, by
Corollary 2.2.5, {ϕi | i ∈ I} is consistent with T and can be extended to a type
p, which then belongs to all [ϕi ].
Lemma 4.2.3. All clopen subsets of Sn (T ) have the form [ϕ].

Proof. It follows from Exercise 3.1.1 that we can separate any two disjoint closed
subsets of Sn (T ) by a basic open set.
Remark. The Stone duality theorem asserts that the map

X 7→ {C | C clopen subset of X}

yields an equivalence between the category of 0-dimensional compact spaces and

the category of Boolean algebras. The inverse map assigns to every Boolean
algebra B its Stone space S(B), the set of all ultrafilters (see Exercise 1.2.4) of
B. For more on Boolean algebras see [21].

Definition 4.2.4. A map f from a subset of a structure A to a structure B is

elementary if it preserves the truth of formulas; i.e., f : A0 → B is elementary
if for every formula ϕ(x1 , . . . , xn ) and a ∈ A0 we have

A |= ϕ(a) ⇒ B |= ϕ(f (a)).

Note that the empty map is elementary if and only if A and B are elementarily
equivalent. An elementary embedding of A is an elementary map which is
defined on all of A.
CHAPTER 4. COUNTABLE MODELS 50

Lemma 4.2.5. Let A and B be L-structures, A0 and B0 subsets of A and B,

respectively. Any elementary map A0 → B0 induces a continuous surjective
map Sn (B0 ) → Sn (A0 ).
Proof. If q(x) ∈ Sn (B0 ), we define

S(f )(q) = {ϕ(x1 , . . . , xn , a) | a ∈ A0 , ϕ(x1 , . . . , xn , f (a)) ∈ q}.

It is easy to see that S(f ) defines a map from Sn (B0 ) to Sn (A0 ). Moreover it is
surjective since {ϕ(x1 , . . . , xn , f (ā)) | ϕ(x1 , . . . , xn , a) ∈ p} is finitely satisfiable
for all p ∈ Sn (A0 ). And S(f ) is continuous since [ϕ(x1 , . . . , xn , f (a))] is the
preimage of [ϕ(x1 , . . . , xn , a)] under S(f ).
There are two main cases:
• An elementary bijection f : A0 → B0 defines a homeomorphism
Sn (A0 ) → Sn (B0 ). We write f (p) for the image of p.
• If A = B and A0 ⊆ B0 , the inclusion map induces the restriction 3
Sn (B0 ) → Sn (A0 ). We write q  A0 for the restriction of q to A0 . We call
q an extension of q  A0 .
We leave the following lemma as an exercise (see Exercise 4.2.1).
Lemma 4.2.6. A type p is isolated in T if and only if p is an isolated point in
Sn (T ). In fact, ϕ isolates p if and only if [ϕ] = {p}. That is, [ϕ] is an atom in
the Boolean algebra of clopen subsets of Sn (T ).
We call a formula ϕ(x) complete if

{ψ(x) | T ` ∀x (ϕ(x) → ψ(x))}

is a type. We have shown:

Corollary 4.2.7. A formula isolates a type if and only if it is complete.
Exercise 4.2.1. Show that a type p is isolated if and only if it is isolated as an
element in the Stone space.
Exercise 4.2.2. a) Closed subsets of Sn (T ) have the form {p ∈ Sn (T ) | Σ ⊆
p}, where Σ is any set of formulas.
b) Let T be countable and consistent. Then any meagre4 subset X of Sn (T )
can be omitted, i.e., there is model which omits all p ∈ X.
Exercise 4.2.3. Consider the space Sω (T ) of all complete types in variables
v0 , v1 , . . .. Note that Sω (T ) is again a compact space and therefore not meagre
by Baire’s theorem.
3 “restriction
of parameters”.
4A subset of a topological space is nowhere dense if its closure has no interior. A countable
union of nowhere dense sets is meagre.
CHAPTER 4. COUNTABLE MODELS 51

1. Show that {tp(a0 , a2 , . . .) | the ai enumerate a model of T } is comeagre

in Sω (T ).
2. Use this to give a purely topological proof the Omitting Types Theorem
(4.1.3).

Exercise 4.2.4. Let L ⊆ L0 , T an L-theory, T 0 an L0 -theory and T ⊆ T 0 . Show

that there is a natural continuous map Sn (T 0 ) → Sn (T ). This map is surjective
if and only if T 0 /T is a conservative extension, i.e., if T 0 and T prove the same
L-sentences.
Exercise 4.2.5. Let B be a subset of A. Show that the restriction 5 map
Sm+n (B) → Sn (B) is open, continuous and surjective. Let a be an n-tuple in
A. Show that the fibre over tp(a/B) is canonically homeomorphic to Sm (aB).
Exercise 4.2.6. A theory T has quantifier elimination if and only if every type
is implied by its quantifier-free part.
Exercise 4.2.7. Consider the structure M = (Q, <). Determine all types in
S1 (Q). Which of these types are realised in R? Which extensions does a type
over Q have to a type over R?

4.3 ℵ0 -categorical theories

In this section, we consider theories with a unique countable model (up to
isomorphism, of course). These theories can be characterised by the fact that
they have only finite many n-types for each n, see Exercise 4.3.3. We show the
following equivalent statement.
Theorem 4.3.1 (Ryll–Nardzewski). Let T be a countable complete theory.
Then T is ℵ0 -categorical if and only if for every n there are only finitely many
formulas ϕ(x1 , . . . , xn ) up to equivalence relative to T .
The proof will make use of the following notion.
Definition 4.3.2. An L-structure A is ω-saturated if all types over finite subsets
of A are realised in A.
The types in the definition are meant to be 1-types. On the other hand, it
is not hard to see that an ω-saturated structure realises all n-types over finite
sets (see Exercise 4.3.9), for all n ≥ 1. The following lemma is a generalisation
of the ℵ0 -categoricity of DLO. The proof is essentially the same, see p. 26.

Lemma 4.3.3. Two elementarily equivalent, countable and ω-saturated struc-

tures are isomorphic.
5 “restriction of variables”.
CHAPTER 4. COUNTABLE MODELS 52

Proof. Suppose A and B are as in the lemma. We choose enumerations A =

{a0 , a1 , . . . } and B = {b0 , b1 , . . .}. Then we construct an ascending sequence
f0 ⊆ f1 ⊆ · · · of finite elementary maps

f i : Ai → B i

between finite subsets of A and B. We will choose the fi in such a way that A
is the union of the Ai and B the union of the Bi . The union of the fi is then
the desired isomorphism between A and B.
The empty map f0 = ∅ is elementary since A and B are elementarily equiv-
alent. Assume that fi is already constructed. There are two cases:
i = 2n: We will extend fi to Ai+1 = Ai ∪ {an }.Consider the type

p(x) = tp(an /Ai ).

Since fi is elementary, fi (p)(x) is in B a type over Bi . Since B is ω-saturated,

there is a realisation b0 of this type. So for a ∈ Ai

A |= ϕ(an , a) ⇒ B |= ϕ(b0 , fi (a)).

This shows that fi+1 (an ) = b0 defines an elementary extension of fi .

i = 2n + 1: We exchange A and B: since A is ω-saturated, we find an
elementary map fi+1 with image Bi+1 = Bi ∪ {bn }.
Proof of Theorem 4.3.1. Assume that there are only finitely many ϕ(x1 ,
. . . , xn ) relative to T for every n. By Lemma 4.3.3 it suffices to show that
all models of T are ω-saturated. Let M be a model of T and A an n-element
subset. If there are only N many formulas, up to equivalence, in the variables
x1 , . . . , xn+1 , there are, up to equivalence in M, at most N many L(A)-formulas
ϕ(x). Thus, each type p(x) ∈ S(A) is isolated (with respect to Th(MA )) by a
“smallest” formula ϕp (x). Each element of M which realises ϕp (x) also realises
p(x), so M is ω-saturated.
Conversely, if there are infinitely many ϕ(x1 , . . . , xn ) modulo T for some n,
then – as the type space Sn (T ) is compact – there must be some non-isolated
type p. By the Omitting Types Theorem (4.1.2) there is a countable model of
T in which this type is not realised. On the other hand, there also exists a
countable model of T realising this type. So T is not ℵ0 -categorical.
Remark 4.3.4. The proof shows that a countable complete theory with infinite
models is ℵ0 -categorical if and only if all countable models are ω-saturated.
In Theorem 5.2.11 this characterisation will be extended to theories categor-
ical in uncountable cardinalities.

Remark 4.3.5. The proof of Lemma 4.3.3 also shows that ω-saturated models
are ω-homogeneous in the following sense.
CHAPTER 4. COUNTABLE MODELS 53

Definition 4.3.6. An L-structure M is ω-homogeneous if for every elementary

map f0 defined on a finite subset A of M and for any a ∈ M there is some
b ∈ M such that
f = f0 ∪ {ha, bi}
is elementary.
Note that f = f0 ∪ {ha, bi} is elementary if and only if b realises
f0 (tp(a/A)).

Corollary 4.3.7. Let A be a structure and a1 , . . . , an elements of A. Then

Th(A) is ℵ0 -categorical if and only if Th(A, a1 , . . . , an ) is ℵ0 -categorical.
Examples. The following theories are ℵ0 -categorical:
• Infset, the theory of infinite sets.

• For every finite field Fq , the theory of infinite Fq -vector spaces. (Indeed,
this theory is categorical in all infinite cardinals. This follows directly from
the fact that vector spaces over the same field and of the same dimension
are isomorphic.)
• The theory RG of the random graph (see Exercise 3.3.1): this follows from
Theorem 4.3.1 since RG has quantifier elimination and for any n there are
only finitely many graphs on n elements.
• The theory DLO of dense linear orders without endpoints. This follows
from Theorem 4.3.1 since DLO has quantifier elimination: for every n
there are only finitely many (say Nn ) ways to order n (not necessarily
distinct) elements. For n = 2 for example there are the three possibilities
a1 < a2 , a1 = a2 and a2 < a1 . Each of these possibilities corresponds
to a complete formula ψ(x1 , . . . , xn ). Hence there are, up to equivalence,
exactly 2Nn many formulas ϕ(x1 , . . . , xn ).
Next we study the existence of countable ω-saturated structures.

Definition 4.3.8. A theory T is small if Sn (T ) are at most countable for all

n < ω.
A countable theory with at most countably many non-isomorphic at most
countable models is always small. The converse is not true.

Lemma 4.3.9. A countable6 complete theory is small if and only if it has a

countable ω-saturated model.
Proof. If T has a finite model A, T is small and A is ω-saturated. So we may
assume that T has infinite models.
If all types can be realised in a single countable model, there can be at most
countably many types.
6 The statement is true even for uncountable L.
CHAPTER 4. COUNTABLE MODELS 54

If conversely all Sn+1 (T ) are at most countable, then over any n-element
subset of a model of T there are at most countably many types. We construct
an elementary chain
A0 ≺ A1 ≺ · · ·
of models of T . For A0 we take any countable model. If Ai is already con-
structed, we use Corollary 2.2.9 and Theorem 2.3.1.1 to construct a countable
model Ai+1 in such a way that all types over finite subsets of Ai are realised in
Ai+1 . This S
can be done since there are only countable many such types. The
union A = i∈ω Ai is countable and ω-saturated since every type over a finite
subset B of A is realised in Ai+1 if B ⊆ Ai .
Theorem 4.3.10 (Vaught). A countable complete theory cannot have exactly
two countable models.
Proof. We can assume that T is small and not ℵ0 -categorical. We will show
that T has at least three non-isomorphic countable models. First, T has an
ω-saturated countable model A and there is a non-isolated type p(x), which
can be omitted in a countable model B. Let p(x) be realised in A by a. Since
Th(A, a) is not ℵ0 -categorical, Th(A, a) has a countable model (C, c) which is
not ω-saturated. Then C is not ω-saturated and therefore not isomorphic to A.
But C realises p(x) and is therefore not isomorphic to B.
Exercise 4.3.5 shows that for any n 6= 2, n ≤ ω, there is a countable
complete theory with exactly n countable models. Vaught’s Conjecture states
that if a complete countable theory has fewer than continuum many countable
non-isomorphic models, the number of countable models is at most countable
(see [49] for a survey on what is currently known).
Exercise 4.3.1. 1. If T is ℵ0 -categorical, then in any model M the algebraic
closure of a finite set is finite (see Definition p. 79). In particular, M is
locally finite, i.e., any substructure generated by a finite subset is finite.
(In many-sorted structures we mean that in each sort the trace of the
algebraic closure is finite.)
2. There is no ℵ0 -categorical theory of fields, i.e., if T is a complete LRing -
theory containing Field, then T is not ℵ0 -categorical.
Exercise 4.3.2. A theory T is small exactly if T has at most countably many
completions, each of which is small.
Exercise 4.3.3. Show that T is ℵ0 -categorical if and only if Sn (T ) is finite for
all n.
Exercise 4.3.4. Write down a theory with exactly two countable models.
Exercise 4.3.5. Show that for every n > 2 there is a countable complete
theory with exactly n countable models. (Consider (Q, <, P0 , . . . , Pn−2 , c0 , c1 ,
. . .), where the Pi form a partition of Q into dense subsets and the ci are an
increasing sequence of elements of P0 .)
CHAPTER 4. COUNTABLE MODELS 55

Exercise 4.3.6. Give an example of an uncountable complete theory with

exactly one countable model which does not satisfy the conclusion of Theo-
rem 4.3.1.
Exercise 4.3.7. Suppose M is countable and ℵ0 -categorical. Show that if
X ⊆ M n is invariant under all automorphisms of M, then X is definable.

Exercise 4.3.8. Let M be a structure and assume that for some n only finitely
many n-types are realised in M. Then any structure elementarily equivalent to
M satisfies exactly the same n-types.
Exercise 4.3.9. If A is ω-saturated, all n-types over finite sets are realised.
More generally prove the following: If A is κ-saturated i.e., if all 1-types over
sets of cardinality less than κ are realised in A, then the same is true for all
n-types. See also Exercise 6.1.6.
Exercise 4.3.10. Show:
1. The theory of (R, 0, +) has exactly two 1-types but ℵ0 many 2-types.

2. The theory of (R, 0, +, <) has exactly three 1-types but 2ℵ0 many 2-types.
Exercise 4.3.11. Show that all models of an ℵ0 -categorical theory are partially
isomorphic.
Exercise 4.3.12. Show that two countable partially isomorphic structures are
isomorphic.
Exercise 4.3.13. Let A be ω-saturated. Show that B is partially isomorphic
to A if and only if B is ω-saturated and elementarily equivalent to A.

4.4 The amalgamation method

In this section we will present one of the main methods for constructing new
and interesting examples of first order structures. It goes back to Fraı̈ssé, but
has more recently been modified by Hrushovski [28]. We here focus mainly on
the ℵ0 -categorical examples and return to the fancier version in Section 10.4.

Definition 4.4.1. For any language L, the skeleton 7 K of an L-structure M

is the class of all finitely-generated L-structures which are isomorphic to a sub-
structure of M. We say that an L-structure M is K-saturated if its skeleton
is K and if for all A, B in K and all embeddings f0 : A → M and f1 : A → B
there is an embedding g1 : B → M with f0 = g1 ◦ f1 .

Theorem 4.4.2. Let L be a countable language. Any two countable K-saturated

structures are isomorphic.
7 This is also called the age of M.
CHAPTER 4. COUNTABLE MODELS 56

Proof. Let M and N be countable L-structures with the same skeleton K, and
assume that M and N are K-saturated. As in the proof of Lemma 4.3.3 we
construct an isomorphism between M and N as the union of an ascending se-
quence of isomorphisms between finitely-generated substructures of M and N .
This can be done because if f1 : A → N is an embedding of a finitely-generated
substructure A of M into N, and a is an element of M, then by K-saturation
f1 can be extended to an embedding g1 : A0 → N where A0 = hAaiM . Now
interchange the roles of M and N.
Remark 4.4.3. The proof shows that any countable K-saturated structure M
is ultrahomogeneous i.e., any isomorphism between finitely generated substruc-
tures extends to an automorphism of M.
Theorem 4.4.4. Let L be a countable language and K a countable class of
finitely-generated L-structures. There is a countable K-saturated L-structure M
if and only if
a) (Heredity) If A0 belongs to K, then all elements of the skeleton of A0 also
belong to K.
b) (Joint Embedding) For B0 , B1 ∈ K there are some D ∈ K and embeddings
gi : Bi → D.
c) (Amalgamation) If A, B0 , B1 ∈ K and fi : A → Bi , (i = 0, 1) are embed-
dings, there is some D ∈ K and two embeddings gi : Bi → D such that
g0 ◦ f0 = g1 ◦ f1 .
D
7
 S
o
g0 Sg1
 S
B0 B1
o
S 7

f0S f1
S 
A

In this case, M is unique up to isomorphism and is called the Fraı̈ssé limit

of K.
Proof. Let K be the skeleton of a countable K-saturated structure M. Clearly,
K has the Hereditary Property. To see that K has the Amalgamation Property
let A, B0 , B1 , f0 and f1 be as in c). We may assume that B0 ⊆ M and
f0 is the inclusion map. Furthermore we can assume A ⊆ B1 and that f1 is
the inclusion map. Now the embedding g1 : B1 → M is the extension of the
isomorphism f0 : A → f0 (A) to B1 and satisfies f0 = g1 ◦ f1 . For D we choose
a finitely-generated substructure of M which contains B0 and the image of g1 .
For g0 : B0 → D take the inclusion map. The Joint Embedding Property is
proved similarly.
CHAPTER 4. COUNTABLE MODELS 57

For the converse assume that K has properties a), b), and c). Choose an
enumeration (Bi )i∈ω of all isomorphism types in K. We construct M as the
union of an ascending chain

M0 ⊆ M1 ⊆ · · · ⊆ M

of elements of K. Suppose that Mi is already constructed. If i = 2n is even, we

choose Mi+1 as the top of a diagram

Mi+1
7
 S
o
g0 Sg1
 S
Mi Bn

where we can assume that g0 is the inclusion map. If i = 2n + 1 is odd, let A

and B from K and two embeddings f0 : A → Mi and f1 : A → B be given. We
construct Mi+1 using the diagram

Mi+1
7
 S
o
g0 Sg1
 S
Mi B
o
S 7

f0S f1
S 
A

To ensure that M is K-saturated we have in the odd steps to make the right
choice of A, B, f0 and f1 . Assume that we have A, B ∈ K and embeddings
f0 : A → M and f1 : A → B. For large j the image of f0 will be contained in Mj .
During the construction of the the Mi , in order to guarantee the K-saturation
of M, we have to ensure that eventually, for some odd i ≥ j, the embeddings
f0 : A → Mi and f1 : A → B were used in the construction of Mi+1 . This can be
done since for each j there are – up to isomorphism – at most countably many
possibilities. Thus there exists an embedding g1 : B → Mi+1 with f0 = g1 ◦ f1 .
Clearly, K is the skeleton of M: the finitely-generated substructures of M are
the substructures of the Mi . Since the Mi belong to K, their finitely-generated
substructures also belong to K. On the other hand each Bn is isomorphic to a
substructure of M2n+1 .
Uniqueness follows from Theorem 4.4.2
CHAPTER 4. COUNTABLE MODELS 58

For finite relational languages L, any non-empty finite subset is itself a

(finitely-generated) substructure. For such languages, the construction yields
ℵ0 -categorical structures. We now take a closer look at ℵ0 -categorical theories
with quantifier elimination in a finite relational language.
Remark 4.4.5. A complete theory T in a finite relational language with quan-
tifier elimination is ℵ0 -categorical. So all its models are ω-homogeneous by
Remarks 4.3.4 and 4.3.5.
Proof. For every n, there is only a finite number of non-equivalent quantifier-
free formulas ρ(x1 , . . . , xn ). If T has quantifier elimination, this number is also
the number of all formulas ϕ(x1 , . . . , xn ) modulo T and so T is ℵ0 -categorical
by Theorem 4.3.1.
Clearly, if a theory has quantifier elimination, any isomorphism between
substructures is elementary. For relational languages we can say more.
Lemma 4.4.6. Let T be a complete theory in a finite relational language and
M an infinite model of T . The following are equivalent:

a) T has quantifier elimination.

b) Any isomorphism between finite substructures is elementary.
c) The domain of any isomorphism between finite substructures can be extended
to any further element.

Proof. a) ⇒ b) is clear.
b) ⇒ a): If any isomorphism between finite substructures of M is elementary,
all n-tuples a which satisfy in M the same quantifier-free type

tpqf (a) = {ρ(x) | M |= ρ(a), ρ(x) quantifier-free}

satisfy the same simple existential formulas. We will show from this that ev-
ery simple existential formula ϕ(x1 , . . . , xn ) = ∃y ρ(x1 , . . . , xn , y) is, modulo T ,
equivalent to a quantifier-free formula. Let r1 (x), . . . , rk−1 (x) be the quantifier-
free types of all n-tuples in M which satisfy ϕ(x). Let ρi (x) be equivalent to
the conjunction of all formulas from ri (x). Then
_

T ` ∀x ϕ(x) ↔ ρi (x) .
i<k

a) ⇒ c): The theory T is ℵ0 -categorical and hence all models are ω-homo-
geneous. Since any isomorphism between finite substructures is elementary by
the equivalence of a) and b) the claim follows.
c) ⇒ b): If the domain of any finite isomorphism can be extended to any
further element, it is easy to see that every finite isomorphism is elementary.
We have thus established the following.
CHAPTER 4. COUNTABLE MODELS 59

Theorem 4.4.7. Let L be a finite relational language and K a class of finite

L-structures. If the Fraı̈ssé limit of K exists, its theory is ℵ0 -categorical and has
quantifier elimination.
Example. The class of finite linear orders obviously has the Amalgamation
Property. Their Fraı̈ssé limit is the dense linear order without endpoints.
Exercise 4.4.1. Show that two K-saturated structures are partially isomorphic.
Exercise 4.4.2. Prove Remark 4.4.3.
Exercise 4.4.3. Let K be the class of finite graphs. Show that its Fraı̈ssé limit
is the countable random graph. This yields another proof that the theory of the
random graph has quantifier elimination.

4.5 Prime models

Some, but not all, theories have models which are smallest in the sense that they
elementarily embed into any other model of the theory. For countable complete
theories these are the models realising only the ‘necessary’ types. If they exist,
they are unique and ω-homogeneous.
In this section – unless explicitly stated otherwise – we let T be a countable
complete theory with infinite models.
Definition 4.5.1. Let T be a countable theory with infinite models, not nec-
essarily complete.
1. We call A0 a prime model of T if A0 can be elementarily embedded into
all models of T .
2. A structure A is called atomic if all n-tuples a of elements of A are atomic.
This means that the types tp(a) are isolated in SA n (∅).

Prime models need not exist, see the example on p. 61. By Corollary 4.2.7,
a tuple a is atomic if and only if it satisfies a complete formula. For the termi-
nology see Lemma 4.2.6.
Since T has countable models, prime models must be countable and since
non-isolated types can be omitted in suitable models by Theorem 4.1.2, only
isolated types can be realised in prime models. Thus, one direction of the
following theorem is clear.
Theorem 4.5.2. A model of T is prime if and only if it is countable and
atomic.
Proof. As just noted, a prime model has to be countable and atomic. For the
converse let M0 be a countable and atomic model of T and M any model of T .
We construct an elementary embedding of M0 to M as a union of an ascending
sequence of elementary maps
f: A→B
CHAPTER 4. COUNTABLE MODELS 60

between finite subsets A of M0 and B of M . We start with the empty map,

which is elementary since M0 and M are elementarily equivalent.
It is enough to show that every f can be extended to any given A ∪ {a}. Let
p(x) be the type of a over A and f (p) the image of p under f (see Lemma 4.2.5).
We will show that f (p) has a realisation b ∈ M . Then f ∪ {ha, bi} is an elemen-
tary extension of f .
Let a be a tuple which enumerates the elements of A and ϕ(x, x) an L-
formula which isolates the type of aa. Then p is isolated by ϕ(x, a): clearly
ϕ(x, a) ∈ tp(a/a) and if ρ(x, a) ∈ tp(a/a), we have ρ(x, y) ∈ tp(a, a). This
implies M0 |= ∀x, y (ϕ(x, y) → ρ(x, y)) and M |= ∀x (ϕ(x, a) → ρ(x, a)). Thus
f (p) is isolated by ϕ(x, f (a)) and, since ϕ(x, f (a)) can be realised in M, so can
be f (p).
Theorem 4.5.3. All prime models of T are isomorphic.
Proof. Let M0 and M00 be two prime models. Since prime models are atomic,
elementary maps between finite subsets of M0 and M00 can be extended to all
finite extensions. Since M0 and M00 are countable, it follows exactly as in the
proof of Lemma 4.3.3 that M0 and M00 are isomorphic.
The previous proof also shows the following.
Corollary 4.5.4. Prime models are ω-homogeneous.
Proof. Let M0 be prime and a any tuple of elements from M0 . By Theo-
rem 4.5.2, (M0 , a) is a prime model of its theory. The claim follows now from
Theorem 4.5.3.
Definition 4.5.5. The isolated types are dense in T if every consistent L-
formula ψ(x1 , . . . , xn ) belongs to an isolated type p(x1 , . . . , xn ) ∈ Sn (T ).
Remark 4.5.6. By Corollary 4.2.7 this definition is equivalent to asking that
every consistent L-formula ψ(x1 , . . . , xn ) contains a complete formula ϕ(x1 , . . . , xn ):
T ` ∀x (ϕ(x) → ψ(x)).
Theorem 4.5.7. T has a prime model if and only if the isolated types are dense.
Proof. Suppose T has a prime model M (so M is atomic by Theorem 4.5.2).
Since consistent formulas ψ(x) are realised in all models of T , ψ(x) is realised
by an atomic tuple a and ψ(x) belongs to the isolated type tp(a).
For the other direction notice that a structure M0 is atomic if and only if
for all n the set
Σn (x1 , . . . , xn ) = {¬ϕ(x1 , . . . , xn ) | ϕ(x1 , . . . , xn ) complete}
is not realised in M0 . Hence, by Corollary 4.1.3, it is enough to show that the
Σn are not isolated in T . This is the case if and only if for every consistent
ψ(x1 , . . . , xn ) there is a complete formula ϕ(x1 , . . . , xn ) with T 6` ∀x (ψ(x) →
¬ϕ(x)). Since ϕ(x) is complete, this is equivalent to T ` ∀x (ϕ(x) → ψ(x)). We
conclude that Σn is not isolated if and only if the isolated n-types are dense.
CHAPTER 4. COUNTABLE MODELS 61

Notice that the last part shows in fact the equivalence directly. (Because if Σn
is isolated for some n, then it is realised in every model and no atomic model
can exist.)
Example. Let L be the language having a unary predicate Ps for every finite
0–1-sequence s ∈ <ω 2. The axioms of Tree say that the Ps , s ∈ <ω 2, form a
binary decomposition of the universe:
• ∀x P∅ (x)
• ∃x Ps (x)
• ∀x ((Ps0 (x) ∨ Ps1 (x)) ↔ Ps (x))

• ∀x ¬(Ps0 (x) ∧ Ps1 (x)).

Tree is complete and has quantifier elimination. There are no complete formulas
and no prime model.
<ω
Definition 4.5.8. A family of formulas ϕs (x), s ∈ 2, is a binary tree if for
all s ∈ <ω 2 the following holds:


a) T ` ∀x (ϕs0 (x) ∨ ϕs1 (x)) → ϕs (x)


b) T ` ∀x ¬ ϕs0 (x) ∧ ϕs1 (x) .
Theorem 4.5.9. Let T be a complete theory.
1. If T is small, it has no binary tree of consistent L-formulas. If T is
countable, the converse holds as well.

2. If T has no binary tree of consistent L-formulas, the isolated types are

dense.


Proof. 1. Let ϕs (x1 , . . . , xn ) be a binary tree of consistent formulas. Then,
for all η ∈ ω 2, the set  
ϕs (x)  s ⊆ η
is consistent and therefore is contained in some type pη (x) ∈ Sn (T ). The pη (x)
are all different showing that T is not small. We leave the converse as Exer-
cise 4.5.1.
2. If the isolated types are not dense, there is a consistent ϕ(x1 , . . . , xn )
which does not contain a complete formula. Call such a formula perfect. Since
perfect formulas are not complete, they can be decomposed into disjoint8 con-
sistent formulas, which again have to be perfect. This allows us to construct a
binary tree of perfect formulas.

Exercise 4.5.1. Countable theories without a binary tree of consistent formulas

are small.
8 We call two formulas disjoint if their conjunction is not consistent with T .
CHAPTER 4. COUNTABLE MODELS 62

Exercise 4.5.2. Show that isolated types being dense is equivalent to isolated
types being (topologically) dense in the Stone space Sn (T ).
Exercise 4.5.3. Let T be the theory of (R, <, Q) where Q is a predicate for
the rational numbers. Does T have a prime model?
Chapter 5

ℵ1-categorical theories

We have already seen examples of ℵ0 -categorical theories (e.g., the theory of

dense linear orderings without endpoints) and of theories categorical in all in-
finite κ (e.g., the theory of infinite dimensional vector spaces over finite fields)
and all uncountably infinite κ (e.g., the theory of algebraically closed fields of
fixed characteristic).
The aim of this chapter is to understand the structure of ℵ1 -categorical the-
ories and to prove, in Corollary 5.8.2, Morley’s theorem that a countable theory
categorical in some uncountable cardinality is categorical in all uncountable
cardinalities (but not necessarily countably categorical).
As in the case of ℵ0 -categorical theories, we will see that the number of
complete types in an ℵ1 -categorical theory is rather small (the theory is ω-
stable) albeit not always finite. We will define a geometry associated to a
strongly minimal set whose dimension determines the isomorphism type of a
model of such a theory. This then implies Morley’s theorem.

5.1 Indiscernibles
In this section we begin with a few facts about ‘indiscernible’ elements. We will
see that structures generated by them realise only few types.
Definition 5.1.1. Let I be a linear order and A an L-structure. A family
(ai )i∈I of elements1 of A is called a sequence of indiscernibles if for all L-formulas
ϕ(x1 , . . . , xn ) and all i1 < · · · < in and j1 < · · · < jn from I

A |= ϕ(ai1 , . . . , ain ) ↔ ϕ(aj1 , . . . , ajn ).

If two of the ai are equal, all ai are the same. Therefore it is often assumed
that the ai are distinct.
Sometimes sequences of indiscernibles are also called order indiscernible to
distinguish them from totally indiscernible sequences in which the ordering of
1 or, more generally, of tuples of elements, all of the same length.

63
CHAPTER 5. ℵ1 -CATEGORICAL THEORIES 64

the index set does not matter. However, in stable theories (see Section 5.2
and Chapter 8), these notions coincide. So if nothing else is said, indiscernible
elements will always be order indiscernible in the sense just defined.
Definition 5.1.2. Let I be an infinite linear order and I = (ai )i∈I a sequence
of k-tuples in M, A ⊆ M . The Ehrenfeucht–Mostowski type EM(I/A) of I over
A is the set of L(A)-formulas ϕ(x1 , . . . , xn ) with M |= ϕ(ai1 , . . . , ain ) for all
i1 < · · · < in ∈ I, n < ω.
Lemma 5.1.3 (The Standard Lemma). Let I and J be two infinite linear orders
and I = (ai )i∈I a sequence of elements of a structure M. Then there is a struc-
ture N ≡ M with an indiscernible sequence (bj )j∈J realising the Ehrenfeucht–
Mostowski type EM(I) of I.
Corollary 5.1.4. Assume that T has an infinite model. Then, for any linear
order I, T has a model with a sequence (ai )i∈I of distinct indiscernibles.
For the proof of the Standard Lemma we need Ramsey’s Theorem. Let [A]n
denote the set of all n-element subsets of A.

Theorem 5.1.5 (Ramsey). Let A be infinite and n ∈ ω. Partition the set of

n-element subsets [A]n into subsets C1 , . . . , Ck . Then there is an infinite subset
of A whose n-element subsets all belong to the same subset Ci .
Proof. Thinking of the partition as a colouring on [A]n , we are looking for an
infinite subset B of A such that [B]n is monochromatic. We prove the theorem
by induction on n. For n = 1, the statement is evident from the pigeonhole
principle. Assuming the theorem is true for n, we now prove it for n + 1.
Let a0 ∈ A. Then any colouring of [A]n+1 induces a colouring of the n-element
subsets of A0 = A\{a0 }: just colour x ∈ [A0 ]n by the colour of {a0 }∪x ∈ [A]n+1 .
By the induction hypothesis, there exists an infinite monochromatic subset B1 of
A0 in the induced colouring. Thus, all the (n+1)-element subsets of A consisting
of a0 and n elements of B1 have the same colour. Now pick any a1 ∈ B1 . By the
same argument we obtain an infinite subset B2 of B1 with the same properties.
Inductively, we thus construct an infinite sequence A = B0 ⊃ B1 ⊃ B2 ⊃ · · · ,
and elements ai ∈ Bi \ Bi+1 such that the colour of each (n + 1)-element subset
{ai(0) , ai(1) , . . . , ai(n) } with i(0) < i(1) < · · · < i(n) depends only on the value
of i(0). Again by the pigeonhole principle there are infinitely many values of
i(0) for which this colour will be the same. These ai(0) then yield the desired
monochromatic set.
Proof of Lemma 5.1.3. Choose a set C of new constants with an ordering iso-
morphic to J. Consider the theories

T 0 = {ϕ(c) | ϕ(x) ∈ EM(I)} and

T 00 = {ϕ(c) ↔ ϕ(d) | c, d ∈ C}.
CHAPTER 5. ℵ1 -CATEGORICAL THEORIES 65

Here the ϕ(x) are L-formulas and c, d tuples in increasing order. We have to
show that T ∪ T 0 ∪ T 00 is consistent. It is enough to show that

TC ,∆ = T ∪ {ϕ(c) ∈ T 0 | c ∈ C0 } ∪ ϕ(c) ↔ ϕ(d) ¯  ϕ(x) ∈ ∆, c, d¯ ∈ C0

 
0

is consistent for finite sets C0 and ∆. We can assume that the elements of ∆
are formulas with free variables x1 , . . . , xn and that all tuples c and d¯ have the
same length n.
For notational simplicity we assume that all ai are different. So we may
consider A = {ai | i ∈ I} as an ordered set. We define an equivalence relation
on [A]n by

a ∼ b ⇔ M |= ϕ(a) ↔ ϕ(b) for all ϕ(x1 , . . . , xn ) ∈ ∆

where a, b are tuples in increasing order. Since this equivalence relation has at
most 2|∆| many classes, by Ramsey’s Theorem there is an infinite subset B of
A with all n-element subsets in the same equivalence class. We interpret the
constants c ∈ C0 by elements bc in B ordered in the same way as the c. Then
(M, bc )c∈C0 is a model of TC0 ,∆ .
Lemma 5.1.6. Assume L is countable. If the L-structure M is generated by
a well-ordered sequence (ai ) of indiscernibles, then M realises only countably
many types over every countable subset of M .
Proof. If A = {ai | i ∈ I}, then every element b of M has the form b = t(a),
where t is an L-term and a is a tuple from A.
Consider a countable subset S of M . Write
n
S = {tM
n (a ) | n ∈ ω}.

Let A0 = {ai | i ∈ I0 } be the (countable) set of elements of A which occur in

the an . Then every type tp(b/S) is determined by tp(b/A0 ) since every L(S)-
formula
n1
ϕ x, tM


n1 (a , . . .)
can be replaced by the L(A0 )-formula ϕ(x, tn1 (an1 ), . . .).
Now the type of b = t(a) over A0 depends only on t(x) (countably many pos-
sibilities) and the type tp(a/A0 ). Write a = aī for a tuple ī from I. Since the
ai are indiscernible, the type depends only on the quantifier-free type tpqf (ī/I0 )
in the structure (I, <). This type again depends on tpqf (ī) (finitely many pos-
sibilities) and on the types p(x) = tpqf (i/I0 ) of the elements i of ī. There are
three kinds of such types:
1. i is bigger than all elements of I0 .
2. i is an element i0 of I0 .
3. For some i0 ∈ I0 , i is smaller than i0 but bigger than all elements of
{j ∈ I0 | j < i0 }.
CHAPTER 5. ℵ1 -CATEGORICAL THEORIES 66

There is only one type in the first case, in the other cases the type is determined
by i0 . This results in countably many possibilities for each component of ī.
Definition 5.1.7. Let L be a language. A Skolem theory Skolem(L) is a theory
in a bigger language LSkolem with the following properties:
a) Skolem(L) has quantifier elimination.
b) Skolem(L) is universal.
c) Every L-structure can be expanded to a model of Skolem(L).
d) |LSkolem | ≤ max(|L|, ℵ0 ).
Theorem 5.1.8. Every language L has a Skolem theory.
Proof. We define an ascending sequence of languages

L = L0 ⊆ L1 ⊆ L2 ⊆ · · · ,

by introducing for every quantifier-free Li -formula ϕ(x1 , . . . , xn , y) a new n-

place Skolem function 2 fϕ and defining Li+1 as the union of Li and the set of
these function symbols. The language LSkolem is the union of all Li . We now
define the Skolem theory as
n
 o
Skolem = ∀x ∃y ϕ(x, y) → ϕ(x, fϕ (x))  ϕ(x, y) q.f. LSkolem -formula .

Corollary 5.1.9. Let T be a countable theory with an infinite model and let κ
be an infinite cardinal. Then T has a model of cardinality κ which realises only
countably many types over every countable subset.
Proof. Consider the theory T ∗ = T ∪ Skolem(L). Then T ∗ is countable, has an
infinite model and quantifier elimination.
Claim. T ∗ is equivalent to a universal theory.
Proof of Claim. Modulo Skolem(L) every axiom ϕ of T is equivalent to a
quantifier-free LSkolem -sentence ϕ∗ . Therefore T ∗ is equivalent to the universal
theory {ϕ∗ | ϕ ∈ T } ∪ Skolem(L).
Let I be a well-ordering of cardinality κ and N∗ a model of T ∗ with in-
discernibles (ai )i∈I . The claim implies that the substructure M∗ generated
by the ai is a model of T ∗ and M∗ has cardinality κ. Since T ∗ has quan-
tifier elimination, M∗ is an elementary substructure of N∗ and (ai ) is indis-
cernible in M∗ . By Lemma 5.1.6, there are only countably many types over
every countable set realised in M∗ . The same is then true for the reduct M =
M∗  L.
Exercise 5.1.1. A sequence of elements in (Q, <) is indiscernible if and only if
it is either constant, strictly increasing or strictly decreasing.
2 If n = 0, fϕ is a constant.
CHAPTER 5. ℵ1 -CATEGORICAL THEORIES 67

Exercise 5.1.2. Prove Ramsey’s Theorem 5.1.5 by induction on n similarly to

the proof of C.3.2 using a non-principal ultrafilter on A. (For ultrafilters see
Exercise 1.2.4. An ultrafilter is non-principal if it contains no finite sets.)

5.2 ω-stable theories

In this section we fix a complete theory T with infinite models.
In the previous section we saw that we may add indiscernible elements to
a model without changing the number of realised types. We will now use this
to show that ℵ1 -categorical theories have a small number of types, i.e., they
are ω-stable. Conversely, with few types it is easier to be saturated and since
saturated structures are unique we find the connection to categorical theories.
Definition 5.2.1. Let κ be an infinite cardinal. We say T is κ-stable if in each
model of T , over every set of parameters of size at most κ, and for each n, there
are at most κ many n-types, i.e.,

|A| ≤ κ ⇒ | Sn (A)| ≤ κ.

Note that if T is κ-stable, then – up to logical equivalence – we have |T | ≤ κ,

see Exercise 5.2.6.
Lemma 5.2.2. T is κ-stable if and only if T is κ-stable for 1-types, i.e.,

|A| ≤ κ ⇒ | S(A)| ≤ κ.

Proof. Assume that T is κ-stable for 1-types. We show that T is κ-stable for
n-types by induction on n. Let A be a subset of the model M and |A| ≤ κ. We
may assume that all types over A are realised in M. Consider the restriction map
π : Sn (A) → S1 (A). By assumption the image S1 (A) has cardinality at most κ.
Every p ∈ S1 (A) has the form tp(a/A) for some a ∈ M . By Exercise 4.2.5 the
fibre π −1 (p) is in bijection with Sn−1 (aA) and so has cardinality at most κ by
induction. This shows | Sn (A)| ≤ κ.
Example 5.2.3 (Algebraically closed fields). The theories ACFp for p a
prime or 0 are κ-stable for all κ.

Note that by Theorem 5.2.6 below it would suffice to prove that the theories
ACFp are ω-stable. The converse holds as well: any infinite ω-stable field is in
fact algebraically closed (see [38]).
Proof. Let K be a subfield of an algebraically closed field. By quantifier elimi-
nation the type of an element a over K is determined by the isomorphism type
of the extension K[a]/K. If a is transcendental over K, K[a] is isomorphic to
the polynomial ring K[X]. If a is algebraic with minimal polynomial f ∈ K[X],
then K[a] is isomorphic to K[X]/(f ). So there is one more 1-type over K than
there are irreducible polynomials.
CHAPTER 5. ℵ1 -CATEGORICAL THEORIES 68

That ACFp is κ-stable for n-types has a direct algebraic proof: the iso-
morphism type of K[a1 , . . . , an ]/K is determined by the vanishing ideal P of
a1 , . . . , an (see Lemma B.3.6). By Hilbert’s Basis Theorem, P is finitely gener-
ated. So, if K has cardinality κ, the polynomial ring K[X1 , . . . , Xn ] has only κ
many ideals.

Theorem 5.2.4. A countable theory T which is categorical in an uncountable

cardinal κ is ω-stable 3 .
Proof. Let N be a model and A ⊆ N countable with S(A) uncountable. Let
(bi )i∈I be a sequence of ℵ1 many elements with pairwise distinct types over
A. (Note that we can assume that all types over A are realised in N.) We
choose first an elementary substructure M0 of cardinality ℵ1 which contains A
and all bi . Then we choose an elementary extension M of M0 of cardinality κ.
The model M is of cardinality κ and realises uncountably many types over the
countable set A. By Corollary 5.1.9, T has another model in which this is not
the case. So T cannot be κ-categorical.
Definition 5.2.5. A countbale theory T is totally transcendental if it has no
model M with a binary tree of consistent L(M )-formulas.
Theorem 5.2.6. 1. ω-stable theories are totally transcendental.
2. Totally transcendental theories are κ-stable for all κ ≥ |T |.
It follows that a countable theory T is ω-stable if and only if it is totally
transcendental.
Proof. 1. Let M be a model with a binary tree of consistent L(M )-formulas with
free variables among x1 , . . . , xn . The set A of parameters which occur in the
tree’s formulas is countable but Sn (A) has cardinality 2ℵ0 . (see Theorem 4.5.9).
2. Assume that there are more than κ many n-types over some set A of
cardinality κ. Let us call an L(A)-formula ϕ(x) big if it belongs to more than κ
many types over A and thin otherwise. By assumption the true formula is big.
If we can show that each big formula decomposes into two big formulas, we can
construct a binary tree of big formulas, which finishes the proof.
So assume that ϕ is big. Since each thin formula belongs to at most κ types
and since there are at most κ formulas, there are at most κ types which contain
thin formulas. Therefore ϕ belongs to two distinct types p and q which contain
only big formulas. If we separate p and q by ψ ∈ p and ¬ψ ∈ q, we decompose
ϕ into the big formulas ϕ ∧ ψ and ϕ ∧ ¬ψ.
The proof and Lemma 5.2.2 show that T is totally transcendental if and
only if there is no binary tree of consistent formulas in one free variable. This
is clear for countable T ; the general case follows from Exercise 5.2.5.
The following definition generalises the notion of ω-saturation.
3 ω-stable and ℵ0 -stable are synonymous.
CHAPTER 5. ℵ1 -CATEGORICAL THEORIES 69

Definition 5.2.7. Let κ be an infinite cardinal. An L-structure A is κ-saturated

if in A all types over sets of cardinality less than κ are realised. An infinite
structure A is saturated if it is |A|-saturated.
Even though saturation requires only that 1-types are realised, as in the
ω-saturated case this easily implies that all n-types are realised as well (see
Exercise 4.3.9).
Lemma 4.3.3 generalises to sets.
Lemma 5.2.8. Elementarily equivalent saturated structures of the same cardi-
nality are isomorphic.

Proof. Let A and B be elementarily equivalent saturated structures each of

cardinality κ. We choose enumerations (aα )α<κ and (bα )α<κ of A and B and
construct an increasing sequence of elementary maps fα : Aα → Bα . Assume
that the fβ are constructed for all β < α. The union of the fβ is an elemen-
tary map fα∗ : A∗α → Bα∗ . The construction will imply that A∗α and Bα∗ have
cardinality at most |α|, which is smaller than κ.
We write α = λ + n (as in p. 183) and distinguish two cases:
n = 2i: In this case we consider p(x) = tp(aλ+i /A∗α ). Realise fα∗ (p) by b ∈ B
and define
fα = fα∗ ∪ {haλ+i , bi}.
n = 2i + 1: Similarly. We find an extension

fα = fα∗ ∪ {ha, bλ+i i}.

S
Then α<κ fα is the desired isomorphism between A and B.

Lemma 5.2.9. If T is κ-stable, then for all regular λ ≤ κ there is a model of

cardinality κ which is λ-saturated.
Proof. By Exercise 5.2.6 we may assume that |T | ≤ κ. Consider a model
M of cardinality κ. Since S(Mα ) has cardinality κ, Corollary 2.2.9 and the
Löwenheim–Skolem Theorem give an elementary extension of cardinality κ in
which all types over M are realised. So we can construct a continuous elemen-
tary chain
M0 ≺ M1 · · · ≺ Mα ≺ · · · (α < λ),
of models of T with cardinality κ such that all p ∈ S(Mα ) are realised in Mα+1 .
Let M be the union of this chain. Then M is S λ-saturated. In fact, if |A| < λ
and if a ∈ A is contained in Mα(a) then Λ = a∈A α(a) is an initial segment
of λ of smaller cardinality than λ. So Λ has an upper bound µ < λ. It follows
that A ⊆ Mµ and all types over A are realised in Mµ+1 .

Remark 5.2.10. If T is κ-stable for a regular cardinal κ, the previous lemma

yields a saturated model of cardinality κ. Harnik [22] showed that this holds in
fact for arbitrary κ. See also Corollary 6.1.3 for more general constructions.
CHAPTER 5. ℵ1 -CATEGORICAL THEORIES 70

Theorem 5.2.11. A countable theory T is κ-categorical if and only if all models

of cardinality κ are saturated.
Proof. If all models of cardinality κ are saturated, it follows from Lemma 5.2.8
that T is κ-categorical.
Assume, for the converse, that T is κ-categorical. For κ = ℵ0 the theorem
follows from (the proof of) Theorem 4.3.1. So we may assume that κ is un-
countable. Then T is totally transcendental by Theorems 5.2.4 and 5.2.6 and
therefore κ-stable by Theorem 5.2.6.
By Lemma 5.2.9, all models of T of cardinality κ are µ+ -saturated for all
µ < κ. i.e., κ-saturated.
Exercise 5.2.1. Use Exercise 8.2.8 to show that a theory with an infinite
definable linear ordering (like DLO and RCF) cannot be κ-stable for any κ.
Exercise 5.2.2. Show that the theory of an equivalence relation with two
infinite classes has quantifier elimination and is ω-stable. Is it ℵ1 -categorical?
Exercise 5.2.3. Let L be at most countable, A0 , A1 , . . . a sequence
Q of L-
structures and F a non-principal ultrafilter on ω. Show that i<ω Ai /F is
ℵ1 -saturated. If we assume the Continuum Hypothesis, this implies that if
A and B are two countable and elementarily equivalent L-structures, the two
ultrapowers Aω /F and Bω /F are isomorphic.
Shelah has shown in [52] that for any two elementarily equivalent struc-
tures there is a set I and an ultrafilter F on I such that AI /F and BI /F are
isomorphic.
Exercise 5.2.4. If A is κ-saturated, then all definable subsets are either finite
or have cardinality at least κ.
Exercise 5.2.5. If T is an L-theory and K is a sublanguage of L, the reduct
T  K is the set of all K–sentences which follow from T . Show that T is totally
transcendental if and only if T  K is ω-stable for all at most countable K ⊆ L.
Exercise 5.2.6. If T is κ-stable, then essentially (i.e., up to logical equivalence)
|T | ≤ κ.

5.3 Prime extensions

As with prime models, prime extensions are the smallest ones in the sense of
elementary embeddings. We will see here (and in Sections 9.2 and 9.3) that
prime extensions, if they exist, share a number of important properties with
prime models.
Definition 5.3.1. Let M be a model of T and A ⊆ M .
1. M is a prime extension of A (or prime over A) if every elementary map
A → N extends to an elementary map M → N.
CHAPTER 5. ℵ1 -CATEGORICAL THEORIES 71

M - N
6 
idA

2. B ⊆ M is constructible over A if B has an enumeration

B = {bα | α < λ},

where each bα is atomic over A ∪ Bα , with Bα = {bµ | µ < α}.

So M is a prime extension of A if and only if MA is a prime model of

Th(MA ).
Notice the following.
Lemma 5.3.2. If a model M is constructible over A, then M is prime over A.

Proof. Let (mα )α<λ an enumeration of M , such that each mα is atomic over
A ∪ Mα . Let f : A → N be an elementary map. We define inductively an
increasing sequence of elementary maps fα : A ∪ Mα+1 → N with f0 = f .
Assume that fβ is defined for all β < α. The union of these fβ is an elementary
map fα0 : A∪Mα → N. Since p(x) = tp(aα /A∪Mα ) is isolated, fα0 (p) ∈ S(fα0 (A∪
Mα )) is also isolated and has a realisation b in N. We set fα = fα0 ∪ {haα , bi}.
Finally, the union of all fα (α < λ) is an elementary embedding M → N.
We will see below that in totally transcendental theories prime extensions
are atomic.
Theorem 5.3.3. If T is totally transcendental, every subset of a model of T
has a constructible prime extension.

We will see in Section 9.2 that in totally transcendental theories, prime

extensions are unique up to isomorphism (see Theorem 4.5.3).
For the proof we need the following lemma which generalises Theorem 4.5.7.
Lemma 5.3.4. If T is totally transcendental, the isolated types are dense over
every subset of any model.
Proof. Consider a subset A of a model M. Then Th(MA ) has no binary tree
of consistent formulas. By Theorem 4.5.9, the isolated types in Th(MA ) are
dense.
We can now prove Theorem 5.3.3.
CHAPTER 5. ℵ1 -CATEGORICAL THEORIES 72

Proof. By Lemma 5.3.2 it suffices to construct an elementary substructure M0 ≺

M which contains A and is constructible over A. An application of Zorn’s
Lemma gives us a maximal construction (aα )α<λ , which cannot be prolonged
by an element aλ ∈ M \ Aλ . Clearly A is contained in Aλ . We show that Aλ is
the universe of an elementary substructure M0 using Tarski’s Test. So assume
that ϕ(x) is an L(Aλ )-formula and M |= ∃x ϕ(x). Since isolated types over Aλ
are dense by Lemma 5.3.4, there is an isolated p(x) ∈ S(Aλ ) containing ϕ(x).
Let b be a realisation of p(x) in M. We can prolong our construction by aλ = b;
thus b ∈ Aλ by maximality and ϕ(x) is realised in Aλ .
To prove that in totally transcendental theories prime extensions are atomic,
we need the following.
Lemma 5.3.5. Let a and b be two finite tuples of elements of a structure M.
Then tp(ab) is atomic if and only if tp(a/b) and tp(b) are atomic.
Proof. First assume that ϕ(x, y) isolates tp(a, b). As in the proof of Theo-
rem 4.5.2, ϕ(x, b) isolates tp(a/b) and we claim that ∃x ϕ(x, y) isolates p(y) =
tp(b): we have ∃x ϕ(x, y) ∈ p(y) and if σ(y) ∈ p(y), then

M |= ∀x, y (ϕ(x, y) → σ(y)).

Hence M |= ∀y (∃xϕ(x, y) → σ(y)).

Now, conversely, assume that ρ(x, b) isolates tp(a/b) and that σ(y) isolates
p(y) = tp(b). Then ρ(x, y) ∧ σ(y) isolates tp(a, b). For, clearly, we have ρ(x, y) ∧
σ(y) ∈ tp(a, b). If, on the other hand, ϕ(x, y) ∈ tp(a, b), then ϕ(x, b) belongs to
tp(a/b) and
M |= ∀x (ρ(x, b) → ϕ(x, b)).
Hence
∀x (ρ(x, y) → ϕ(x, y)) ∈ p(y)
and it follows that

M |= ∀y (σ(y) → ∀x (ρ(x, y) → ϕ(x, y))).

Thus M |= ∀x, y (ρ(x, y) ∧ σ(y) → ϕ(x, y)).

Corollary 5.3.6. Constructible extensions are atomic.
Proof. Let M0 be a constructible extension of A and let a be a tuple from
M0 . We have to show that a is atomic over A. We can clearly assume that
the elements of a are pairwise distinct and do not belong to A. We can also
permute the elements of a so that

a = aα b

for some tuple b ∈ Aα . Let ϕ(x, c) be an L(Aα )-formula which is complete over
Aα and satisfied by aα . Then aα is also atomic over A ∪ {bc}. Using induction,
we know that bc is atomic over A. By Lemma 5.3.5 applied to (M0 )A , aα bc is
atomic over A, which implies that a = aα b is atomic over A.
CHAPTER 5. ℵ1 -CATEGORICAL THEORIES 73

Corollary 5.3.7. If T is totally transcendental, prime extensions are atomic.

Proof. Let M be a model of T and A ⊆ M . Since A has at least one constructible
extension M0 and since all prime extensions of A are contained in M0 4 , all prime
extensions are atomic.

A structure M is called a minimal extension of the subset A if M has no

proper elementary substructure which contains A.
Lemma 5.3.8. Let M be a model of T and A ⊆ M . If A has a prime ex-
tension and a minimal extension, they are isomorphic over A, i.e., there is an
isomorphism fixing A elementwise.

Proof. A prime extension embeds elementarily in the minimal extension. This

embedding must be surjective by minimality.
Exercise 5.3.1. For Theorem 5.3.3 we used only that isolated types are dense
in all S1 (A). Prove for arbitrary T that this implies that the isolated types are
dense in all Sn (A).

Exercise 5.3.2. For every countable T the following are equivalent (see Theo-
rem 4.5.7):
a) Every parameter set has a prime extension. (We say that T has prime
extensions.)

b) Over every countable parameter set the isolated types are dense.
c) Over every parameter set the isolated types are dense.
Exercise 5.3.3. Lemma 5.3.5 follows from Exercise 4.2.5 and the following
observation: let π : X → Y be a continuous open map between topological
spaces. Then a point x ∈ X is isolated if and only if π(x) is isolated in Y and
x is isolated in π −1 (π(x)).

5.4 Lachlan’s Theorem

Using the fact (established in Section 5.2) that uncountably categorical theories
are totally transcendental, we will prove the downward direction of Morley’s the-
orem. We use Lachlan’s result that, in totally transcendental theories, models
have arbitrary large elementary extensions realising few types.
Theorem 5.4.1 (Lachlan). ([2] Lemma 10) Let T be totally transcendental
and M an uncountable model of T . Then M has arbitrarily large elementary
extensions which omit every countable set of L(M )-formulas that is omitted
in M.
4 More precisely, they are isomorphic over A to elementary substructures of M0 .
CHAPTER 5. ℵ1 -CATEGORICAL THEORIES 74

Proof. For the proof, we call an L(M )-formula ϕ(x) large if its realisation set
ϕ(M) is uncountable. Since there is no infinite binary tree of large formulas,
there exists a minimal large formula ϕ0 (x). This means that for every L(M )-
formula ψ(x) either ϕ0 (x) ∧ ψ(x) or ϕ0 (x) ∧ ¬ψ(x) is at most countable. Now
it is easy to see that

p(x) = {ψ(x) | ϕ0 (x) ∧ ψ(x) large}

is a type in S(M ).
.
Clearly p(x) contains no formula of the form x = a for a ∈ M , so p(x) is
not realised in M . On the other hand, every countable subset Π(x) of p(x) is
realised in M: since ϕ0 (M) \ ψ(M) is at most countable for every ψ(x) ∈ Π(x),
the elements of ϕ0 (M) which do not belong to the union of these sets realise
Π(x).
Let a be a realisation of p(x) in a (proper) elementary extension N. By
Theorem 5.3.3 we can assume that N is atomic over M ∪ {a}.
Fix b ∈ N . We have to show that every countable subset Σ(y) of tp(b/M )
is realised in M .
Let χ(x, y) be an L(M )-formula such that χ(a, y) isolates q(y) = tp(b/M ∪
{a}). If b realises an L(M )-formula σ(y), we have N |= ∀y (χ(a, y) → σ(y)).
Hence the formula
σ ∗ (x) = ∀y (χ(x, y) → σ(y))
belongs to p(x). Note that ∃y χ(x, y) belongs also to p(x).
Choose an element a0 ∈ M which satisfies

{σ ∗ (x) | σ ∈ Σ} ∪ {∃y χ(x, y)}

and choose b0 ∈ M with M |= χ(a0 , b0 ). Since σ ∗ (a0 ) is true in M, σ(b0 ) is true

in M. So b0 realises Σ(y).
We have shown that M has a proper elementary extension which realises no
new countable set of L(M )-formulas. By iteration we obtain arbitrarily long
chains of elementary extensions with the same property.
The corollary is the downwards part of Morley’s Theorem, p. 63.
Corollary 5.4.2. A countable theory which is κ-categorical for some uncount-
able κ, is ℵ1 -categorical.
Proof. Let T be κ-categorical and assume that T is not ℵ1 -categorical. Then T
has a model M of cardinality ℵ1 which is not saturated. So there is a type p over
a countable subset of M which is not realised in M. By Theorems 5.2.4 and 5.2.6
T is totally transcendental. Theorem 5.4.1 gives an elementary extension N of
M of cardinality κ which omits all countable sets of formulas which are omitted
in M. Thus also p is omitted. Since N is not saturated, T is not κ-categorical,
a contradiction.
Exercise 5.4.1. Prove in a similar way: if a countable theory T is κ-categorical
for some uncountable κ, it is λ-categorical for every uncountable λ ≤ κ.
CHAPTER 5. ℵ1 -CATEGORICAL THEORIES 75

5.5 Vaughtian pairs

A crucial fact about uncountably categorical theories is the absence of definable
sets whose size is independent of the size of the model in which they live (cap-
tured in the notion of a Vaughtian pair). In fact, in an uncountably categorical
theory each model is prime over any infinite definable subset. This will allow us
in Section 5.7 to attach a dimension to the models of an uncountably categorical
theory. In this section, T is a countable complete theory with infinite models.
Definition 5.5.1. We say that T has a Vaughtian pair if there are two models
M ≺ N and an L(M )-formula ϕ(x) such that
a) M 6= N,
b) ϕ(M) is infinite,
c) ϕ(M) = ϕ(N).
If ϕ(x) does not contain parameters, we say that T has a Vaughtian pair for
ϕ(x).
Remark. Notice that T does not have a Vaughtian pair if and only if every
model M is a minimal extension of ϕ(M) ∪ A for any formula ϕ(x) with param-
eters in A ⊆ M which defines an infinite set in M.
Let N be a model of T where ϕ(N) is infinite but has smaller cardinality
than N. The Löwenheim–Skolem Theorem yields an elementary substructure M
of N which contains ϕ(N) and has the same cardinality as ϕ(N). Then M ≺ N
is a Vaughtian pair for ϕ(x). The next theorem shows that a converse of this
observation is also true.
Theorem 5.5.2 (Vaught’s Two-cardinal Theorem). If T has a Vaughtian
pair, it has a model M of cardinality ℵ1 with ϕ(M) countable for some formula
ϕ(x) ∈ L(M̄ ).
For the proof of Theorem 5.5.2 we need the following.
Lemma 5.5.3. Let T be complete, countable, and with infinite models.
1. Every countable model of T has a countable ω-homogeneous elementary
extension.
2. The union of an elementary chain of ω-homogeneous models is ω-homo-
geneous.
3. Two ω-homogeneous countable models of T realising the same n-types for
all n < ω are isomorphic.
Proof. 1. Let M0 be a countable model of T . We realise the countably many
types
{f (tp(a/A)) | a, A ⊆ M0 , A finite, f : A → M0 elementary}
CHAPTER 5. ℵ1 -CATEGORICAL THEORIES 76

in a countable elementary extension M1 . By iterating this process we obtain an

elementary chain
M0 ≺ M1 ≺ · · · ,
whose union is ω-homogeneous.
2. Clear.
3. Suppose A and B are ω-homogeneous, countable and realise the same n-
types. We show that we can extend any finite elementary map f : {a1 , . . . , ai } →
{b1 , . . . , bi }; aj 7→ bj to any a ∈ A \ Ai . Realise the type tp(a1 , . . . , ai , a) by
some tuple b0 = b01 , . . . , b0i+1 in B. Using the ω-homogeneity of B we may extend
the finite partial isomorphism g = {(b0j , bj ) | 1 ≤ j ≤ i} by (b0i+1 , b) for some
b ∈ B. Then fi+1 = fi ∪ {(a, b)} is the required extension. Reversing the roles
of B and A we construct the desired isomorphism.
Proof. (of Theorem 5.5.2) Suppose that the Vaughtian pair is witnessed (in
certain models) by some formula ϕ(x). For simplicity we assume that ϕ(x) does
not contain parameters (see Exercise 5.5.4). Let P be a new unary predicate.
It is easy to find an L(P )-theory TVP whose models (N, M ) consist of a model
N of T and a subset M defined by the new predicate P which is the universe of
an elementary substructure M which together with N forms a Vaughtian pair
for ϕ(x). The Löwenheim–Skolem Theorem applied to TVP yields a Vaughtian
pair M0 ≺ N0 for ϕ(x) with M0 , N0 countable.
We first construct an elementary chain
(N0 , M0 ) ≺ (N1 , M1 ) ≺ · · ·
of countable Vaughtian pairs, with the aim that both components of the union
pair
(N, M )
are ω-homogeneous and realise the same n-types. If (Ni , Mi ) is given, we first
choose a countable elementary extension (N0 , M 0 ) such that M0 realises all n-
types which are realised in Ni . Then we choose as in the proof of Lemma 5.5.3(1)
a countable elementary extension (Ni+1 , Mi+1 ) of (N0 , M 0 ) for which Ni+1 and
Mi+1 are ω-homogeneous.
It follows from Lemma 5.5.3(3) that M and N are isomorphic.
Next we construct a continuous elementary chain
M0 ≺ M1 ≺ · · · ≺ Mα ≺ · · · (α < ω1 )
with (M α+1 α
,M ) =∼ (N, M ) for all α. We start with M0 = M. If Mα is con-
structed, we choose an isomorphism M → Mα and extend it to an isomorphism
N → Mα+1 (see Lemma 1.1.8). For a countable limit ordinal λ, Mλ is the union
of the Mα (α < λ). So Mλ is isomorphic to M by Lemma 5.5.3(2) and 5.5.3(3).
Finally we set [
M= Mα .
α<ω1

Since the Mα are growing, M has cardinality ℵ1 while ϕ(M) = ϕ(Mα ) = ϕ(M0 )
is countable.
CHAPTER 5. ℵ1 -CATEGORICAL THEORIES 77

Corollary 5.5.4. If T is categorical in an uncountable cardinality, it does not

have a Vaughtian pair.
Proof. If T has a Vaughtian pair, then by Theorem 5.5.2 it has a model M of
cardinality ℵ1 such that for some ϕ(x) ∈ L(M ) the set ϕ(M) is countable. On
the other hand, if T is categorical in an uncountable cardinal, it is ℵ1 -categorical
by Corollary 5.4.2 and by Theorem 5.2.11, all models of T of cardinality ℵ1 are
saturated. In particular, each formula is either satisfied by a finite number or
by ℵ1 many elements, a contradiction.
Corollary 5.5.5. Let T be categorical in an uncountable cardinal, M a model,
and ϕ(M) infinite and definable over A ⊆ M . Then M is – the unique – prime
extension of A ∪ ϕ(M).
Proof. By Corollary 5.5.4, T does not have a Vaughtian pair, so M is minimal
over A ∪ ϕ(M). If N is a prime extension over A ∪ ϕ(M), which exists by
Theorem 5.3.3, N is isomorphic to M over A ∪ ϕ(M) by Lemma 5.3.8.
Definition 5.5.6. We say that T eliminates the quantifier ∃∞ x, there are in-
finitely many x, if for every L-formula ϕ(x, y) there is a finite bound nϕ such
that in all models M of T and for all parameters a ∈ M ,

ϕ(M, a)

is either infinite or has or at most nϕ elements.

Remark. This means that for all ϕ(x, y) there is a ψ(y) such that in all models
M of T and for all a ∈ M

M |= ∃∞ x ϕ(x, a) ⇐⇒ M |= ψ(a).

We denote this by
T ` ∀y ∃∞ x ϕ(x, y) ↔ ψ(y) .

Proof. If nϕ exists, we can use ψ(y) = ∃>nϕ x ϕ(x, y) (there are more than nϕ
many x such that ϕ(x, y)). If, conversely, ψ(y) is a formula which is implied by
∃∞ x ϕ(x, y), a compactness argument shows that there must be a bound nϕ
such that

T ` ∃>nϕ x ϕ(x, y) → ψ(y).

Lemma 5.5.7. A theory T without Vaughtian pair eliminates the quantifier

∃∞ x.
Proof. Let P be a new unary predicate and c1 , . . . , cn new constants. Let T ∗
be the theory of all L ∪ {P, c1 , . . . , cn }-structures

(M, N, a1 , . . . , an ),
CHAPTER 5. ℵ1 -CATEGORICAL THEORIES 78

where M is a model of T , N the universe of a proper elementary substructure,

a1 , . . . , an elements of N and ϕ(M, a) ⊆ N . Suppose that the bound nϕ does
not exist. Then, for any n, there is a model N of T and a ∈ N such that ϕ(N, a)
is finite, but has more than n elements. Let M be a proper elementary extension
of N. Then ϕ(M, a) = ϕ(N, a) and the pair (M, N, a) is a model of T ∗ . This
shows that the theory

T ∗ ∪ {∃>n x ϕ(x, c) | n = 1, 2, . . .}

is finitely satisfiable. A model of this theory gives a Vaughtian pair for T .

Exercise 5.5.1. If T is totally transcendental and has a Vaughtian pair for
ϕ(x), then it has, for all uncountable κ, a model of cardinality κ with countable
ϕ(M). Prove Corollary 5.5.4 from this. (Use Theorem 5.4.1.)
Exercise 5.5.2. Show directly (without using Lemma 5.2.9) that a theory T
which is categorical in some uncountable cardinality, has a model M of cardi-
nality ℵ1 in which each L(M )-formula is either satisfied by a finite number or
by ℵ1 many elements.
Exercise 5.5.3. Show that the theory RG of the random graph has a Vaughtian
pair.
Exercise 5.5.4. Let T be a theory, M a model of T and a ⊆ M a finite tuple
of parameters. Let q(x) be the type of a in M. Then for new constants c, the
L(c)-theory
T (q) = Th(M, a) = T ∪ {ϕ(c)| ϕ(x) ∈ q(x)}
is complete. Show that T is λ-stable (or without Vaughtian pair etc.) if and
only if T (q) is. For countable languages, this implies that T is categorical in
some uncountable cardinal if and only if T (q) is.
Exercise 5.5.5. If T eliminates ∃∞ , then T eliminates for every n the quantifier
“there are infinitely many n-tuples x1 , . . . , xn ”.
Exercise 5.5.6. Assume that T eliminates the quantifier ∃∞ . Then for every
formula ϕ(x1 , . . . , xn , y) there is a formula θ(y) such that in all models M of
T a tuple b satisfies θ if and only if M has an elementary extension M0 with
elements a1 , . . . , an ∈ M 0 \ M such that M0 |= ϕ(a1 , . . . , an , b).
Exercise 5.5.7. Let T1 and T2 be two model complete theories in disjoint
languages L1 and L2 . Assume that both theories eliminate ∃∞ . Then T1 ∪ T2
has a model companion.

5.6 Algebraic formulas

Formulas defining a finite set are called algebraic. In this section we collect a
few facts and a bit of terminology around this concept which will be crucial in
the following sections.
CHAPTER 5. ℵ1 -CATEGORICAL THEORIES 79

Definition 5.6.1. Let M be a structure and A a subset of M . A formula

ϕ(x) ∈ L(A) is called algebraic if ϕ(M) is finite. An element a ∈ M is algebraic
over A if it realises an algebraic L(A)-formula. We call an element algebraic if
it is algebraic over the empty set. The algebraic closure of A, acl(A), is the set
of all elements of M algebraic over A, and A is called algebraically closed if it
equals its algebraic closure.

Remark. Note that the algebraic closure of A does not grow in elementary
extensions of M because an L(A)-formula which defines a finite set in M defines
the same set in every elementary extension. As a special case we have that
elementary substructures are algebraically closed.

It is easy to see that

| acl(A)| ≤ max(|T |, |A|) (5.1)

(see Theorem 2.3.1).

In algebraically closed fields, an element a is algebraic over A precisely if a
is algebraic (in the field-theoretical sense) over the field generated by A. This
follows easily from quantifier elimination in ACF.
We call a type p(x) ∈ S(A) algebraic if (and only if) p contains an algebraic
formula. Any algebraic type p is isolated by an algebraic formula ϕ(x) ∈ L(A),
namely by any ϕ ∈ p having the minimal number of solutions in M. This
number is called the degree deg(p) of p. As isolated types are realised in every
model, the algebraic types over A are exactly those of the form tp(a/A) where
a is algebraic over A. The degree of a over A deg(a/A) is the degree of tp(a/A).
Lemma 5.6.2. Let p ∈ S(A) be non-algebraic and A ⊆ B. Then p has a
non-algebraic extension q ∈ S(B).
Proof. The extension q0 (x) = p(x) ∪ {¬ψ(x) | ψ(x) algebraic L(B)-formula}
is finitely satisfiable. For otherwise there are ϕ(x) ∈ p(x) and algebraic L(B)-
formulas ψ1 (x), . . . , ψn (x) with

M |= ∀x (ϕ(x) → ψ1 (x) ∨ · · · ∨ ψn (x)).

But then ϕ(x) and hence p(x) is algebraic. So we can take for q any type
containing q0 .

Remark 5.6.3. Since algebraic types are isolated by algebraic formulas, an

easy compactness argument shows that a type p ∈ S(A) is algebraic if and only
if p has only finitely many realisations (namely deg(p) many) in all elementary
extensions of M.
Lemma 5.6.4. Let M and N be two structures and f : A → B an elementary
bijection between two subsets. Then f extends to an elementary bijection between
acl(A) and acl(B).
CHAPTER 5. ℵ1 -CATEGORICAL THEORIES 80

Proof. Let g : A0 → B 0 a maximal extension of f to two subsets of acl(A) and

acl(B). Let a be an element of acl(A). Since a is algebraic over A0 , a is atomic
over A0 . We can therefore realise the type g(tp(a/A0 )) in N – by an element
b ∈ acl(B) – and obtain an extension g ∪ {ha, bi} of g. It follows that a ∈ A0 .
So g is defined on the whole of acl(A). Interchanging A and B shows that g is
surjective. (See Lemma 6.1.9 for an alternative proof.)
The algebraic closure operation will be used to study models of ℵ1 -categori-
cal theories in further detail.
Definition 5.6.5. A pregeometry 5 (or matroid ) (X, cl) is a set X with a closure
operator cl : P(X) → P(X), where P denotes the power set, such that for all
A ⊆ X and a, b ∈ X:
a) (Reflexivity) A ⊆ cl(A).
b) (Finite character) cl(A) is the union of all cl(A0 ), where the A0 range
over all finite subsets of A.
c) (Transitivity) cl(cl(A)) = cl(A).
d) (Exchange) a ∈ cl(Ab) \ cl(A) ⇒ b ∈ cl(Aa).
A set A is called closed (or cl-closed) if A = cl(A). Note that the closure
cl(A) of A is the smallest cl-closed set containing A. So a pregeometry is given
by the system of cl-closed subsets.
The operator cl(A) = A for all A ⊆ X is a trivial example of a pregeometry.
The three standard examples from algebra are vector spaces with the linear
closure operator, for a field K with prime field F , the relative algebraic closure
cl(A) = F (A)alg ∩ K,6 and for a field K of characteristic p, the p-closure given
by cl(A) = K p (A) (see Remark C.1.1).
Lemma 5.6.6. If X is the universe of a structure, acl satisfies Reflexivity,
Finite character and Transitivity.
Proof. Reflexivity and Finite Character are clear. For Transitivity
assume that c is algebraic over b1 , . . . , bn and the bi are algebraic over A.
We have to show that c is algebraic over A. Choose an algebraic formula
ϕ(x, b1 , . . . , bn ) satisfied by c and algebraic L(A)-formulas ϕi (y) satisfied by
the bi . Let ϕ(x, b1 , . . . , bn ) be satisfied by exactly k elements. Then the L(A)-
formula

∃ y1 . . . yn (ϕ1 (y1 ) ∧ · · · ∧ ϕn (yn ) ∧ ∃≤k zϕ(z, y1 , . . . , yn ) ∧ ϕ(x, y1 , . . . , yn ))

is algebraic and realised by c.

Exercise 5.6.1. If A ⊆ M and M is |A|+ -saturated, then p ∈ S(A) is algebraic
if and only if p(M ) is finite.
5 Pregeometries were introduced by van der Waerden (1930) and Whitney (1934).
6 Lalg denotes the algebraic closure of the field L.
CHAPTER 5. ℵ1 -CATEGORICAL THEORIES 81

Exercise 5.6.2 (P. M. Neumann). Let A, B be subsets of M and (c0 , . . . ,

cn ) a sequence of elements which are not algebraic over A. If M is |A ∪ B|+ -
saturated, the type tp(c0 , . . . , cn /A) has a realisation which is disjoint from B.
(Hint: Use induction on n. Distinguish between whether or not ci is algebraic over
Acn for some i < n.)

5.7 Strongly minimal sets

Strongly minimal theories defined below turn out to be uncountably categorical:
the isomorphism type of the model is determined by the dimension of an asso-
ciated geometry. While this appears to be a very special case of such theories,
we will see in the next section that we can always essentially reduce to this
situation.
Throughout this section we fix a complete theory T with infinite models. For
Corollary 5.7.9 we have to assume T countable.
Definition 5.7.1. Let M be a model of T and ϕ(x) a non-algebraic L(M )-
formula.
1. The set ϕ(M) is called minimal in M if for all L(M )-formulas ψ(x) the
intersection ϕ(M) ∧ ψ(M) is either finite or cofinite in ϕ(M).
2. The formula ϕ(x) is strongly minimal if ϕ(x) defines a minimal set in
all elementary extensions of M. In this case, we also call the definable
set ϕ(M) strongly minimal. A non-algebraic type containing a strongly
minimal formula is called strongly minimal.
.
3. A theory T is strongly minimal if the formula x = x is strongly minimal.
Clearly, strong minimality is preserved under definable bijections; i.e., if A
and B are definable subsets of Mk , Mm defined by ϕ and ψ, respectively, such
that there is a definable bijection between A and B, then if ϕ is strongly minimal
so is ψ.
Examples. 1. The following theories are strongly minimal, which is easily
seen in each case using quantifier elimination.

• Infset, the theory of infinite sets. The sets which are definable over
a parameter set A in a model M are the finite subsets S of A and
their complements M \ S.
• For a field K, the theory of infinite K-vector spaces. The sets defin-
able over a set A are the finite subsets of the subspace spanned by A
and their complements.
• The theories ACFp of algebraically closed fields of fixed characteristic.
The definable sets of any model K are Boolean combinations of zero-
sets
{a ∈ K | f (a) = 0}
CHAPTER 5. ℵ1 -CATEGORICAL THEORIES 82

of polynomials f (X) ∈ K[X]. Zero-sets are finite or, if f = 0, all of

K.
2. If K is a model ACFp , for any a, b ∈ K, the formula ax1 + b = x2 defining
an affine line A in K 2 is strongly minimal as there is a definable bijection
between A and K.
3. For any strongly minimal formula ϕ(x1 , . . . , xn ), the induced theory T  ϕ
is strongly minimal. Here, for any model M of T , the induced theory is the
theory of ϕ(M) with the structure given by all intersections of 0-definable
subsets of M nm with ϕ(M)m for all m ∈ ω. This theory depends only on
T and ϕ, not on M.
Whether ϕ(x, a) is strongly minimal depends only on the type of the pa-
rameter tuple a and not on the actual model: observe that ϕ(x, a) is strongly
minimal if and only if for all L-formulas ψ(x, z) the set

Σψ (z, a) = ∃>k x (ϕ(x, a) ∧ ψ(x, z))∧



∃>k x (ϕ(x, a) ∧ ¬ψ(x, z))  k = 1, 2, . . .



cannot be realised in any elementary extension. This means that for all ψ(x, z)
there is a bound kψ such that

M |= ∀z ∃≤kψ x (ϕ(x, a) ∧ ψ(x, z)) ∨ ∃≤kψ x (ϕ(x, a) ∧ ¬ψ(x, z)) .

This is an elementary property of a, i.e., expressible by a first-order formula. So

it makes sense to call ϕ(x, a) a strongly minimal formula without specifying a
model.
Lemma 5.7.2. If M is ω-saturated, or if T eliminates the quantifier ∃∞ , any
minimal formula is strongly minimal. If T is totally transcendental, every infi-
nite definable subset of Mn contains a minimal set ϕ(M).
Proof. If M is ω-saturated and ϕ(x, a) not strongly minimal, then for some
L-formula ψ(x, z) the set Σψ (z, a) is realised in M, so ϕ is not minimal.
If on the other hand ϕ(x, a) is minimal and T eliminates the quantifier ∃∞ ,
then for all L-formulas, ψ(x, z)

¬ ∃∞ x(ϕ(x, a) ∧ ψ(x, z)) ∧ ∃∞ x(ϕ(x, a) ∧ ¬ψ(x, z))

is an elementary property of z.
If ϕ0 (M) does not contain a minimal set, one can construct from ϕ0 (x) a
binary tree of L(M )-formulas defining infinite subsets of M. This contradicts
ω-stability.
From now on we will only consider strongly minimal formulas in one variable.
It should be clear how to extend everything to the more general context.
Lemma 5.7.3. The formula ϕ(M) is minimal if and only if there is a unique
non-algebraic type p ∈ S(M ) containing ϕ(x).
CHAPTER 5. ℵ1 -CATEGORICAL THEORIES 83

Proof. If ϕ(M) is minimal, then clearly

p = {ψ | ψ(x) ∈ L(M ) such that ϕ ∧ ¬ψ is algebraic}

is the unique non-algebraic type in S(M ) containing it. If ϕ(M) is not minimal,
there is some L-formula ψ with both ϕ ∧ ψ and ϕ ∧ ¬ψ non-algebraic. By
Lemma 5.6.2 there are at least two non-algebraic types in S(M ) containing
ϕ.
Corollary 5.7.4. A strongly minimal type p ∈ S(A) has a unique non-algebraic
extension to all supersets B of A in elementary extensions of M. Conse-
quently, the type of m realisations a1 , . . . , am of p with ai ∈
/ acl(a1 . . . ai−1 A),
i = 1, . . . , m, is uniquely determined.
Proof. Existence of non-algebraic extensions follows from Lemma 5.6.2, which
also allows us to assume that B is a model. Uniqueness now follows from
Lemma 5.7.3 applied to any strongly minimal formula of p. The last sentence
follows by induction.

Theorem 5.7.5. If ϕ(x) is a strongly minimal formula in M without parame-

ters, the operation
cl : P(ϕ(M)) → P(ϕ(M))
defined by
cl(A) = aclM (A) ∩ ϕ(M)
is a pregeometry (ϕ(M), cl).
Proof. We have to verify Exchange. For notational simplicity we assume A =
∅. Let a ∈ ϕ(M) be not algebraic over ∅ and b ∈ ϕ(M) not algebraic over a.
By Corollary 5.7.4, all such pairs a, b have the same type p(x, y). Let A0 be an
infinite set of non-algebraic elements realising ϕ (which exists in an elementary
extension of M) and b0 non-algebraic over A0 . Since all a0 ∈ A0 have the same
type p(x, b0 ) over b0 , no a0 is algebraic over b0 . Thus also a is not algebraic over
b.
The same proof shows that algebraic closure defines a pregeometry on the set
of realisations of a minimal type, i.e., a non-algebraic type p ∈ S1 (A) having a
unique non-algebraic extension to all supersets B of A in elementary extensions
of M. Here is an example to show that a minimal type need not be strongly
minimal.
Let T be the theory of M = (M, Pi )i<ω in which the Pi form a proper
descending sequence of subsets. The type p = {x ∈ Pi | i < ω} ∈ S1 (∅) is
minimal. If all Pi+1 are coinfinite in Pi , then p does not contain a minimal
formula and is not strongly minimal.
In pregeometries there is a natural notion of independence and dimension
(see Definition C.1.2), so in light of Theorem 5.7.5 we may define the following.
CHAPTER 5. ℵ1 -CATEGORICAL THEORIES 84

If ϕ(x) is strongly minimal without parameters, the ϕ-dimension of a model

M of T is the dimension of the pregeometry (ϕ(M), cl)

dimϕ (M).

If M is the model of a strongly minimal theory, we just write dim(M).

If ϕ(x) is defined over A0 ⊆ M , the closure operator of the pregeometry
ϕ(MA0 ) is given by
cl(A) = aclM (A0 ∪ A) ∩ ϕ(M)
and
dimϕ (M/A0 ) := dimϕ (MA0 )
is called the ϕ-dimension of M over A0 .
Lemma 5.7.6. Let ϕ(x) be defined over A0 and strongly minimal, and let
M and N be models containing A0 . Then there exists an A0 -elementary map
between ϕ(M) and ϕ(N) if and only if M and N have the same ϕ-dimension
over A0 .
Proof. An A0 -elementary map between ϕ(M) and ϕ(N) maps bases to bases,
so one direction is clear.
For the other direction we use Corollary 5.7.4: if ϕ(M) and ϕ(N) have the
same dimension over A0 , let U and V be bases of ϕ(M) and ϕ(N), respectively,
and let f : U → V be a bijection. By Corollary 5.7.4, f is A0 -elementary and
by Lemma 5.6.4 f extends to an elementary bijection g : acl(A0 U ) → acl(A0 V ).
Thus, g  ϕ(M) is an A0 -elementary map from ϕ(M) to ϕ(N).
We now turn to showing that strongly minimal theories are categorical in
all uncountable cardinals. For reference we first note the following special cases
of the preceding lemmas.
Corollary 5.7.7. 1. A theory T is strongly minimal if and only if over every
parameter set there is exactly one non-algebraic type.
2. In models of a strongly minimal theory the algebraic closure defines a
pregeometry.
3. Bijections between independent subsets of models of a strongly minimal
theory are elementary. In particular, the type of n independent elements
is uniquely determined.
If T is strongly minimal, by the preceding we have

| S(A)| ≤ | acl(A)| + 1.

Strongly minimal theories are therefore λ-stable for all λ ≥ |T |. Also there can
be no binary tree of finite or cofinite sets. So by the remark after the proof of
Theorem 5.2.6 T is totally transcendental. If ϕ(M) is cofinite and N a proper
elementary extension of M, then ϕ(N) is a proper extension of ϕ(M). Thus
strongly minimal theories have no Vaughtian pairs.
CHAPTER 5. ℵ1 -CATEGORICAL THEORIES 85

Theorem 5.7.8. Let T be strongly minimal. Models of T are uniquely deter-

mined by their dimensions. The set of possible dimensions is an end segment
of the cardinals. A model M is ω-saturated if and only if dim(M) ≥ ℵ0 . All
models are ω-homogeneous.
Proof. Let M0 , M1 be models of the same dimension, and let B0 , B1 be bases for
M0 and M1 , respectively. Then any bijection f : B0 → B1 is an elementary map
by Corollary 5.7.7, which extends to an isomorphism of the algebraic closures
M0 and M1 by Lemma 5.6.4.
The next claim implies that the possible dimensions form an end segment of
the cardinals.
Claim. Every infinite algebraically closed subset S of M is the universe of
an elementary substructure.
Proof of Claim. By Theorem 2.1.2 it suffices to show that every consistent
L(S)-formula ϕ(x) can be realised in S. If ϕ(M) is finite, all realisations are
algebraic over S and belong therefore to S. If ϕ(M) is cofinite, ϕ(M) meets all
infinite sets.
Let A be a finite subset of M and p the non-algebraic type in S(A). Thus,
p is realised in M exactly if M 6= acl(A), i.e., if and only if dim(M) > dim(A).
Since all algebraic types over A are always realised in M, this shows that M is
ω-saturated if and only if M has infinite dimension.
It remains to show that all models are ω-homogeneous. Let f : A → B be
an elementary bijection between two finite subsets of M . By Lemma 5.6.4, f
extends to an elementary bijection between acl(A) and acl(B). If a ∈ M \acl(A),
then p = tp(a/A) is the unique non-algebraic type over A and f (p) is the
unique non-algebraic type over B. Since dim(A) = dim(B), the argument in
the previous paragraph shows that f (p) is realised in M.
Corollary 5.7.9. If T is countable and strongly minimal, it is categorical in all
uncountable cardinalities.
Proof. Let M1 and M2 be two models of cardinality κ > ℵ0 . Choose two bases
B1 and B2 of M1 and M2 respectively. By p. 79, equation (5.1), B1 and B2
both have cardinality κ. Then any bijection f : B1 → B2 is an elementary map
by Corollary 5.7.7, which extends to an isomorphism of the algebraic closures
M1 and M2 by Lemma 5.6.4.
Exercise 5.7.1. If M is minimal and ω-saturated, then Th(M) is strongly
minimal.
Exercise 5.7.2. Show that the theory of an infinite set equipped with a bijec-
tion without finite cycles is strongly minimal and that the associated geometry
is trivial.
Exercise 5.7.3. Show directly that strongly minimal theories eliminate ∃∞
(without using Corollaries 5.5.4 and 5.7.9.)
Exercise 5.7.4. A type is minimal if and only if its set of realisations in any
model is minimal (i.e., has no infinite and coinfinite relatively definable subsets).
CHAPTER 5. ℵ1 -CATEGORICAL THEORIES 86

Exercise 5.7.5. Show that acl defines a pregeometry on µ(M) if µ(M) is min-
imal. In fact the following is true: if b ∈ µ(M), a ∈ acl(Ab), b 6∈ acl(Aa), then
a ∈ acl(A). Furthermore we have deg(a/A) = deg(a/Ab).

5.8 The Baldwin–Lachlan Theorem

In this section, we present the characterisation of uncountably categorical theo-
ries due to Baldwin and Lachlan [2]. Since this characterisation is independent
of the uncountable cardinal, it implies Morley’s Theorem. The crucial point is
the existence of a strongly minimal formula ϕ in a totally transcendental the-
ory. By Corollary 5.5.5, each model M is prime over the set of realisations ϕ(M)
whose dimension determines the isomorphism type of the model.
Theorem 5.8.1 (Baldwin–Lachlan). Let κ be an uncountable cardinal. A
countable theory T is κ-categorical if and only if T is ω-stable and has no Vaugh-
tian pairs.

Proof. If T is categorical in some uncountable cardinal, then T is ω-stable by

Theorem 5.2.4 and has no Vaughtian pair by Corollary 5.5.4.
For the other direction we first obtain a strongly minimal formula: since T
is totally transcendental, it has a prime model M0 . (This follows from Theo-
rems 4.5.7 and 4.5.9 or from Theorem 5.3.3.) Let ϕ(x) be a minimal formula in
L(M0 ), which exists by Lemma 5.7.2. Since T has no Vaughtian pairs, ∃∞ can be
eliminated by Lemma 5.5.7 and hence ϕ(x) is strongly minimal by Lemma 5.7.2.
Let M1 , M2 be models of cardinality κ. We may assume that M0 is an
elementary submodel of both M1 and M2 . Since T has no Vaughtian pair, Mi
is a minimal extension of M0 ∪ ϕ(Mi ), i = 1, 2. Therefore, ϕ(Mi ) has cardi-
nality κ and hence (since κ is uncountable) we conclude that dimϕ (M1 /M0 ) =
κ = dimϕ (M2 /M0 ). By Lemma 5.7.6 there exists an M0 -elementary map from
ϕ(M0 ) to ϕ(M1 ), which by Lemma 5.3.8 extends to an isomorphism from M1
to M2 .
Corollary 5.8.2 (Morley). Let κ be an uncountable cardinal. Then T is ℵ1 -
categorical if and only if T is κ-categorical.
Notice that the proof of Theorem 5.8.1 shows in fact the following.

Corollary 5.8.3. Suppose T is ℵ1 -categorical, M1 , M2 are models of T , ai ∈

Mi and ϕ(x, ai ) strongly minimal, i = 1, 2, with tp(a1 ) = tp(a2 ). If M1 and
M2 have the same respective ϕ-dimension, then they are isomorphic.
For uncountable models, the ϕ-dimension equals the cardinality of the model,
so clearly does not depend on the realisation of tp(ai ). We will show in Sec-
tion 6.3 that the converse to Corollary 5.8.3 holds also for countable models, i.e.,
if ϕ(x, a0 ) is strongly minimal, then the ϕ-dimension of M is the same for all a
realising tp(a0 ). Thus, also the countable models of an uncountably categorical
theory are in one-to-one correspondence with the possible ϕ-dimensions.
Chapter 6

Morley rank

In this chapter we collect a number of further results about totally transcen-

dental theories, in particular we will introduce Morley rank. We then finish the
analysis of the countable models of uncountably categorical theories.
For convenience, we first introduce the ‘monster model’ (for arbitrary theo-
ries), a very large, very saturated, very homogeneous model. From now on, all
models we consider will be elementary submodels of this monster model.
Furthermore, we often simplify the notation by assuming that we are working
in a many-sorted structure where for each n ∈ ω we have an extra sort for n-
tuples of elements. While we are then working in a many-sorted language in
which we have to specify the sorts for all variables involved in a formula, this
allows us to treat n-tuples exactly like 1-tuples, i.e., elements. We emphasise
that this is purely a notational convention. In Section 8.4 we will show how to
systematically extend a structure by introducing new sorts in a big way without
changing those properties of the theories we are interested in.

6.1 Saturated models and the monster

The importance of saturated structures was already visible in Section 5.2 where
we showed that saturated structures of fixed cardinality are unique up to iso-
morphism. Saturated structures need not exist (think about why not), but
by considering special models instead, we can preserve many of the important
properties – and prove their existence.
Definition 6.1.1. A structure M of cardinality κ ≥ ω is special if M is the
union of an elementary chain Mλ where λ runs over all cardinals less than κ
and each Mλ is λ+ -saturated.
We call (Mλ ) a specialising chain.
Remark. Saturated structures are special. If |M| is regular, the converse is
true.

87
CHAPTER 6. MORLEY RANK 88

Lemma 6.1.2. Let λ be an infinite cardinal ≥ |L|. Then every L-structure M

of cardinality 2λ has a λ+ -saturated elementary extension of cardinality 2λ .
Proof. Every set of cardinality 2λ has 2λ many subsets of cardinality at most
λ. This allows us to construct a continuous elementary chain

M = M0 ≺ M 1 · · · ≺ Mα ≺ · · · (α < λ+ )

of structures of cardinality 2λ such that all p ∈ S(A), for A ⊆ Mα , |A| ≤ λ, are

realised in Mα+1 . The union of this chain has the desired properties.
Corollary 6.1.3. Let κ > |L| be an uncountable cardinal. Assume that

λ < κ ⇒ 2λ ≤ κ (6.1)

Then every infinite L-structure M of cardinality smaller than κ has a special

extension of cardinality κ.
Let α be a limit ordinal. Then for any cardinal µ, κ = iα (µ) satisfies (6.1)
and we have cf(κ) = cf(α) (see p. 186).
The following is a generalisation of Lemma 5.2.8
Theorem 6.1.4. Two elementarily equivalent special structures of the same
cardinality are isomorphic.
Proof. The proof is a refined version of the proof of Lemma 5.2.8. Let A and
B be two elementarily equivalent special structures of cardinality κ with spe-
cialising chains (Aλ ) and (Bλ ), respectively. The well-ordering defined in the
proof of Lemma A.3.7 can be used to find enumerations (aα )α<κ and (bα )α<κ
of A and B such that aα ∈ A|α| and bα ∈ B|α| . We construct an increasing
sequence of elementary maps f α : Aα → B α such that for all α which are zero
or limit ordinals we have aα+i ∈ Aα+2i , bα+i ∈ B α+2i+1 , and also |Aα | ≤ |α|,
Aα ⊆ A|α| , |B α | ≤ |α|, B α ⊆ B|α| .
Definition 6.1.5. A structure M is
• κ-universal if every structure of cardinality < κ which is elementarily
equivalent to M can be elementarily embedded into M.

• κ-homogeneous if for every subset A of M of cardinality smaller than κ

and for every a ∈ M , every elementary map A → M can be extended to
an elementary map A ∪ {a} → M .
• strongly κ-homogeneous if for every subset A of M of cardinality less than
κ, every elementary map A → M can be extended to an automorphism of
M.
Theorem 6.1.6. Special structures of cardinality κ are κ+ -universal and
strongly cf(κ)-homogeneous.
CHAPTER 6. MORLEY RANK 89

Proof. Let M be a special structure of cardinality κ. The κ+ -universality of

M can be proved in the same way as Theorem 6.1.4. Let A be a subset of
M of cardinality less than cf(κ) and let f : A → M an elementary map. Fix
a specialising sequence (Mλ ). For λ0 sufficiently large, Mλ0 contains A. The
sequence (
∗ (Mλ , a)a∈A , if λ0 ≤ λ
Mλ =
(Mλ0 , a)a∈A , if λ < λ0
is then a specialising sequence of (M, a)a∈A . For the same reason (M,f(a))a∈A
is special. By Theorem 6.1.4 these two structures are isomorphic under an
automorphism of M which extends f .

The monster model.

Let T be a complete theory with infinite models. For convenience, we would like
to work in a very large saturated structure, large enough so that any model of T
can be considered as an elementary substructure. If T is totally transcendental,
by Remark 5.2.10 we can choose such a monster model as a saturated model of
cardinality κ where κ is a regular cardinal greater than all the models we ever
consider otherwise. Using Exercise 8.2.7, this also works for stable theories and
regular κ with κ|T | = κ. For any infinite λ, κ = (λ|T | )+ has this property.
In order to construct the monster model C for an arbitrary theory T we
will work in BGC (Bernays–Gödel + Global Choice). This is a conservative
extension of ZFC (see Appendix A) which adds classes to ZFC. Then being a
model of T is interpreted as being the union of an elementary chain of (set-size)
models of T . The universe of our monster model C will be a proper class.
Theorem 6.1.7 (BGC). There is a class-size model C of T such that all types
over all subsets of C are realised in C. Moreover C is uniquely determined up to
isomorphism.
Proof. Global choice allows us to construct a long continuous elementary chain
(Mα )α∈On of models of T such that all types over Mα are realised in Mα+1 . Let
C be the union of this chain. The uniqueness is proved as in Lemma 5.2.8.
We call C the monster model of T . Note that Global Choice implies that C
can be well-ordered.
Corollary 6.1.8.
• C is κ-saturated for all cardinals κ.
• Any model of T is elementarily embeddable in C

• Any elementary bijection between two subsets of C can be extended to an

automorphism of C.
CHAPTER 6. MORLEY RANK 90

We say that two elements are conjugate over some parameter set A if there
is an automorphism of C fixing A elementwise and taking one to the other. Note
that a, b ∈ C are conjugate over A if and only if they have the same type over A.
We call types p ∈ S(A), q ∈ S(B) conjugate over D if there is an automorphism f
of C fixing D and taking A to B and such that q = {ϕ(x, f (a)) | ϕ(x, a) ∈ p}.
Note that strictly speaking Aut(C) is not an object in Bernays–Gödel Set theory
but we will nevertheless use this term as a way of talking about automorphisms.
Readers who mistrust set theory can fix a regular cardinal γ bigger than the
cardinality of all models and parameter sets they want to consider. For C they
may then use a special model of cardinality κ = iγ (ℵ0 ). This is κ+ -universal
and strongly γ-homogeneous.
We will use the following convention throughout the rest of this book:
• Any model of T is an elementary substructure of C. We identify models
with their universes and denote them by M , N ,. . . .
• Parameter sets A, B, . . . are subsets of C.
• Formulas ϕ(x) with parameters define a subclass F = ϕ(C) of C. Two
formulas are equivalent if they define the same class.

• We write |= ϕ for C |= ϕ.
• A set of formulas with parameters from C is consistent if it is realised in C.
• If π(x) and σ(x) are partial types we write π ` σ for π(C) ⊆ σ(C).
• A global type is a type p over C; we denote this by p ∈ S(C).

This convention changes the flavour of quite a number of proofs. As an

example look at the following
Lemma 6.1.9. An elementary bijection f : A → B extends to an elementary
bijection between acl(A) and acl(B).

Proof. Extend f to an automorphism f 0 of C. Clearly f 0 maps acl(A) to acl(B).

This implies Lemma 5.6.4 and the second claim in the proof of Theorem 5.7.6.
Note that by the remark on p. 79 for any model M and any A ⊆ M the algebraic
closure of A in the sense of M equals the algebraic closure in the sense of C.

Lemma 6.1.10. Let D be a definable class and A a set of parameters. Then

the following are equivalent:
a) D is definable over A.
b) D is invariant under all automorphisms of C which fix A pointwise.
CHAPTER 6. MORLEY RANK 91

Proof. Let D be defined by ϕ, defined over B ⊃ A. Consider the maps

τ π
C −→ S(B) −→ S(A),

where τ (c) = tp(c/B) and π is the restriction map. Let Y be the image of D in
S(A). Since Y = π[ϕ], Y is closed.
Assume that D is invariant under all automorphisms of C which fix A point-
wise. Since elements which have the same type over A are conjugate by an
automorphism of C, this means that D-membership depends only on the type
over A i.e., D = (πτ )−1 (Y ).
This implies that [ϕ] = π −1 (Y ), or S(A) \ Y = π[¬ϕ], hence S(A) \ Y is also
closed and we conclude that Y is clopen. By Lemma 4.2.3 Y = [ψ] for some
L(A)-formula ψ. This ψ defines D.
The same proof shows that the same is true for definable relations R ⊆ Cn :
namely, R is A-definable if and only if it is invariant under all α ∈ Aut(C/A).
Definition 6.1.11. The definable closure dcl(A) of A is the set of elements c for
which there is an L(A)-formula ϕ(x) such that c is the unique element satisfying
ϕ. Elements or tuples a and b are said to be interdefinable if a ∈ dcl(b) and
b ∈ dcl(a).
Corollary 6.1.12. 1. a ∈ dcl(A) if and only if a has only one conjugate
over A.
2. a ∈ acl(A) if and only if a has finitely many conjugates over A.
Proof. 1. is clear, since a ∈ dcl(A) means that {a} is A-definable. 2. follows
from Remark 5.6.3, since the realisations of tp(a/A) are exactly the conjugates
of a over A.
Exercise 6.1.1. Finite structures are saturated.
Exercise 6.1.2. acl(A) is the intersection of all models which contain A.
Exercise 6.1.3. Prove Robinson’s Joint Consistency Lemma: extend the
complete L-theory T to an L1 -theory T1 and an L2 -theory T2 such that L =
L1 ∩ L2 . If T1 and T2 are both consistent, show that T1 ∪ T2 is consistent.
Exercise 6.1.4. Prove Beth’s Interpolation Theorem: if ` ϕ1 → ϕ2 for Li -
sentences ϕi , there is an L = L1 ∩ L2 -sentence θ such that ` ϕ1 → θ and
` θ → ϕ2 .
Exercise 6.1.5. A class C of L-structures is a PC∆ -class if there is an extension
L0 of L and an L0 -theory T 0 such that C consists of all reducts to L of models
of T 0 . Show that a PC∆ -class is elementary if and only if it is closed under
elementary substructures.
Exercise 6.1.6. If M is κ-saturated, then over every set of cardinality smaller
than κ every type in κ many variables is realised in M .
CHAPTER 6. MORLEY RANK 92

Exercise 6.1.7. Let κ be an infinite cardinal, not smaller than the cardinality
of L and M an L-structure. Show that the following are equivalent
a) M is κ-saturated
b) M is κ+ -universal and κ-homogeneous

If max(|L|, ℵ0 ) < κ this is also equivalent to

c) M is κ-universal and κ-homogeneous.
Exercise 6.1.8. Let κ be a an uncountable regular cardinal > |L|. We use the
notation 2<κ for supλ<κ 2λ . Show that

1. Every L-structure M of cardinality 2<κ has a κ-saturated elementary ex-

tension of cardinality 2<κ .
2. Assume that λ < κ implies 2λ ≤ κ. Then every L-structure M of cardi-
nality κ has a saturated elementary extension of cardinality κ.

Exercise 6.1.9. Let Pω (N) be the set of all finite subsets of N. Show that the
theory of (Pω (N), ⊆) has a saturated model of cardinality κ if and only if κ is
regular and λ < κ implies 2λ ≤ κ.
Exercise 6.1.10. A type-definable class is the class of all realisations of a set of
formulas. Show that a type-definable class is invariant under all automorphisms
of C which fix A pointwise if and only if it can be defined by a set of L(A)-
formulas. (Use Exercise 4.2.2(a) and the proof of Lemma 6.1.10.)
Exercise 6.1.11. Let A be contained in B. Show that the following are equiv-
alent:
1. B ⊆ dcl(A).

2. Every type over A extends uniquely to B.

Exercise 6.1.12.
1. b is in the definable closure of a if and only if there is a 0-definable class
D containing a and a 0-definable map D → C which maps a to b.

2. a and b are interdefinable if and only if a and b are contained in 0-definable

classes D and E and there is a 0-definable bijection between D and E which
maps a to b.
Exercise 6.1.13. Let K be a model of ACF0 , ACFp for p > 0, of RCF or of DCF0
and let A be a (differential) subfield of K. Prove that the definable closure of
A is
1. (ACF0 ) A itself,
2. (ACFp ) the perfect hull (see Definition B.3.8) of A,
CHAPTER 6. MORLEY RANK 93

3. (RCF) the relative algebraic closure of A,

4. (DCF0 ) A itself.
Exercise 6.1.14. Use Exercise 6.1.13 to show the following:
1. Let K be a model of ACF0 and f : K n → K a definable function. Then
K n can be decomposed into a finite number of definable subsets Xi such
that, on each Xi , f is given by a rational function.
2. The same is true for models of ACFp , p > 0. But on each Xi , f is of the
−m
form hp , for some rational function h.

3. In models of DCF0 , f is given on each Xi by a differential rational function.

Exercise 6.1.15 (P. Neumann). Let X be an infinite set, G ≤ Sym(X), and
B ⊆ X finite. Suppose that the orbit of each of the elements c0 , . . . , cn ∈
X under G is infinite. Then there is some g ∈ G with g(ci ) ∈ / B for i =
0, . . . , n. (Hint: Proceed as in Exercise 5.6.2. In fact, Exercise 5.6.2 follows from this
by compactness.)

Exercise 6.1.16 (B. Neumann). Let G be a group (not necessarily abelian),

and H0 , . . . , Hn , subgroups of infinite index. Show that G is not a finite union
of cosets of the Hi (but see Exercise 6.1.17, which is easy). Deduce Lemma 3.3.9
from this.
Exercise 6.1.17 (B. & P. Neumann). Deduce Exercise 6.1.15 from 6.1.16 and
conversely.

6.2 Morley rank

The Morley rank is a rather natural notion of dimension on the formulas of a
theory T or, equivalently, on the definable subsets of the monster model, defined
inductively very much like the dimension of algebraic sets. It is ordinal valued
for all consistent formulas if and only if T is totally transcendental. In this
section, we let T be a complete (possibly uncountable) theory.
We now define the Morley rank MR for formulas ϕ(x) with parameters in
the monster model. We remind the reader that by the conventions introduced at
the beginning of this chapter, a variable or element in the many-sorted language
described there may refer to an n-tuple of the original single sort. We begin by
defining the relation MR ϕ ≥ α by induction on the ordinal α.1 We remind the
reader that by the conventions introduced at the beginning of this chapter, a
variable or element in the many-sorted language described there may refer to
an n-tuple of the original single sort.
1 See the set-theoretic caveat before Exercise 6.2.1.
CHAPTER 6. MORLEY RANK 94

MR ϕ ≥ 0 if ϕ is consistent;
MR ϕ ≥ β + 1 if there is an infinite family (ϕi (x) | i < ω) of formulas
(in the same variable x) which imply ϕ, are pairwise
inconsistent and such that MR ϕi ≥ β for all i;
MR ϕ ≥ λ (for a limit ordinal λ) if MR ϕ ≥ β for all β < λ.
Definition 6.2.1. To define MR ϕ we distinguish three cases

1. If there is no α with MR ϕ ≥ α, we put MR ϕ = −∞.

2. MR ϕ ≥ α for all α, we put MR ϕ = ∞.
3. Otherwise, by the definition of MR ϕ ≥ λ for limit ordinals λ, there is a
maximal α with MR ϕ ≥ α, and we set MR ϕ = max{α | MR ϕ ≥ α}.

It is easy to see by induction on α that the relation MR ϕ ≥ α implies the

relation MR ϕ ≥ β for β ≤ α. It follows from this that indeed the Morley rank
of ϕ is at least α if and only if the relation MR ϕ ≥ α holds.2
Note that
MR ϕ = −∞ ⇔ ϕ is inconsistent
MR ϕ = 0 ⇔ ϕ is consistent and algebraic.

If a formula has ordinal-valued Morley rank, we also say that this formula
has Morley rank.3 The Morley rank MR(T ) of T is the Morley rank of the
.
formula x = x. The Morley rank of a formula ϕ(x, a) only depends on ϕ(x, y)
and the type of a. It follows that if a formula has Morley rank, then it is less
than (2|T | )+ . We will see in Exercise 6.2.5 that in fact all ordinal ranks are
smaller that |T |+ .
Remark. Clearly, if ϕ implies ψ, then MR ϕ ≤ MR ψ. It is also clear from the
definition that if ϕ has rank α < ∞, then for every β < α there is a formula ψ
which implies ϕ and has rank β.
.
Example 6.2.2. In Infset the formula x1 = a has Morley rank 0. If considered
.
as a formula in two variables, ϕ(x1 , x2 ) = x1 = a, it has Morley rank 1.
The next lemma expresses the fact that the formulas of rank less than α
form an ideal in the Boolean algebra of equivalence classes of formulas.

Lemma 6.2.3.
MR(ϕ ∨ ψ) = max(MR ϕ, MR ψ).
Proof. By the previous remark, we have MR(ϕ ∨ ψ) ≥ max(MR ϕ, MR ψ). For
the other inequality we show by induction on α that

MR(ϕ ∨ ψ) ≥ α + 1 implies max(MR ϕ, MR ψ) ≥ α + 1.

2 Here, of course −∞ is considered as being smaller and ∞ as being bigger than all ordinals.
3 Note that having Morley rank is a nontrivial property of a formula (see Theorem 6.2.7).
CHAPTER 6. MORLEY RANK 95

Let MR(ϕ ∨ ψ) ≥ α + 1. Then there is an infinite family of formulas (ϕi )

that imply ϕ ∨ ψ, are pairwise inconsistent and such that MR ϕi ≥ α. By the
induction hypothesis, for each i we have MR(ϕi ∧ ϕ) ≥ α or MR(ϕi ∧ ψ) ≥ α.
If the first case holds for infinitely many i, then MR ϕ ≥ α + 1. Otherwise
MR ψ ≥ α + 1.
We call ϕ and ψ α-equivalent,

ϕ ∼α ψ,

if their symmetric difference ϕ 4 ψ has rank less than α. By our previous

considerations it is clear that α-equivalence is in fact an equivalence relation.
We call a formula ϕ α-strongly minimal if it has rank α and for any formula
ψ implying ϕ either ψ or ϕ ∧ ¬ψ, has rank less than α, (equivalently, if every
ψ ⊆ ϕ is α-equivalent to ∅ or to ϕ). In particular, ϕ is 0-strongly minimal if
and only if ϕ is realised by a single element and ϕ is 1-strongly minimal if and
only if ϕ is strongly minimal.
Lemma 6.2.4. Each formula ϕ of rank α < ∞ is equivalent to a disjunction
of finitely many pairwise disjoint α-strongly minimal formulas ϕ1 , . . . , ϕd , the
α-strongly minimal components (or just components) of ϕ. The components are
uniquely determined up to α-equivalence.
Proof. Suppose ϕ is a formula of rank α without such a decomposition. Then ϕ
can be written as the disjoint disjunction of a formula ϕ1 of rank α and another
formula ψ1 of rank α not having such a decomposition. Inductively there are
formulas ϕ = ϕ0 , ϕ1 , . . . of rank α and ψ1 , ψ2 , . . . so that ϕi is the disjoint union
of ϕi+1 and ψi+1 . But then the rank of ϕ would be greater than α.
To see the uniqueness of this decomposition, let ψ be an α-strongly minimal
formula implying ϕ and let ϕ1 , . . . , ϕd be the α-strongly minimal components.
Then ψ can be decomposed into the formulas ψ ∧ ϕi , one of which must be
α-equivalent to ψ. So up to α-equivalence the components of ϕ are exactly the
α-strongly minimal formulas implying ϕ.
Definition 6.2.5. For a formula ϕ of Morley rank α < ∞, the Morley degree
MD(ϕ) is the number of its α-strongly minimal components.
The Morley degree is not defined for inconsistent formulas or formulas not
having Morley rank. The Morley degree of a consistent algebraic formula is the
number of its realisations. Strongly minimal formulas are exactly the formulas
of Morley rank and Morley degree 1. As with strongly minimal formulas it is
easy to see that Morley rank and degree are preserved under definable bijections.
Defining MDα (ϕ) as the Morley degree for formulas ϕ of rank α, as 0 for
formulas of smaller rank and as ∞ for formulas ϕ of higher rank, we obtain the
following.
Lemma 6.2.6. If ϕ is the disjoint union of ψ1 and ψ2 , then

MDα (ϕ) = MDα (ψ1 ) + MDα (ψ2 ).

CHAPTER 6. MORLEY RANK 96

Theorem 6.2.7. The theory T is totally transcendental if and only if each

formula has Morley rank.
Proof. Since there are not arbitrarily large ordinal Morley ranks, each formula
ϕ(x) without Morley rank can be decomposed into two disjoint formulas without
Morley rank, yielding a binary tree of consistent formulas in the free variable x.
For the other direction let (ϕs (x) | s ∈ <ω 2) be a binary tree of consistent
formulas. Then none of the ϕs has Morley rank. Otherwise there is a ϕs whose
ordinal rank α is minimal and (among the formulas of rank α) of minimal
degree. Then both ϕs0 and ϕs1 have rank α and therefore smaller degree than
ϕ, a contradiction.

A group is said to have the descending chain condition (dcc) on definable

subgroups, if there is no infinite properly descending chain H0 ⊃ H1 ⊃ H2 ⊃ · · ·
of definable subgroups.
Remark 6.2.8. A totally transcendental group has the descending chain con-
dition on definable subgroups.

Proof. If H is a definable proper subgroup of a totally transcendental group G,

then the Morley rank or the Morley degree of H must be smaller than that of
G since any coset of H has the same Morley rank and degree as H. Therefore
the claim follows from the fact that the ordinals are well-ordered.
The previous remark is a crucial tool in the theory of totally transcendental
groups. For example, it immediately implies that (Z, +) is not totally transcen-
dental.
Corollary 6.2.9. The theory of separably closed fields of degree of imperfection
e > 0 is not totally transcendental.
2 3
Proof. The subfields K ⊃ K p ⊃ K p ⊃ K p ⊃ · · · form an infinite definable
chain of properly descending (additive) subgroups. In fact we will see later that
the proof shows that that separably closed fields are not even superstable (see
Exercise 8.6.10.)
Definition 6.2.10. The Morley rank MR(p) of a type p is the minimal rank of
any formula in p. If MR(p) is an ordinal α, then its Morley degree MD(p) is the
minimal degree of a formula of p having rank α. If p = tp(a/A) we also write
MR(a/A) and MD(a/A).
Algebraic types have Morley rank 0 and

MD(p) = deg(p).

Strongly minimal types are exactly the types of Morley rank and Morley de-
gree 1.
Let p ∈ S(A) have Morley rank α and Morley degree d. Then by definition
there is some ϕ ∈ p of rank α and degree d. Clearly, ϕ is uniquely determined
CHAPTER 6. MORLEY RANK 97

up to α-equivalence since for all ψ we have MR(ϕ∧¬ψ) < α if and only if ψ ∈ p.

Thus, p is uniquely determined by ϕ:

p = {ψ(x) | ψ L(A)-formula, MR(ϕ ∧ ¬ψ) < α} . (6.2)

Obviously, α-equivalent formulas determine the same type (see Lemma 5.7.3).
Thus ϕ ∈ L(A) belongs to a unique type of rank α if and only if ϕ is α-
minimal over A; i.e., if ϕ has rank α and cannot be decomposed as the union of
two L(A)-formulas of rank α.
Lemma 6.2.11. Let ϕ be a consistent L(A)-formula.

1. MR ϕ = max{MR(p) | ϕ ∈ p ∈ S(A)}.
2. Let MR ϕ = α. Then
X 

MD ϕ = MD(p)  ϕ ∈ p ∈ S(A), MR(p) = α .

Proof. 1: If MR ϕ = ∞, then {ϕ} ∪ {¬ψ | ψ L(A)-formula, MR ψ < ∞} is

consistent. Any type over A containing this set of formulas has rank ∞.
If MR ϕ = α, there is a decomposition of ϕ in L(A)-formulas ϕ1 , . . . , ϕk , α-
minimal over A. (Note that k is bounded by MD ϕ.) By (6.2), the ϕi determine
types pi of rank α.
2: The pi are exactly the types of rank α containing ϕ. Furthermore,

MD ϕi = MD(pi ).

Corollary 6.2.12. If p ∈ S(A) has Morley rank and A ⊆ B, then

X 

MD(p) = MD(q)  p ⊆ q ∈ S(B), MR(p) = MR(q)

Corollary 6.2.13. Let p ∈ S(A) have Morley rank and A ⊆ B. Then p ∈ S(A)
has at least one and at most MD(p) many extensions to B of the same rank.
We will later show that extensions of the same Morley rank are a special
case of the non-forking extensions studied in Chapters 7 and 8. For types with
Morley rank those of the same Morley rank are exactly the non-forking ones.
Caveat: Set-theoretically we defined the Morley rank as a function which
maps each α to a class of formulas. In Bernays–Gödel set theory one cannot
in general define functions from ordinals to classes by a recursion scheme. The
more conscientious reader should therefore use a different definition: for each
set A define the relation MRA (ϕ) ≥ α using only formulas with parameters
from A, and put MR ϕ ≥ α if MRA (ϕ) ≥ α for some (sufficiently large) A. The
following exercise shows that if ϕ is defined over an ω-saturated model M , we
have MR ϕ = MRM ϕ.
CHAPTER 6. MORLEY RANK 98

Exercise 6.2.1. Let ϕ be a formula with parameters in the ω-saturated model

M . If MR ϕ > α, show that there is an infinite family of formulas with parame-
ters in M which each imply ϕ, are pairwise inconsistent and have Morley rank
≥ α.
Exercise 6.2.2. Let ϕ be a formula of Morley rank α < ∞ and ψ0 , ψ1 , . . . an
infinite sequence of formulas. Assume that there is a number k such that the
conjunction any k of the ψi has Morley rank smaller than α. Then MR(ϕ∧ψi ) <
α for almost all i.
Exercise 6.2.3. Show that a totally transcendental group G has a connected
component, i.e., a smallest definable subgroup G0 of finite index. Show also that
any finite normal subgroup of G0 lies in the centre of G0 .
Exercise 6.2.4. If T is totally transcendental, then all types over ω-saturated
models have Morley degree 1. (We will see in Corollary 8.5.12 that this is true
without assuming ω-saturation.)
Exercise 6.2.5 (Lachlan). If a type p has Morley rank, then MR(p) < |T |+ .
Hence, if T is totally transcendental, we have MR(T ) < |T |+ .
Exercise 6.2.6. For a topological space X we define recursively on ordinals
X 0 := X, X α+1 := X α \ {x | x isolated in X α } and X λ := α<λ X α if λ is a
T
limit ordinal. The Cantor–Bendixson rank of x ∈ X is equal to α if α is maximal
with x ∈ X α .
Show that on S(C) the Morley rank equals the Cantor–Bendixson rank. Note
that S(C) is not even a class. So, for this exercise we have to ignore set-theoretic
subtleties.
Exercise 6.2.7. Call a function R which associates to every non-empty defin-
able class an ordinal dimensional if R(ϕ ∨ ψ) = max(R(ϕ), R(ψ)). A function
R : S(C) → On is continuous if {p | R(p) ≥ α} is closed for every α. Show that

R(ϕ) = max{R(p) | ϕ ∈ p}
R(p) = min{R(ϕ) | ϕ ∈ p}

defines a bijection between dimensional R and continuous functions R.

Exercise 6.2.8. If p is a type over acl(A), then p and p  A have the same
Morley rank.

6.3 Countable models of ℵ1 -categorical theories

Uncountable models of ℵ1 -categorical theories are determined up to isomor-
phism by their cardinality. Section 5.8 showed that this cardinality coincides
with the dimension of a strongly minimal formula. We here extend this analysis
in order to classify also the countable models of ℵ1 -categorical theories and show
CHAPTER 6. MORLEY RANK 99

in Theorem 6.3.7 that for each possible dimension there is a unique model of
the theory.
Throughout this section we fix a countable ℵ1 -categorical theory T . For mod-
els M ≺ N of T and ϕ(x) ∈ L(M ) a strongly minimal formula, we write
dimϕ (N/M ) for the ϕ-dimension of N over M .

Theorem 6.3.1. Let T be a countable ℵ1 -categorical theory, M ≺ N be models

of T , A ⊆ M and ϕ(x) ∈ L(A) a strongly minimal formula.
1. If b1 , . . . , bn ∈ ϕ(N ) are independent over M and N is prime over M ∪
{b1 , . . . , bn }, then
dimϕ (N/M ) = n, and

2. dimϕ (N ) = dimϕ (M ) + dimϕ (N/M ).

Proof. For ease of notation we assume A = ∅.
1: Let c ∈ ϕ(N ). We want to show that c is algebraic over M ∪ {b1 , . . . , bn }.
Assume the contrary. Then p(x) = tp(c/M ∪ {b1 , . . . , bn }) is strongly minimal
and is axiomatised by
{ϕ(x)} ∪ {¬ϕi (x) | i ∈ I},
where the ϕi range over all algebraic formulas defined over M ∪ {b1 , . . . , bn }.
Since ϕ(M ) is infinite, any finite subset of p(x) is realised by an element of M .
Thus, p(x) is not isolated. But all elements of the prime extension N are atomic
over M ∪ {b1 , . . . , bn } by Corollary 5.3.7, a contradiction.
2: This follows from Remark C.1.8, if we can show that a basis of ϕ(N ) over
ϕ(M ) is also a basis of ϕ(N ) over M . So the proof is complete once we have
established the following lemma.
Lemma 6.3.2. Let T be ω-stable, M ≺ N models of T , ϕ(x) be strongly mini-
mal and bi ∈ ϕ(N ). If the bi are independent over ϕ(M ), they are independent
over M .

Proof. Assume that b1 , . . . , bn are algebraically independent over ϕ(M ) but de-
pendent over a ∈ M . Put b = b1 . . . bn . An argument as in the proof of Theorem
5.5.2 shows that we may assume that M is ω-saturated. Let p be the type of
0 1 2i
b over M . We choose a sequence b , b , . . . in ϕ(M ) such that b is an n-
0 2i−1
tuple of elements algebraically independent over ab . . . b and b̄2i+1 realises
0 2i
p  ab . . . b . Let q be the type of a over the set B of elements of (b̄i ). Since
i
the sequence (b ) is indiscernible, every permutation π of ω defines a type π(q)
over B. If {i | π(2i) even} 6= {i | π 0 (2i) even}, we have π(q) 6= π 0 (q). So there
are uncountably many types over B and T is not ω-stable.

The previous lemma holds for arbitrary theories. This uses the fact that
ϕ(x) is stable in the sense of Exercise 8.3.5 and that by symmetry there are few
types over parameter sets contained in ϕ(C) (see Definition 8.2.1).
CHAPTER 6. MORLEY RANK 100

Corollary 6.3.3. The dimension

dim(N/M ) = dimϕ (N/M )

of N over M does not depend on ϕ: it is the maximal length of an elementary

chain
M = N0  N1  · · ·  Nn = N
between M and N .
Proof. This follows from the previous theorem since T has no Vaughtian pairs.

For the remainder of this section, we let M0 denote the prime model of T . We
also fix a strongly minimal formula ϕ(x, a0 ) ∈ L(M0 ) and put p0 (x) = tp(a0 ).
Note that the type p0 (x) of a0 is isolated by Theorem 4.5.2, hence realised
in every model of T . For any model M and realisation a of p0 in M , let
dimϕ(x,a) (M ) denote the ϕ(x, a)-dimension of M over a. To simplify notation
we assume that a0 is some element a0 rather than a tuple.
Since M0 is ω-homogeneous by Corollary 4.5.4, the dimension

m0 = dimϕ(x,a) (M0 )

does not depend on the realisation a of p0 in M0 . We will show in Lemma 6.3.6

that the same is true for any model of T .
Lemma 6.3.4. A countable model M is saturated if and only if its ϕ(x, a)-
dimension is ω. Hence in this case, the dimension is independent of the reali-
sation of p0 (x) in M . In particular, T is ℵ0 -categorical if and only if m0 = ω.
Proof. In a saturated model the ϕ(x, a)-dimension is infinite. Since there exists
a unique countable saturated model by Lemmas 5.2.8 and 5.2.9, the first claim
follows. This obviously does not depend on the realisation of p0 . The last
sentence now follows from Theorem 5.2.11.
We need the following observation.
Lemma 6.3.5. If M is prime over a finite set and m0 < ω, then dimϕ(x,a) (M )
is finite.
Proof. Suppose M is prime over the finite set C. Let B be a basis of ϕ(M, a)
over M0 . Since M is the minimal prime extension of M0 ∪ B, C is atomic over
M0 ∪ B. Thus there exists a finite subset B0 of B such that C is contained in
the prime extension N of M0 ∪ B0 . As M is prime over Ca, it suffices to show
that dimϕ(x,a) (N ) is finite and this follows by Theorem 6.3.1 from

dimϕ(x,a) (N ) = m0 + |B0 | .

The crucial lemma and promised converse to Corollary 5.8.3 is the following.
CHAPTER 6. MORLEY RANK 101

Lemma 6.3.6. The dimension dimϕ(x,a) (M ) does not depend on the realisation
a of p0 in M .
Proof. The lemma is clear if M is uncountable and also if M is countable with
infinite ϕ-dimension by Lemma 6.3.4. Therefore we may assume that M has
finite ϕ-dimension, which implies that m0 is finite.
For the proof we now introduce the following notion: let a1 and a2 realise
p0 . Choose a model N of finite ϕ-dimension containing a1 and a2 , which exists
by Lemma 6.3.5, and put

diff(a1 , a2 ) = dimϕ(x,a1 ) (N ) − dimϕ(x,a2 ) (N ).

This definition does not depend on the model N : if N 0 ≺ N is prime over

a1 , a2 , then by Theorem 6.3.1 we have for i = 1, 2

dimϕ(x,ai ) (N ) = dimϕ(x,ai ) (N 0 ) + dim(N/N 0 ),

and therefore

diff(a1 , a2 ) = dimϕ(x,a1 ) (N 0 ) − dimϕ(x,a2 ) (N 0 ).

Clearly we have

diff(a1 , a3 ) = diff(a1 , a2 ) + diff(a2 , a3 ).

Note that independence of the model implies that diff(a1 , a2 ) only depends on
tp(a1 , a2 ). We will show that diff(a1 , a2 ) = 0 for all realisations of p0 . This
implies that dimϕ(x,a1 ) (M ) = dimϕ(x,a2 ) (M ) for all models M which contain a1
and a2 . For the proof choose an infinite sequence a1 , a2 , . . . with

tp(ai , ai+1 ) = tp(a1 , a2 )

for all i.
Now we use the fact that ℵ1 -categorical theories are ω-stable, so the type p0
has Morley rank and an extension q0 to {a1 , a2 , . . .} of the same rank. Let b be
a realisation of q0 . Then, by Corollary 6.2.13, there are at most MD(p0 ) many
different types of the form tp(bai ). So let i < j be such that tp(bai ) = tp(baj ).
Then diff(ai , b) = − diff(b, aj ) and

(j − i) diff(a1 , a2 ) = diff(ai , aj ) = diff(ai , b) + diff(b, aj ) = 0,

implying diff(a1 , a2 ) = 0.
Thanks to the previous lemma we obtain a complete account of the models of
an uncountably categorical theory.
Theorem 6.3.7 (Baldwin–Lachlan). If T is uncountably categorical, then for
any cardinal m ≥ m0 there is a unique model M of T with dimϕ(x,a) (M ) = m.
These models are pairwise non-isomorphic.
CHAPTER 6. MORLEY RANK 102

Proof. If m = m0 + β, choose M prime over M0 ∪ {bi | i < β} where the

bi ∈ ϕ(C, a0 ) are independent over M0 . Uniqueness follows from Corollary 5.8.3
and non-isomorphism from Lemma 6.3.6.
Exercise 6.3.1. All models of an ℵ1 -categorical theory are ω-homogeneous.
Exercise 6.3.2. Let T be strongly minimal and m0 be the dimension of the
prime model. Show that m0 is the smallest number n such Sn+1 (T ) is infinite.

6.4 Computation of Morley rank

In this section we show that the Morley rank agrees with the dimension of
the pregeometry on strongly minimal sets and give some examples of how to
compute it in ω-stable fields. We start with some general computations and
continue to assume that T is a countable complete theory with infinite models.
Lemma 6.4.1. If b is algebraic over aA, we have MR(b/A) ≤ MR(a/A).
Proof. Let MR(a/A) = α. We prove MR(b/A) ≤ α by induction on α. Let d =
MD(b/Aa). Choose an L(A)-formula ϕ(x, y) in tp(ab/A) such that MR(∃yϕ(x, y)) =
α and |ϕ(a0 , C)| ≤ d for all a0 .
We show that the Morley rank of χ(y) = ∃xϕ(x, y) is bounded by α. For
this consider an infinite family χi (C) of disjoint subclasses of χ(C) defined over
some extension A0 of A. Put ψi (x) = ∃y(ϕ(x, y) ∧ χi (y)). Since any d + 1 of
the ψi have empty intersection, some ψi (x) has Morley rank β < α. Let b0 be
any realisation of χi (y). Choose a0 such that |= ϕ(a0 , b0 ). Then b0 is algebraic
over a0 A and since a0 realises ψi (x), we have MR(a0 /A0 ) ≤ β. So by induction
we conclude MR(b0 /A0 ) ≤ β, which shows MR χi ≤ β. So χ does not contain
an infinite family of disjoint formulas of Morley rank greater or equal to α. So
MR χ ≤ α.
Theorem 6.4.2. Let ϕ(x) be a strongly minimal formula defined over B and
a1 , . . . , an a sequence of realisations. Then

MR(a1 , . . . , an /B) = dimϕ (a1 , . . . , an /B).

Proof. By the lemma we may assume that a1 , . . . , an are independent over B.

Let a1 , . . . , an realise the L(B)-formula ψ(x1 , . . . , xn ). By induction we have
MR(a1 , a2 , . . . , an /Ba1 ) = n − 1. So the formula
.
χa1 (x̄) = ψ(x1 , . . . , xn ) ∧ x1 = a1

has rank at least n−1. The infinitely many conjugates of χa1 over B are disjoint
and have rank n − 1 as well. This shows that MR ψ ≥ n.
Let B 0 ⊃ B be an extension of B. By Corollary 5.7.7 there is only one type
p ∈ Sn (B 0 ) which is realised by an B 0 -independent sequence of elements of ϕ(C).
So by induction, there is only one n-type of elements of ϕ(C) of rank ≥ n. This
implies that ϕ(x1 ) ∧ · · · ∧ ϕ(xn ) has rank n.
CHAPTER 6. MORLEY RANK 103

Corollary 6.4.3. Let ϕ(x) be a strongly minimal formula and ψ(x1 , . . . , xn ) be

defined over B such that ψ implies ϕ(xi ) for all i. Then

MR ψ = max{dimϕ (a/B) | C |= ψ(a)}.

On strongly minimal sets, Morley rank is definable.

Corollary 6.4.4. For any strongly minimal formula ϕ(x) and any formula
ψ(x1 , . . . , xn , y) which implies ϕ(xi ) for all i, we have that

{b | MR ψ(x1 , . . . , xn , b) = k}

is a definable class for every k.

Proof. We show that MR ψ(x1 , . . . , xn , b) ≥ k is an elementary property of b
by induction on n. The case n = 1 follows from the fact that MR ψ(x1 , b) ≥ 1
is equivalent to ∃∞ x1 ψ(x1 , b). This is an elementary property of b since ϕ is
strongly minimal (see the discussion on page 82). For the induction step we
conclude from Corollary 6.4.3 that MR ψ(x1 , . . . , xn , b) ≥ k if and only if one of
the following is true:
• there is an a1 such that MR ψ(a1 , x2 , . . . , xn , b̄) ≥ k,
• there is an a1 which is is not algebraic over b such that
MR ψ(a1 , x2 , . . . , xn , b) ≥ k − 1.
The first part is an elementary property of b by induction. For the second part
note that by induction MR ψ(a1 , x2 , . . . , xn , b) ≥ k − 1 can be expressed by a
formula θ(a1 , b). The second condition is then equivalent to ∃∞ x1 θ(x1 , b).
For algebraically closed fields, these considerations translate into the follow-
ing statement.
Corollary 6.4.5. Let K be a subfield of a model of ACFp and let a be a tuple
of elements. Then the Morley rank of a over K equals the transcendence degree
of K(a) over K.
Note that by quantifier elimination definable sets in algebraically closed fields
are exactly the constructible sets in algebraic geometry. The previous corollary
expresses the important fact that for a definable set in an algebraically closed
field the Morley rank equals the dimension of its Zariski closure in the sense of
algebraic geometry (see e.g., [51]).
We now turn to the theory of differentially closed fields of characteristic 0,
DCF0 .
Let K ⊆ F be an extension of differential fields and a an element of F .
The dimension of a over K is defined as the transcendence degree of K{a} over
K. There is a unique quantifier-free type over K of infinite dimension.
Remark 6.4.6. If the dimension of a over K equals n, then the type of a over
K is determined by the d-minimal polynomial f of a over K: so f is irreducible
in K[x0 , . . . , xn ] and f (a, . . . , dn a) = 0 (see Remark B.3.7).
CHAPTER 6. MORLEY RANK 104

Corollary 6.4.7. DCF0 is ω-stable. For a differential field K and elements a

we have
MR(a/K) ≤ dim(a/K).
If a has infinite dimension, then the type of a over K has Morley rank ω.
Proof. There are at most |K| many d-minimal polynomials over K, so at most
|K| many 1-types. Thus, DCF0 is ω-stable.
We may assume that K is ℵ1 -saturated (otherwise we take a extension of
tp(a/K) to an ℵ1 -saturated field with the same Morley rank. Then the Morley
rank stays the same and the dimension does not increase.) We show MR(a/K) ≤
dim(a/K) by induction on dim(a/K). If dim(a/K) = 0, then a is algebraic
over K and the Morley rank is 0. Let dim(a/K) = n and let f be the minimal
polynomial of a over K. Apart from tp(a/K), all other types over K containing
.
f (x, . . . , dn x) = 0 have dimension, and hence Morley rank, less than n. Since
K is sufficiently saturated, this implies that the Morley rank of tp(a/K) is at
most n.
By the next remark there are types of rank n for every n. This implies that
there must be 1-types of rank ≥ ω. Since there is only one type p∞ of infinite
dimension, it follows4 that p∞ has rank ω.
Lemma 6.4.8. If a, . . . , dn−1 a are algebraically independent over K and
dn a ∈ K, then MR(a/K) = n.
.
Proof. We prove this by induction on n. Consider the formula ϕ(x) = (dn (x) =
n n−1 .
d (a)). If the claim is true for n − 1, all formulas ϕb (x) = (d (x) = b) have
rank n − 1. For all constants c the ϕdn−1 (a)+c (x) are contained in ϕ(x). So ϕ(x)
has rank n. All a0 which realise ϕ(x) have either dimension at most n − 1 or
have the same type as a over K. This shows that a has Morley rank n over
K.
Dimension in pregeometries is additive, i.e., we have dim(ab/B) =
dim(a/B) + dim(b/aB). This translates into additivity of Morley rank for ele-
ments in the algebraic closure of strongly minimal sets.

Proposition 6.4.9. Let ϕ be a strongly minimal formula defined over B and

a, b algebraic over ϕ(C) ∪ B. Then MR(ab/B) = MR(a/B) + MR(b/aB).
Proof. Assume B = ∅ for notational simplicity. Then a is algebraic over some
finite set of elements of ϕ(C). We can split this set into a sequence f¯ of elements
which are independent over a and a tuple a which is algebraic over f¯a. By tak-
ing non-forking extensions if necessary (see Corollary 6.2.13 and the discussion
thereafter), we may assume that f¯ is independent from ab. Now a and a are
interalgebraic over f¯. In the same way we find tuples ḡ and b in ϕ(C) such
4 This is immediate if K is ω-saturated, see Exercise 6.2.1.
CHAPTER 6. MORLEY RANK 105

F = f¯ḡ is independent from ab and b is interalgebraic with b over aF . The

claim now follows from

MR(ab) = MR(ab/F )
= MR(ab/F ) = MR(a/F ) + MR(b̄/aF )
= MR(a/F ) + MR(b/aF ) = MR(a) + MR(b/a).

In fact, Exercise 6.4.6 shows that if F is any infinite B-independent subset

F of ϕ(C) then every element of acl(ϕ(C) ∪ B) is interalgebraic over F B with a
tuple in ϕ(C).
Exercise 6.4.1. Let T be strongly minimal. Show that a finite tuple a is
geometrically independent from B over C (in the sense of a pregeometry, see
page 205 and Exercise C.1.1) if and only if MR(a/BC) = MR(a/C).
Exercise 6.4.2. Let be ψ a formula without parameters. Assume that ψ is
almost strongly minimal , i.e., that there is a strongly minimal formula ϕ defined
over some set B such that all elements of ψ(C) are algebraic over ϕ(C)∪B. Then
for all a, b in ψ(C) and any set C we have MR(ab/C) = MR(a/C) + MR(b/aC).
The following exercise shows that for arbitrary totally transcendental theo-
ries the Morley rank need not be additive.
Exercise 6.4.3. Consider the following theory in a two-sorted language having
sorts A and B and a function f : B → A. Assume that sort A is split into
infinitely many infinite predicates A1 , A2 , . . . such that any a ∈ An has exactly
n preimages under f . Let a be a generic element of A, i.e., an element such
that MR(a) is maximal, and choose f (b) = a. Show that MR(ab) = MR(a) = 2,
MR(b/a) = 1.
Exercise 6.4.4. Let f : B → A be a definable map. Prove the following:
1. If f is surjective, then MR(A) ≤ MR(B).
2. If f has finite fibres, then MR(B) ≤ MR(A).
3. Let A have Morley rank α. Call a fibre f −1 (a) generic if MR(a/C) = α
where C is a set of parameters over which A, B and f are defined. Now
assume that the rank of all fibres is bounded by β > 0 and the rank of the
generic fibres is bounded by βgen . Prove5

MR(B) ≤ β · α + βgen .

(Hint: Use induction on βgen and α. The slightly weaker inequality MR(B) ≤
β · (α + 1) is due to Shelah ([54], Thm. V 7.8) and Erimbetov [18].)
5 We use here ordinal addition and multiplication: α + β is the order type of α followed

by β and β · α is the order type of the lexicographical ordering of α × β.

CHAPTER 6. MORLEY RANK 106

Exercise 6.4.5. A theory has the definable multiplicity property if for all
ϕ(x, y), n and k the class { b | MR ϕ(x, b) = n, MD ϕ(x, b) = k} is defin-
able. Find an example of a theory T which has definable Morley rank but not
the definable multiplicity property.
Exercise 6.4.6. Let ϕ be a strongly minimal formula without parameters and
F an infinite independent subset of ϕ(C). Then every element of acl ϕ(C) is
interalgebraic over F with a tuple in ϕ(C).
Chapter 7

Simple theories

So far, we have mainly studied totally transcendental theories, a small subclass

of the class of stable theories, indeed the most stable ones. Before we turn
to stable theories in general, we consider simple (but possibly unstable) theo-
ries, a generalisation which, after their first introduction by Shelah [56], gained
new attention following the fundamental work of Kim and Pillay [34]. Interest
in simple theories increased with Hrushovski’s results on pseudo-finite fields,
see [29]. The presentation given here owes much to Casanovas, see [14].

7.1 Dividing and forking

We will characterise simple theories by the existence of a well-behaved notion
of independence, a relation on types satisfying certain properties. To this end
we here define forking and dividing. In the context of totally transcendental
theories, these concepts correspond to type extensions of smaller Morley rank.
Throughout this section, we work in a countable complete theory T with infinite
models.
We begin with a reformulation of the Standard Lemma 5.1.3 on indis-
cernibles.

Lemma 7.1.1 (The Standard Lemma). Let A be a set of parameters, I an

infinite sequence of tuples and J a linear order. Then there is a sequence of
indiscernibles over A of order type J realising EM(I/A).
Definition 7.1.2. We say ϕ(x, b) divides over A (with respect to k) if there
is a sequence (bi )i<ω of realisations of tp(b/A) such that (ϕ(x, bi ))i<ω is k-
inconsistent.1 A set of formulas π(x) divides over A if π(x) implies some ϕ(x, b)
which divides over A. There is no harm in allowing ϕ(x, y) to contain parameters
from A.
1 A family (ϕ (x))
i x∈I is k-inconsistent if for every k-element subset K of I the set {ϕi |
i ∈ K} is inconsistent.

107
CHAPTER 7. SIMPLE THEORIES 108

If ϕ(x, a) implies ψ(x, a0 ) and ψ(x, a0 ) divides over A, then ϕ(x, a) divides
over A. Thus ϕ divides over A if and only if {ϕ} divides over A. Also a set
π divides over A if and only if a conjunction of formulas from π divides over
A. Note that it makes sense to say that π(x) divides over A for x an infinite
sequence of variables as we may use dummy variables without changing the
meaning of dividing.
Example. In the theory DLO, the formula b1 < x < b2 divides over the empty
set (for k = 2). The type p = {x > a | a ∈ Q} does not divide over the empty
set for any k.
The following is easy to see.
Remark 7.1.3. 1. If a 6∈ acl(A), then tp(a/Aa) divides over A.
2. If π(x) is consistent and defined over acl(A), then π(x) does not divide
over A.
Lemma 7.1.4. The set π(x, b) divides over A if and only if there
S is a sequence
(bi )i<ω of indiscernibles over A with tp(b0 /A) = tp(b/A) and i<ω π(x, bi ) in-
consistent.
We may replace ω by any infinite linear order. Note also that b may be a
tuple of infinite length.
Proof.
S If (bi )i<ω is a sequence of indiscernibles over A with tp(b0 /A) = tp(b/A)
and i<ω π(x, bi ) inconsistent there is a conjunction ϕ(x, b) of formulas from
π(x, b) for which Σ(x) = {ϕ(x, bi ) | i < ω} is inconsistent. So Σ contains some
k-element inconsistent subset. This implies that (ϕ(x, bi ))i<ω is k-inconsistent.
Assume conversely that π(x, b) divides over A. Then some finite conjunction
ϕ(x, b) of formulas from π(x, b) divides. Let (bi )i<ω be a sequence of realisa-
tions of tp(b/A) such that (ϕ(x, bi ) | i < ω) is k-inconsistent. We
S may assume
by Lemma 7.1.1 that (bi )i<ω is indiscernible over A. Clearly, i<ω π(x, bi ) is
inconsistent.
Corollary 7.1.5. The following are equivalent:
1. tp(a/Ab) does not divide over A.
2. For any infinite sequence of A-indiscernibles I containing b, there exists
some a0 with tp(a0 /Ab) = tp(a/Ab) and such that I is indiscernible over
Aa0 .
3. For any infinite sequence of A-indiscernibles I containing b, there exists
I 0 with tp(I 0 /Ab) = tp(I/Ab) and such that I 0 is indiscernible over Aa.
Proof. 2) ⇔ 3): this is clear by considering appropriate automorphisms. It is
also easy to see that the conclusion of 2) and 3) is equivalent to:
There exist a0 and I 0 with tp(a0 /Ab) = tp(a/Ab) and tp(I 0 /Ab) =
(∗)
tp(I/Ab) such that I 0 is indiscernible over Aa0 .
CHAPTER 7. SIMPLE THEORIES 109

1) ⇒ (∗): Let I = (bi )i∈I be an infinite

S sequence of indiscernibles with bi0 =
b. Let p(x, y) = tp(ab/A). Then i∈I p(x, bi ) is consistent by Lemma 7.1.4.
Let a0 be a realisation. By Lemma 7.1.1, there is I 00 = (b00i )i∈I indiscernible
over Aa0 and realising EM(I/Aa0 ). Since |= p(a0 , b00i0 ), there is an automorphism
α ∈ Aut(C/Aa0 ) taking b00i0 to b. Put I 0 = α(I 00 ).
2) ⇒ 1): Let p(x, y) = tp(ab/A) and let (bi )i<ω be a sequence S of indis-
cernibles over A with tp(b0 /A) = tp(b/A). We have to show that i<ω p(x, bi )
is consistent. By assumption there is a0 with tp(a0 /Ab) = tp(a/Ab) such that I
is indiscernible over Aa0 . As |= p(a0 , b), a0 is a realisation of i<ω p(x, bi ).
S

The next proposition states a transitivity property of dividing. See Corol-

lary 7.2.17, its proof and Exercise 7.2.5.
Proposition 7.1.6. If tp(a/B) does not divide over A ⊆ B and tp(c/Ba) does
not divide over Aa, then tp(ac/B) does not divide over A.
Proof. Let b ∈ B be a finite tuple and I an infinite sequence of A-indiscernibles
containing b. If tp(a/B) does not divide over A, there is some I 0 with tp(I 0 /Ab) =
tp(I, Ab) and indiscernible over Aa. If tp(c/Ba) does not divide over Aa, there
is I 00 with tp(I 00 /Aab) = tp(I 0 /Aab) and indiscernible over Aac proving the
claim.
Definition W7.1.7. The set of formulas π(x) forks over A if π(x) implies a
disjunction `<d ϕ` (x) of formulas ϕ` (x) each dividing over A.
Thus, if π(x) divides over A, it forks over A. The converse need not be true
in general (see Exercise 7.1.6). By definition (and compactness), we immediately
see the following.
Remark 7.1.8 (Non-forking is closed). If p ∈ S(B) forks over A, there is some
ϕ(x) ∈ p such that any type in S(B) containing ϕ(x) forks over A.
Corollary 7.1.9 (Finite character). If p ∈ S(B) forks over A, there is a finite
subset B0 ⊆ B such that p  AB0 forks over A.
Lemma 7.1.10. If π is finitely satisfiable in A, then π does not fork over A.
W
Proof. If π(x) implies the disjunction `<d ϕ` (x, b), then some ϕ` has a realisa-
tion a in A. If the bi , i < ω, realise tp(b/A), then {ϕ` (x, bi ) | i < ω} is realised
by a. So ϕ` does not divide over A.
Lemma 7.1.11. Let A ⊆ B and let π be a partial type over B. If π does not
fork over A, it can be extended to some p ∈ S(B) which does not fork over A.
Proof. Let p(x) be a maximal set of L(B)-formulas containing π(x) which does
not fork over A. Clearly, p is consistent. Let ϕ(x) ∈ L(B). If neither ϕ nor ¬ϕ
belongs to p, then both p ∪ {ϕ} and p ∪ {¬ϕ} fork over A, and hence p forks
over A. Thus p is complete.
Exercise 7.1.1. If I is an infinite sequence of indiscernibles over A, then there
is a model M extending A over which I is still indiscernible.
CHAPTER 7. SIMPLE THEORIES 110

Exercise 7.1.2. 1. Let ϕ(x) be a formula over A with Morley rank and let
ψ(x) define a subclass of ϕ(C). If ψ forks over A, it has smaller Morley
rank than ϕ.
2. Let p be a type with Morley rank and q an extension of p. If q forks over
A, it has smaller Morley rank than p.
We will see in Exercise 8.5.5 that in both statements the converse is also
true.
Exercise 7.1.3. Let p be a type over the model M and A ⊆ M . Assume that
M is |A|+ -saturated. Show that p forks over A if and only if p divides over A.

Exercise 7.1.4. A global type which is A-invariant, i.e., invariant under all
α ∈ Aut(C/A), does not fork over A.
Exercise 7.1.5. Let M be a κ-saturated and strongly κ-homogeneous model.
If p ∈ S(M ) forks over a subset A of cardinality smaller than κ, then p has κ
many conjugates under Aut(M/A).

Exercise 7.1.6. Define the cyclical order on Q by

cyc(a, b, c) ⇔ (a < b < c) ∨ (b < c < a) ∨ (c < a < b).

Show:
1. (Q, cyc) has quantifier elimination.

2. For a 6= b, cyc(a, x, b) divides over the empty set.

3. The unique type over the empty set forks (but of course does not divide)
over the empty set.
Exercise 7.1.7. If tp(a/Ab) does not divide over A and ϕ(x, b) divides over A
with respect to k, then ϕ(x, b) divides over Aa with respect to k.

7.2 Simplicity
In this section, we define simple theories and the notion of forking independence
whose properties characterise such theories. By the absence of binary trees of
consistent formulas, totally transcendental theories are simple. We will see in
the next chapter that in fact all stable theories are simple. Recall that by
our convention (see page 87), variables x and y may belong to different sorts
representing nx and ny -tuples of elements, respectively. We continue to denote
by T a countable complete theory with infinite models.

Definition 7.2.1. 1. A formula ϕ(x, y) has the tree property (with respect
to k) if there is a tree of parameters (as | ∅ 6= s ∈ <ω ω) such that:
a) For all s ∈ <ω ω, (ϕ(x, asi ) | i < ω) is k-inconsistent.
CHAPTER 7. SIMPLE THEORIES 111

b) For all σ ∈ ω ω, {ϕ(x, as ) | ∅ =

6 s ⊆ σ} is consistent.
2. A theory T is simple if there is no formula ϕ(x, y) with the tree property.
Clearly, for the notion of simplicity, it suffices to consider formulas ϕ(x, y) with-
out parameters.
Remark 7.2.2. It is not hard to see that in totally transcendental theories no
formula has the tree property. This is immediate for k = 2. The general case
follows from Exercise 6.2.2.
Definition 7.2.3. Let ∆ be a finite set of formulas ϕ(x, y) without parameters.
A ∆–k-dividing sequence over A is a sequence (ϕi (x, ai ) | i < δ) such that
1. ϕi (x, y) ∈ ∆.,
2. ϕi (x, ai ) divides over A ∪ {aj | j < i} with respect to k.
3. {ϕi (x, ai ) | i < δ} is consistent.
Lemma 7.2.4. 1. If ϕ has the tree property with respect to k, then for every
A and µ there exists a ϕ–k-dividing sequence over A of length µ.
2. If no ϕ ∈ ∆ has the tree property with respect to k, there is no infinite
∆–k-dividing sequence over ∅.
Proof. 1): Note first that we may assume that µ is a limit ordinal. A compact-
ness argument shows that for every µ and κ there is a tree (as | ∅ 6= s ∈ <µ κ)
such that all families (ϕ(x, asi ) | i < κ) are k-inconsistent and for all σ ∈ µ κ,
{ϕ(x, as ) | ∅ =6 s ⊆ σ} is consistent. If κ is bigger than 2max(|T |,|A|,µ) , we recur-
sively construct a path σ such that for all s ∈ σ, infinitely many asi have the
same type over A ∪ {at | t ≤ s}. Now (ϕ(x, aσi+1 ) | i < µ) is a ϕ–k-dividing
sequence over A.
2): Suppose there is an infinite ∆–k-dividing sequence over ∅. If ϕ appears
infinitely many times in this sequence, there is an infinite ϕ–k-dividing sequence
(ϕ(x, ai ) | i < ω). For each i we choose a sequence (ani | n < ω) with tp(ani /{aj |
j < i}) = tp(ai /{aj | j < i}) such that (ϕ(x, ani ) | n < ω) is k-inconsistent.
Then we find parameters bs showing that ϕ has the tree property with respect
to k as follows: assume s ∈ i+1 ω and ~b = (bs1 , . . . , bsi ) have been defined such
that tp(a1 , . . . , ai−1 ) = tp(~b). Choose α ∈ Aut(C) with α(a1 , . . . , ai−1 ) = ~b and
s(i)
put bs = α(ai ).
It is easy to see that in simple theories for every finite set ∆ and all k there
exists a finite bound on the possible lengths of ∆–k-dividing sequences.
Proposition 7.2.5. Let T be a complete theory. The following are equivalent.
a) T is simple.
b) (Local Character) For all p ∈ Sn (B) there is some A ⊆ B with |A| ≤ |T |
such that p does not divide over A.
CHAPTER 7. SIMPLE THEORIES 112

c) There is some κ such that for all models M and p ∈ Sn (M ) there is some
A ⊆ M with |A| ≤ κ such that p does not divide over A.
Proof. a) ⇒ b): If b) does not hold, there is a sequence (ϕi (x, bi ) | i < |T |+ )
of formulas from p(x) such that every ϕi (x, bi ) divides over {bj | j < i} with
respect to ki . There is an infinite subsequence for which all ϕi (x, y) equal ϕ(x, y)
and all ki = k yielding a ϕ–k-dividing sequence.
b) ⇒ c): Clear.
c) ⇒ a): If ϕ has the tree property, there are ϕ–k-dividing sequences
(ϕ(x, bi ) | i < κ+ ). It is easy to construct an ascending sequence of models
Mi , (i < κ+ ) such that bj ∈ Mi for j < i and ϕ(x, bi ) dividesS over Mi . Extend
the set of ϕ(x, bi ) to some type p(x) ∈ S(M ) where M = i<κ+ Mi . Then p
divides over each Mi .
Corollary 7.2.6. Let T be simple and p ∈ S(A). Then p does not fork over A.
W
Proof. Suppose p forks over A, so p implies some disjunction l<d ϕl (x, b) of
formulas all of which divide over A with respect to k. Put ∆ = {ϕl (x, y) | l < d}.
We will show by induction that for all n there is a ∆–k-dividing sequence
over A of length n. This contradicts the remark after Lemma 7.2.4. We will
assume also that the dividing sequence is consistent with p(x).
Suppose that (ψi (x, ai ) | i < n) is a ∆–k-dividing sequence over A, consis-
tent with p(x). By Exercise 7.2.4 we can replace b with a conjugate b0 over A
such that (ψi (x, ai ) | i < n) is a dividing sequence over Ab0 . Now one of the
formulas ϕl (x, b0 ), say ϕ0 (x, b0 ), is consistent with p(x) ∪ {ψi (x, ai ) | i < n}.
So ϕ0 (x, b0 ), ψ0 (x, a0 ), . . . , ψn−1 (x, an−1 ) is a ∆–k-dividing sequence over A and
consistent with p(x).
Let p be a type over A and q an extension of p. We call p a forking extension
if q forks over A.
Corollary 7.2.7 (Existence). If T is simple, every type over A has a non-
forking extension to any B containing A.
Proof. This follows from Corollary 7.2.6 and Lemma 7.1.11.
Definition 7.2.8. The set A is independent from B over C, written

| B,
A^
C

if for every finite tuple a from A, the type tp(a/BC) does not fork over C.
If C is empty, we may omit it and write A ^ | B.
This definition makes sense since forking of tp(a/BC) does not depend on the
enumeration of a and since tp(a/BC) forks over C if the type of a subsequence
of a forks over C. So this is the same as saying that tp(A/BC) does not fork
over C.
Definition 7.2.9. Let I be a linear order. A sequence (ai )i∈I is called
CHAPTER 7. SIMPLE THEORIES 113

| A {aj | j < i} for all i;

1. independent over A if ai ^
2. a Morley sequence over A if it is independent and indiscernible over A;
3. a Morley sequence in p(x) over A if it is a Morley sequence over A con-
sisting of realisations of p.
Example 7.2.10. Let q be a global type invariant over A. Then any sequence
(bi )i∈I where each bi realises q  A ∪ {bj | j < i} is a Morley sequence over A.
Proof. Let us call such sequences good. Clearly a subsequence of a good se-
quence is good again. So for indiscernibility it suffices to show that all finite
good sequences b0 . . . bn and b00 . . . b0n have the same type over A. Indeed, using
induction, we may assume that b0 . . . bn−1 and b00 . . . b0n−1 have the same type
and so α(b0 . . . bn−1 ) = b00 . . . b0n−1 for some α ∈ Aut(C/A). Then
α(tp(bn /Ab0 . . . bn−1 )) = α(q  Ab0 . . . bn−1 ) = q  Ab00 . . . b0n−1
= tp(b0n /Ab00 . . . b0n−1 ),
which proves our claim. Independence follows from Exercise 7.1.4.
We call such a sequence (bi )i∈I a Morley sequence of q over A. Note that
our proof shows that the type of a Morley sequence of q over A is uniquely
determined by its order type.
Lemma 7.2.11. If (ai )i∈I is independent over A and J < K are subsets of I,
then tp((ak )k∈K /A{aj | j ∈ J}) does not divide over A.
Proof. We may assume that K is finite. The claim now follows from Proposi-
tion 7.1.6 by induction on |K|.
Lemma 7.2.12 (Shelah). For all A there is some λ such that for any linear
order I of cardinality λ and any family (ai )i∈I there exists an A-indiscernible
sequence (bj )j<ω such that for all j1 < · · · < jn < ω there is a sequence i1 <
· · · < in in I with tp(ai1 . . . ain /A) = tp(bj1 . . . bjn /A).
Proof. We only need that λ satisfies the following. Let τ = supn<ω | Sn (A)|.
1. cf(λ) > τ
2. For all κ < λ and all n < ω there is some κ0 < λ with κ0 → (κ)nτ (see
Definition C.3.1).
By Erdős–Rado (see Theorem C.3.2) we may take λ = iτ + .
We now construct a sequence of types p1 (x1 ) ⊆ p2 (x1 , x2 ) ⊆ · · · with pn ∈
Sn (A) such that for all κ < λ there is some I 0 ⊆ I with |I 0 | = κ such that
tp(ai1 . . . ain ) = pn for all i1 < · · · < in from I 0 . S
Then we can choose the (bi )i<ω as a realisation of i<ω pi .
If pn−1 has been constructed and we are given κ < λ, we choose κ0 < λ
with κ0 → (κ)nτ and some I 0 ⊆ I with |I 0 | = κ0 such that tp(ai1 . . . ain−1 /A) =
pn−1 for all i1 < · · · < in−1 from I 0 . Thus there are I 00 ⊆ I 0 and pκn with
tp(ai1 . . . ain ) = pκn for all i1 < · · · < in from I 00 . Since cf(λ) > τ , there is some
pn with pκn = pn for cofinally many κ.
CHAPTER 7. SIMPLE THEORIES 114

The existence of a Ramsey cardinal κ > τ (see p. 208) would directly imply
that any sequence of order type κ contains a countable indiscernible subsequence
(in fact even an indiscernible subsequence of size κ).
Lemma 7.2.13. If p ∈ S(B) does not fork over A, there is an infinite Morley
sequence in p over A which is indiscernible over B. In particular, if T is simple,
for every p ∈ S(A), there is an infinite Morley sequence in p over A.
Proof. Let a0 be a realisation of p. By Lemma 7.1.11 there is a non-forking
extension p0 of p to Ba0 . Let a1 be a realisation of p0 . Continuing in this
way we obtain a sequence (ai )i<λ with ai ^ | A B(aj )j<i for arbitrary λ. By
Lemma 7.2.12 we obtain a sequence of length ω with the same property and
indiscernible over B. The last sentence is immediate by Corollary 7.2.6.
Proposition 7.2.14. Let T be simple and π(x, y) beSa partial type over A.
Let (bi )i<ω be an infinite Morley sequence over A and i<ω π(x, bi ) consistent.
Then π(x, b0 ) does not divide over A.
Proof. By Lemma 7.1.1, for every linear order S I there is a Morley sequence
(bi )i∈I in tp(b0 /A) over A such that Σ(x) = i∈I π(x, bi ) is consistent. Choose
I having the inverse order type of |T |+ . Let c be a realisation of Σ. By Proposi-
tion 7.2.5(b) there is some i0 such that tp(c/A∪{bi | i ∈ I}) does not divide over
A ∪ {bi | i > i0 }. This implies that tp(c/A ∪ {bi | i ≥ i0 }) does not divide over
A ∪ {bi | i > i0 }. By Lemma 7.2.11, tp((bi | i > i0 )/Abi0 ) does not divide over
A. Hence tp(c (bi | i > i0 )/Abi0 ) does not divide over A by Proposition 7.1.6.
This implies that π(x, bi0 ) does not divide over A.
Proposition 7.2.15. Let T be simple. Then π(x, b) divides over A if and only
if it forks over A.
Proof. By definition, if π(x, b) divides over A, it forksWover A. For the converse
assume π(x, b) does not divide over A. So if ψ(x, b) = l<d ϕl (x, b) is implied by
π(x, b), it does not divide over A. Let (bi )i<ω be a Morley sequence in tp(b/A)
over A, which exists since T is simple. So {ψ(x, bi ) | i ∈ ω} is consistent. By
the pigeon-hole principle there must be some l and some infinite I ⊆ ω such
that {ϕl (x, bi ) | i ∈ I} is consistent. By Proposition 7.2.14, ϕl (x, b) does not
divide over A. Hence π(x, b) does not fork over A.

Proposition 7.2.16 (Symmetry). In simple theories, independence is symmet-

ric.
Proof. Assume A ^ | C B and consider finite tuples a ∈ A and b ∈ B. Since
a^ | C b, Lemma 7.2.13 gives an infinite Morley sequence (ai )i<ω S in tp(a/Cb)
over C, indiscernible over Cb. Let p(x, y) = tp(ab/C). Then i<ω p(ai , y) is
consistent because it is realised by b. Thus, by Proposition 7.2.14, p(a, y) does
not divide over C. This proves b ^ | C a. Since this holds for all a ∈ A, b ∈ B, it
follows B ^ | C A by Finite Character.
CHAPTER 7. SIMPLE THEORIES 115

Corollary 7.2.17 (Monotonicity and Transitivity). Let T be simple, B ⊆ C ⊆

D. Then we have A ^| B D if and only if A ^
| B C and A ^| C D.
Proof. One direction of this equivalence, Monotonicity, holds for arbitrary the-
ories and follows easily from the definition. For Transitivity, the other direction,
note that by Proposition 7.2.15 we may read Proposition 7.1.6 after replacing
finite tuples by infinite ones as
A0 ^
| B and C ^
| B ⇒ CA0 ^
| B.
A 0 AA A

Swapping the left and the right hand sides, this is exactly the transitivity. Hence
the claim follows from Proposition 7.2.16.
Corollary 7.2.18. That (ai )i∈I is independent over A does not depend on the
ordering of I.
Proof. Let i be an element of I and J, K two subsets such that J < i < K.
Write aJ = {aj | j ∈ J} and aK = {ak | k ∈ K}. We have to show that
| A aJ aK . Now by Lemma 7.2.11 we have aK ^
ai ^ | A aJ ai . Monotonicity yields
aK ^| Aa ai and by Symmetry we have ai ^ | Aa aK . The claim follows now
J J
from ai ^| A aJ and Transitivity.
So we can define a family (ai | i ∈ I) to be independent over A if it is
independent for some ordering of I. Clearly (ai )i∈I is independent over A if
| A {aj | j 6= i} for all i. One calls a set B independent over A if
and only if ai ^
b^| A (B \ {b}) for all b ∈ B.
The following lemma is a generalisation of Proposition 7.2.14.
Lemma 7.2.19. Let T be simple and I be an infinite Morley sequence over A.
If I is indiscernible over Ac, then c ^
| A I.
Proof. We may assume I = (ai )i<ω . Consider any ϕ(x, a0 , . . . , an−1 )
∈ tp(c/AI). Put bi = (ani , . . . , ani+n−1 ). Then by Lemma 7.2.11 (bi )i<ω is
again a Morley sequence over A and {ϕ(x, bi ) | i ∈ ω} is consistent since re-
alised by c. We see from Proposition 7.2.14 that ϕ(x, a0 , . . . , an−1 ) does not
fork over A.
Exercise 7.2.1. If T is simple, there does not exist an ascending chain
(pα )α∈|T |+ of forking extensions. Hence, there do not exist an A-independent
6 | A c for all α.
sequence (bα )α∈|T |+ and a finite tuple c such that bα ^
Exercise 7.2.2 (Diamond Lemma). Assume T to be simple and p ∈ S(A). Let
q be a non-forking extension of p and r any extension of p. Then there is an
A-conjugate r0 of r with a non-forking extension s which also extends q.
s
 S
 S nf
 S
q r0
S 
nfS 
S 
p
CHAPTER 7. SIMPLE THEORIES 116

We can choose r0 in such a way that the domains of r0 and q are independent
over A.
Exercise 7.2.3. If T is simple and (ai )i∈I is an A-independent sequence, then

aX |
^ aY
AaX∩Y

for all X, Y ⊆ I where aX = {ai | i ∈ X}.

Exercise 7.2.4. If ϕ(x, b) divides over A and A ⊆ B, there is some A-conjugate
B 0 of B such that ϕ(x, b) divides over B 0 .
Exercise 7.2.5. If T is simple, then

| B ⇐⇒ a ^
ab ^ | B and b ^
| B.
A A Aa

Exercise 7.2.6. Assume that T simple and b1 . . . bn ^ | A C. Then the sequence

b1 , . . . , bn is independent over A if and only if it is independent over AC.

| B C, then a ∈ acl(ABC) implies

Exercise 7.2.7. If T is simple and Aa ^
a ∈ acl(AB).
Exercise 7.2.8. Use Proposition 7.1.6 to show that T is simple if no formula
ϕ(x, y) for a single variable x has the tree property or, equivalently, if every
1-type does not divide over a set of cardinality at most |T |.

7.3 The independence theorem

The core of this section is the characterisation of simple theories in terms of
a suitable notion of independence. This is due to Kim and Pillay and will
be applied to pseudo-finite fields in Section 7.5. We will later specialise this
characterisation to stable theories. Unless explicitly stated otherwise, we assume
throughout this section that T is a simple theory.
Definition 7.3.1. For any set A we write ncA (a, b) if a and b start an infinite
sequence of indiscernibles over A.

A formula θ(x, y) is called thick if there are no infinite antichains, i.e., se-
quences (ci )i<ω where ¬θ(ci , cj ) for all i < j < ω. By compactness this says
that there is a bound k < ω on the length of finite antichains. See Exercise 7.3.2
for an explanation of the terminology.
Lemma 7.3.2. (T arbitrary.) For any set A and n-tuples a, b the following are
equivalent:
a) ncA (a, b).
b) |= θ(a, b) for all thick θ(x, y) defined over A.
CHAPTER 7. SIMPLE THEORIES 117

In particular, ncA is type-definable.

Proof. Let p(x, y) = tp(ab/A). By 5.1.3, a and b start an infinite sequence of
indiscernibles if and only if there is a sequence (ci )i<ω with |= p(ci , cj ) for i < j
if and only if for all ϕ ∈ p the complement of ϕ(C) contains arbitrarily long
antichains, and so
6|= ψ(a, b) ⇒ ψ is not thick.

Corollary 7.3.3. ( T arbitrary.) If a and b have the same type over a model
M , there is some c such that ncM (a, c) and ncM (c, b).
Proof. We have to show |= ∃z(ϕ(a, z)∧ϕ(z, b)) for every thick formula ϕ(x, y) ∈
L(M ). We may assume that ϕ is symmetric.2 Since M is a model, there is a
maximal antichain a0 , . . . , ak−1 of ϕ in M . Thus for some i we have |= ϕ(a, ai )
and hence |= ϕ(b, ai ).
In Exercise 8.1.2 below we give a different proof of the corollary, independent
of Lemma 7.3.2.
Lemma 7.3.4. ( T arbitrary.) Let (bi )i<ω be indiscernible over A and (bi )1≤i<ω
indiscernible over Aa0 b0 . Then there is some a1 such that ncA (a0 b0 , a1 b1 ).
Proof. Choose ai with tp(ai bi bi+1 . . . /A) = tp(a0 b0 b1 . . . /A). Let a00 b0 ,
a01 b1 , . . . be a sequence of indiscernibles realising the EM-type of a0 b0 , a1 b1 , . . .
over A. Since (bi )1≤i<ω is indiscernible over Aa0 b0 , we have
tp(a0i1 , bi1 , bi2 , . . . , bin /A) = tp(a0 , b0 , b1 , . . . , bn /A)
for all i1 < · · · < in . So tp(a00 , b0 , b1 , . . . /A) = tp(a0 , b0 , b1 , . . . /A) and we may
assume that a0 b0 , a1 b1 , . . . is indiscernible over A.
Lemma 7.3.5. Let I be indiscernible over A and J an infinite initial segment
without last element. Then I \ J is a Morley sequence over AJ .
Proof. Let I = (ai )i∈I and J = (ai )i∈J . It suffices to show ai ^
| AJ aX for all
i ∈ I \ J and all finite X ⊆ I with X < i. But this follows from Lemma 7.1.10
as tp(aX /AJ ai ) is finitely satisfiable in AJ .
Proposition 7.3.6. If ϕ(x, a) does not fork over A and ncA (a, b), then ϕ(x, a)∧
ϕ(x, b) does not fork over A.
Proof. Let I be an infinite sequence of indiscernibles over A containing a and b.
We extend I by an infinite initial segment J without last element. Let c be a
realisation of ϕ(x, a) independent from J a over A. By Corollary 7.1.5 we may
assume I to be indiscernible over AJ c.
It follows from Lemma 7.3.5 that I is a Morley sequence over AJ . So by
Lemma 7.2.19 we have c ^ | AJ I. Transitivity now implies c ^| A J I and hence
the claim.
2 That is, |= ∀x, y (θ(x, y) → θ(y, x)).
CHAPTER 7. SIMPLE THEORIES 118

Lemma 7.3.7. Let ncA (b, b0 ) and a ^

| Ab b0 . Then there is some a0 with
ncA (ab, a0 b0 ).
Proof. Let (bi )i<ω indiscernible over A, b = b0 and b0 = b1 . By Corollary 7.1.5
we may assume (bi )1≤i<ω to be indiscernible over Aab. The claim now follows
from Lemma 7.3.4.
Proposition 7.3.8. If ϕ(x, a) ∧ ψ(x, b) does not fork over A, ncA (b, b0 ) and
a^| Ab b0 , then neither does ϕ(x, a) ∧ ψ(x, b0 ) fork over A.
Proof. By Lemma 7.3.7 there is some a0 such that ncA (ab, a0 b0 ). Proposi-
tion 7.3.6 implies that ϕ(x, a) ∧ ψ(x, b) ∧ ϕ(x, a0 ) ∧ ψ(x, b0 ) does not fork over
A.
| A b0 b
Corollary 7.3.9. Assume that ϕ(x, a) ∧ ψ(x, b) does not fork over A, a ^
0
and that b and b have the same type over some model containing A. Then
ϕ(x, a) ∧ ψ(x, b0 ) does not fork over A.
Proof. By Corollary 7.3.3 there is some c such that ncA (b, c) and ncA (c, b0 ). By
replacing a, if necessary, by a realisation of a non-forking extension of tp(a/Abb0 )
to Abb0 c, which exists by Corollary 7.2.7, we may assume that a ^ | A bb0 c. Propo-
sition 7.3.8 yields now first that ϕ(x, a) ∧ ψ(x, c) does not fork over A and then
that ϕ(x, a) ∧ ψ(x, b0 ) does not fork over A.
Corollary 7.3.10. Let a ^| M b, a0 ^
| M a, b0 ^
| M b, |= ϕ(a0 , a) ∧ ψ(b0 , b) and
0 0
assume that a and b have the same type over M . Then ϕ(x, a) ∧ ψ(x, b) does
not fork over M .
Proof. Choose a00 such that tp(a00 a0 /M ) = tp(bb0 /M ) and a00 ^
| M a0 abb0 . Then
by Transitivity we have
a00 ^ | aa0 bb0 .
M
It follows that the sequences a, a , a and a, b, a00 are both independent over A.
0 00

This implies
a0 ^
| aa00 (7.1)
M
and
| a00 b
a^ (7.2)
M
Since |= ψ(a0 , a00 ), (7.1) implies that ϕ(x, a) ∧ ψ(x, a00 ) does not fork over M .
So ϕ(x, a) ∧ ψ(x, b) does not fork over M by (7.2) and Corollary 7.3.9.
Theorem 7.3.11 (Independence Theorem). Suppose that b and c have the same
type over the model M and suppose that
| C, b ^
B^ | B and c ^
| C.
M M M

Then there exists some d with tp(d/B) = tp(b/B) and tp(d/C) = tp(c/C) and
such that
d^| BC.
M
CHAPTER 7. SIMPLE THEORIES 119

Proof. By Corollary 7.3.10, tp(b/B) ∪ tp(c/C) does not fork over M . So we find
some d such that d ^| M BC, tp(d/B) = tp(b/B) and tp(d/C) = tp(c/C).
Corollary 7.3.12. Let Bi , i ∈ I, be independent over M and let bi be such
| M Bi all bi having the same type over M . Then there is some d with
that bi ^
tp(d/Bi ) = tp(bi /Bi ) for all i and

| {Bi | i ∈ I}.
d^
M

Proof. Well-order I and show the existence of d by recursively constructing

pi = tp(d/{Bi | i ∈ I}). The details are left as an exercise.

Theorem 7.3.13 (Kim–Pillay [34]). Let T be a complete theory and a ^ |0 A B a

relation between finite tuples a and sets A and B invariant under automorphisms
and having the following properties:
|0 A BC if and only if a ^
a) (Monotonicity and Transitivity) a ^ |0 A B and
0
a^| AB C.

|0 A b3 if and only if b ^
b) (Symmetry) a ^ |0 A a.

|0 A B if a ^
c) (Finite Character) a ^ |0 A b for all finite tuples b ∈ B.
d) (Local Character) There is a cardinal κ, such that for all a and B there
|0 B B
exists B0 ⊆ B of cardinality less than κ such that a ^
0

e) (Existence) For all a, A and C there is a such that tp(a0 /A) = tp(a/A)
0

and a0 ^
|0 A C.
f ) (Independence over Models) Let M be a model, tp(a0 /M ) = tp(b0 /M )
and
a^|0 b, a0 ^
|0 a, b0 ^
|0 b.
M M M

Then there is some c such that tp(c/M a) = tp(a0 /M a), tp(c/M b) = tp(b0 /M b)
|0 M ab.
and c ^

|0 = ^
Then T is simple and ^ | .
| satisfies the properties
We have seen that in simple theories the relation ^
of the previous theorem (for κ = |T |+ ).

Proof. We may assume κ to be a regular cardinal, otherwise just replace κ by

κ+ . Assume now a ^ |0 A b. We will use Lemma 7.1.4 to show that tp(a/Ab) does
not divide over A. So, let (bi )i<ω be a sequence of A-indiscernibles starting with
b = b0 .
Claim. We can find a model M containing A such that (bi )i<ω is indis-
cernible over M and bi ^ |0 M {bj | j < i} for all i.
3 We here consider a tuple as a finite set.
CHAPTER 7. SIMPLE THEORIES 120

Proof of Claim. By Lemma 7.1.1 we can extend the sequence to (bi )i≤κ .
Furthermore, it is easy to construct an ascending sequence of models A ⊆
M0 ⊆ M1 ⊆ · · · such that for all i < κ all bj , (j < i) is contained in Mi
and (bj )i≤j≤κ is indiscernible over Mi . By Local Character there is some
i0 such that bκ ^|0 M {bj | i0 ≤ j < κ}. From the Indiscernibility it now follows
i0

that bi ^ |0 M {bj | i0 ≤ j < i} for all i. We can take M = Mi0 and the sequence
i0
bi0 , bi0 +1 , . . ..
Claim. We may assume a ^ |0 M b.
Proof of Claim. By Existence we may replace a by a0 with tp(a0 /Ab) =
tp(a/Ab) and a0 ^ |0 Ab M and then apply Monotonicity and Transitivity.

We now find elements a = a0 , a1 , . . . such that ai ^ |0 M {bj | j ≤ i}, qi (x) =

tp(ai+1 /M {bj | j ≤ i}) = tp(ai /M {bj | j ≤ i}) and tp(ai bi /M ) = tp(ab/M ):
given a0 , . . . , ai , choose a0 with tp(a0 bi+1 /M ) = tp(ab/M ). Now apply Inde-
pendence over Models to

{bj | j ≤ i} ^ |0 {bj | j ≤ i} and a0 ^

|0 bi+1 , ai ^ |0 bi+1
M M M

to find ai+1 . S
This implies that i<ω qi (x) is consistent and contains all p(x, bi ) where
p(x, y) = tp(ab/M ). By Lemma 7.1.4, tp(a/Ab) does not divide over A. Sim-
plicity of T now follows from Local Character and Proposition 7.2.5.
It remains to show a ^ |0 A b if tp(a/Ab) does not divide over A. Using Ex-
istence we construct for any λ a sequence (bi )i<λ which is ^ |0 A -independent
and for which tp(bi /A) = tp(b/A). If this sequence is sufficiently long, the
same argument used for Lemma 7.2.13 (but now using Monotonicity and
Finite Character) yields an A-indiscernible sequence (b0i )i<κ which is ^ |0 A -
0 0
independent as well and satisfies tp(bi /A) = tp(b/A) and b = b0 . We now apply
Corollary 7.1.5 and obtain a0 such that tp(a0 /Ab) = tp(a/Ab) and so that (b0i )i<κ
is indiscernible over Aa0 . Local Character and Monotonicity yield the
existence of i0 with
a0 ^ |0 b0i0 .
A{b0i |i<i0 }

Since
b0i0 ^
|0 {b0i | i < i0 }
A
we get
a0 ^
|0 b0i0 and hence a ^
|0 b
A A

from Symmetry and Transitivity using that tp(a0 b0i0 /A) = tp(a0 b00 /A) =
tp(ab/A).
Corollary 7.3.14. The theory of the random graph is simple.
|0 B C by A ∩ C ⊆ B and apply Theorem 7.3.13,
Proof. Define A ^
CHAPTER 7. SIMPLE THEORIES 121

Exercise 7.3.1. Let T be simple. Assume that the partial type π(x, b) does
not fork over A and that I is an infinite sequence of indiscernibles over A
| A I and
containing b. Show that there is a realisation c of π(x, b) such that c ^
I is indiscernible over Ac.
Exercise 7.3.2. A symmetric formula is thick if and only if there is no infinite
anti-clique, i.e., a sequence (ci )i<ω , where |= ¬θ(x, y) for all i 6= j. This explains
the notation ncA . Prove the following, without using Lemma 7.3.2:
1. The conjunction of two thick formulas is thick.
2. If θ(x, y) is thick, then θ∼ (x, y) = θ(y, x) is thick.
3. A formula is thick if and only if it is implied by a symmetric thick formula.
Exercise 7.3.3. Let T be a simple theory and A a set of parameters. Assume
that there is an element b which is algebraic over A but not definable over A.
Then the Independence Theorem does not hold if in its formulation the model
M is replaced by A.
Exercise 7.3.4. Fill in the details of the proof of Corollary 7.3.12.
Exercise 7.3.5. Prove directly from the axioms in Theorem 7.3.13 that

|0 B ⇔ a ^
a^ |0 AB.
A A

Exercise 7.3.6. Let T be simple and p be a type over a model. Then p has
either exactly one non-forking extension to C (p is stationary) or arbitrarily
many.

7.4 Lascar strong types

In this section we will prove a version of the Independence Theorem 7.3.11 over
arbitrary parameter sets A. For this we have to strengthen the assumption that
b and c have the same type over A to having the same Lascar strong type over
A. In what follows T is an arbitrary complete theory.
Definition 7.4.1. Let A be any set of parameters. The group Autf (C/A)
of Lascar strong automorphisms of C over A is the group generated by all
Aut(C/M ) where the M are models containing A. Two tuples a and b have
the same Lascar strong type over A if α(a) = b for some α ∈ Autf (C/A). We
denote this by Lstp(a/A) = Lstp(b/A).
It is easy to see that tuples a and b have the same Lascar strong type over
A if and only if there is a sequence a = b0 , b1 , . . . , bn = b such that for all for all
i < n, bi and bi+1 have the same type over some model containing A.
Lemma 7.4.2. Assume that T is simple. If ϕ(x, a) ∧ ψ(x, b) does not fork over
A, and if Lstp(b/A) = Lstp(b0 /A) and a ^
| A b0 b, then ϕ(x, a) ∧ ψ(x, b0 ) does not
fork over A.
CHAPTER 7. SIMPLE THEORIES 122

Proof. Choose a sequence b = b0 , b1 , . . . , bn = b0 such that for each i < n, bi and

bi+1 have the same type over some model containing A. By the properties of
forking we may assume that a ^| A b0 b1 . . . bn . We thus always have a ^ | A bi bi+1
and the claim follows by induction from Corollary 7.3.9.
As in the previous proof we will repeatedly use Existence (Corollary 7.2.7)
to assume that we have realisations of types which are independent from other
sets. This is also crucial in the following lemma.
Lemma 7.4.3. Assume T to be simple. For all a, A and B there is some a0
such that Lstp(a0 /A) = Lstp(a/A) and a0 ^
| A B.
Proof. By Existence and Symmetry, we can choose a model M ⊃ A with
a^| A M . By Existence again we can find a0 such that tp(a0 /M ) = tp(a/M )
and a0 ^
| M B, so the claim now follows from Transitivity.
A stronger statement will be proved in Exercise 7.4.4.
Corollary 7.4.4. Let Lstp(a/A) = Lstp(b/A). For all a0 , B there exists b0 such
that Lstp(aa0 /A) = Lstp(bb0 /A) and b0 ^
| Ab B.
Proof. Choose bb00 such that Lstp(aa0 /A) = Lstp(bb00 /A) and by the Lemma
there is some b0 such that b0 ^| Ab B and Lstp(b00 /Ab) = Lstp(b0 /Ab). It is easy
to see that this implies Lstp(bb00 /A) = Lstp(bb0 /A).
Corollary 7.4.5. Let a ^ | A b, a0 ^
| A a, b0 ^
| A b, Lstp(a0 /A) = Lstp(b0 /
0 0
A) and |= ϕ(a , a) ∧ ψ(b , b). Then ϕ(x, a) ∧ ψ(x, b) does not fork over A.
Proof. Choose a00 by Corollary 7.4.4 such that Lstp(a00 a0 /A) = Lstp(bb0 /
A) and a00 ^
| Aa0 abb0 . Then proceed as in the proof of Corollary 7.3.10, but
use Lemma 7.4.2 instead of Corollary 7.3.9.
This yields the following generalisation of Theorem 7.3.11.
Theorem 7.4.6 (Independence Theorem). Let T be simple and suppose
Lstp(b/A) = Lstp(c/A),

| C, b ^
B^ | B and c ^
| C.
A A A

| A BC, Lstp(d/B) = Lstp(b/B) and

Then there exists some d such that d ^
Lstp(d/C) = Lstp(c/C).
Proof. By Corollary 7.4.5, tp(b/B) ∪ tp(c/C) does not fork over A. So we find
some d such that d ^| A BC, tp(d/B) = tp(b/B) and tp(d/C) = tp(c/C). The
stronger claim about Lascar strong types is left as Exercise 7.4.3.
Corollary 7.4.7. Assume T to be simple and let Bi , i ∈ I, be independent
| A Bi with all bi having the same Lascar strong type
over A and bi such that bi ^
over A. Then there is some d such that d ^ | A {Bi | i ∈ I} and Lstp(d/Bi ) =
Lstp(bi /Bi ) for all i.
CHAPTER 7. SIMPLE THEORIES 123

Proof. Assume the Bi are models containing A. The proof goes then as the
proof of Corollary 7.3.12.
Lemma 7.4.8. Let T be simple, a ^ | A b and Lstp(a/A) = Lstp(b/A), there is
an infinite Morley sequence over A containing a and b.
Proof. Consider p(x, y) = tp(ab/A). Starting from a0 = a and a1 = b we
recursively construct a long independent sequence (ai ) of elements all having
the same Lascar type over A. If (ai | i < α) is given, the p(ai , y) are realised by
| A ai and Lstp(bi /A) = Lstp(a/A). By Corollary 7.4.7 there
elements bi with bi ^
is some aα with aα ^ | A {ai | i < α}, |= p(ai , aα ) for all i < α and Lstp(aα /A) =
Lstp(a/A). If the sequence is sufficiently long, then by Lemma 7.2.12 there
is an A-indiscernible sequence (a0i )i<ω such that |= p(a0i , a0j ) for all i < j and
furthermore the sequence is independent over A because all types (a0j /A{a0i |
i < j}) appear in (ai ). Since tp(a01 a00 /A) = tp(ba/A), we may assume a00 = a
and a01 = b.
Corollary 7.4.9. We have Lstp(a/A) = Lstp(b/A) if and only if nc2A (a, b). In
particular, the equivalence relation ELA (a, b) defined as Lstp(a/A) = Lstp(b/A)
is type-definable.
| A ab and Lstp(c/A) = Lstp(a/A) = Lstp(b/A). By
Proof. Choose c with c ^
Lemma 7.4.8, we have ncA (c, a) and ncA (c, b). Hence

ELA (x, y) ⇔ ∃z (ncA (x, z) ∧ ncA (z, y))

and this is type-definable by Lemma 7.3.2.

It is an open problem whether in simple theories Lascar strong types are the
same as strong types (see Exercise 8.4.9).
Exercise 7.4.1. Show that an automorphism of C is Lascar strong if and only
if it preserves the Lascar strong type of any tuple of length |T |.
Exercise 7.4.2. Show that ncA (a, b) implies ncacl(A) (a, b).
Exercise 7.4.3. Deduce Theorem 7.4.6 from the weaker version which claims
only the equalities tp(d/B) = tp(b/B) and tp(d/C) = tp(c/C).
Exercise 7.4.4. Show that in arbitrary theories ncA (x, a) does not divide over
A. Use this to prove a stronger version of Lemma 7.4.3: Assume that T is
simple. For all a, A, B there is some a0 such that ncA (a, a0 ) and a0 ^
| A B.
Exercise 7.4.5. If ncA (a, b), there is some model M containing A such that
tp(a/M ) = tp(b/M ). Conclude that ELA is the transitive closure of ncA .
Exercise 7.4.6. A relation R ⊆ Cn × Cn is called bounded if there are no
arbitrarily long antichains, i.e., sequences (cα | α < κ) with ¬R(cα , cβ ) for all
α < β < κ. Show that the intersection of a family Ri , (i ∈ I), of bounded
relations is again bounded.
CHAPTER 7. SIMPLE THEORIES 124

Exercise 7.4.7. We call a relation A-invariant if it is invariant under all au-

tomorphisms in Aut(C/A). Show that ncA is the smallest bounded A-invariant
relation.
Exercise 7.4.8. Show that ELA is the smallest bounded A-invariant equivalence
relation.

7.5 Example: pseudo-finite fields

We now turn to an important example of simple theories, namely those of
pseudo-finite fields. A perfect field K is called pseudo-finite if it is pseudo-
algebraically closed, i.e., if every absolutely irreducible affine variety defined
over K has a K-rational point and if its absolute Galois group is Z, b i.e., K has
a unique extension of degree n for each n ≥ 1. Equivalently, a field is pseudo-
finite if it is elementarily equivalent to an infinite ultraproduct of finite fields,
see Exercise 7.5.2 or [1], Theorem 8. For background on pseudo-finite fields and
profinite groups see Section B.4.
Proposition 7.5.1. Let L1 and L2 be regular procyclic extensions of a field K
and let L2 be pseudo-finite. Then L1 can be regularly embedded over K into an
elementary extension of L2 .
Proof. By Lemma B.4.16, we may assume that N is a common regular procyclic
extension of the Li . As a regular procyclic extension of L2 , N is 1-free (by
Corollary B.4.14). Since L2 is existentially closed in N (see Lemma B.4.2), N
is embeddable over L2 into an elementary extension L02 of L2 . Let N 0 denote
the image of this embedding. By B.4.14, L02 /N 0 is regular.
Theorem 7.5.2. Let L1 and L2 be regular pseudo-finite extensions of K. Then
L1 and L2 are elementarily equivalent over K.
Proof. By Proposition 7.5.1 we obtain an alternating elementary chain. Its
union is an elementary extension of both L1 and L2 .
The absolute part Abs(L) of a field L is the relative algebraic closure in L
of its prime field. Since a perfect field L is a regular extension of Abs(L) (see
Proposition B.4.13) we obtain.
Corollary 7.5.3. The elementary theory of a pseudo-finite field L is determined
by the isomorphism type of Abs(L). A field K algebraic over its prime field is
the absolute part of some pseudo-finite field if and only if it is procyclic. (This
is always true in finite characteristic.)
We now fix the complete theory of a pseudo-finite field and work in its
monster model C.
Corollary 7.5.4. Let K be a subfield of C, and a and b tuples of elements C.
Then a and b have the same type over K if and only if the relative algebraic
closures of K(a) and K(b) in C are isomorphic over K via some isomorphism
taking a to b.
CHAPTER 7. SIMPLE THEORIES 125

Proof. Let A and B be the relative algebraic closures of K(a) and K(b), respec-
tively. If a and b have the same type over K, then A and B are isomorphic in the
required way by Lemma 5.6.4. Conversely, if A and B are isomorphic over K by
such an isomorphism, the claim follows immediately from Theorem 7.5.2.
Theorem 7.5.5. In pseudo-finite fields, algebraic independence has all the prop-
erties of forking listed in Theorem 7.3.13.
Proof. We keep working in C. All properties are clear except (Existence) and
(Independence over Models).
(Existence): Let K be a subfield of C and L and H two extensions of K.
We may assume that all three fields are relatively algebraically closed in C. By
Lemma B.4.16, there is a procyclic extension C of H (not necessarily contained
in C) containing a copy L0 of L/K independent from H over K and such that
C/L0 is regular. By Proposition 7.5.1, C can be regularly over H embedded
into C. Let L00 denote the image of L0 in C. Then L00 and H are independent
over K and L00 and L have the same type over K (see p. 23).
(Independence over Models): Let M be an elementary submodel of C
and let K and L be field extensions independent over M . Assume further that
we are given extensions K 0 and L0 so that K and K 0 as well as L and L0 are
independent over M and such that K 0 and L0 have the same type over M .
We may assume that all these fields are relatively algebraically closed in
C. Then, if σ is a generator of G(C), the relative algebraic closures of KK 0
and LL0 in C are the fixed fields of κ0 = σ  (KK 0 )alg in (KK 0 )alg and of
λ0 = σ  (LL0 )alg , respectively.
We now take another field extension H/M (possibly outside C) isomorphic
to K 0 /M and L0 /M and independent of KL over M . Then KK 0 and KH, and
LL0 and LH, respectively, are isomorphic over M and the isomorphisms are
compatible with the given isomorphism between K 0 and L0 . We transport κ0
and λ0 to (KH)alg and (LH)alg via these isomorphism and call the transported
automorphisms κ and λ. Clearly κ, λ and µ = σ  (KL)alg agree on K alg and
Lalg . They also agree on H alg , since κ  H alg and λ  H alg are both the unique
extension of σ  M alg to G(H).
Since (KH)alg and (LH)alg are independent over H alg , it follows that κ and
λ extend to an automorphism µ0 of (KH)alg (LH)alg which agrees with µ on
K alg Lalg . We will see in Corollary 8.1.8 that

(KH)alg (LH)alg ∩ (KL)alg = KL.

So µ0 and µ have a common extension to some automorphism of

(KH)alg (LH)alg (KL)alg

which again can be extended to some automorphism τ of (KLH)alg . Let C be

the fixed field of τ , so C is procyclic. The relative algebraic closures of KL,
KH and LH in C are isomorphic to the relative algebraic closures of KL, KK 0
and LL0 in C. Let N be the relative algebraic closure of KL in C. Then C is
CHAPTER 7. SIMPLE THEORIES 126

a regular extension of N . So, by Proposition 7.5.1 we find a regular embedding

of C into C over N . The image of H has the required properties.
Note that we did not make use of the fact that M is a model, but only that
M is 1-free and C/M is regular.
We have now proved

Corollary 7.5.6. Pseudo-finite fields are simple. Forking independence

agrees with algebraic independence.
Exercise 7.5.1. Let K be a procyclic field which is algebraic over its prime
field. It is shown in [1] that K is the absolute part of an infinite ultraproduct of
finite fields. While the characteristic 0 case uses Čebotarev’s Density Theorem,
the proof of the characteristic p is an easy exercise.
Exercise 7.5.2. 1. Use the previous exercise to show that every pseudo-
finite field is elementarily equivalent to an infinite ultraproduct of finite
fields.

2. Show that pseudo-finite fields are exactly the infinite models of the theory
of all finite fields (see Exercise 2.1.2.)
Exercise 7.5.3. Show that in pseudo-finite fields every formula ϕ(x̄) is equiv-
.
alent to a Boolean combination of formulas of the form ∃yf (x̄, y) = 0, where
f (X̄, Y ) is a polynomial over Z. (Hint: Use Lemma 3.1.1, Theorem 7.5.2 and
Lemma B.3.13.)
Chapter 8

Stable theories

Recall from Section 5.2, that a theory is κ-stable if there are only κ-many types
over any parameter set of size κ. A theory is stable if it is κ-stable for some κ.
This is equivalent to the definition given in Section 8.2: a theory is stable if no
formula has the order property; the equivalence will be proven in Exercise 8.2.7.
In order to apply the results of the previous chapter to stable theories, we
will eventually show that stable theories are simple and then specialise the
characterisation given in Theorem 7.3.13 to stable theories. But before that
we will introduce some of the classical notions of stability theory, all essentially
describing forking in stable theories.

8.1 Heirs and coheirs

In this section we fix an arbitrary complete theory T . For types over models we
here define some special extensions to supersets, viz. heirs, coheirs, and definable
type extensions. All these extensions have in common that they do not add too
much new information to the given type. For stable theories, we will see in
Section 8.3 that these extensions coincide with the non-forking extension (and
this is in fact unique).
Definition 8.1.1. Let p be a type over a model M of T and q ∈ S(B) an
extension of p to B ⊃ M .
1. We call q an heir of p if for every L(M )-formula ϕ(x, y) such that ϕ(x, b) ∈
q for some b ∈ B there is some m ∈ M with ϕ(x, m) ∈ p.
2. We call q a coheir of p if q is finitely satisfiable in M .
It is easy to see that tp(a/M b) is an heir of tp(a/M ) if and only if tp(b/M a)
is a coheir of tp(b/M ).
The following observation is trivial, but used frequently.
Remark 8.1.2. Suppose q is an heir of p ∈ S(M ). If ϕ(x, b) ∈ q and |= σ(b),
then there is some m ∈ M with |= σ(m) and ϕ(x, m) ∈ p.

127
CHAPTER 8. STABLE THEORIES 128

Lemma 8.1.3. Let q ∈ S(B) be a (co)heir of p ∈ S(M ) and C an extension of

B. Then q can be extended to a type r ∈ S(C) which is again a (co)heir of p.
Proof. Suppose q is an heir of p. We have to show that

s(x) = q(x) ∪ {ϕ(x, c) | c ∈ C, ϕ(x, y) ∈ L(M ),

ϕ(x, m) ∈ p for all m ∈ M }

is consistent. If there are formulas ϕ(x, b), ϕ1 (x, c1 ), . . . , ϕn (x, cn ) ∈ s(x) with
ϕ(x, b) ∈ q(x) whose conjunction is inconsistent, then as M is a model and q
is an heir of p there would be m, m1 , . . . , mn ∈ M with ϕ(x, m̄) ∈ p and its
conjunction with ϕ1 (x, m1 ), . . . , ϕn (x, mn ) inconsistent. Since ϕi (x, mi ) ∈ p,
this is impossible. Any type r(x) ∈ S(C) containing s(x) is then an heir of p(x).
If q is a coheir of p, let r be a maximal set of L(C)-formulas containing q
which is finitely satisfiable in M . Clearly, r is consistent. Let ϕ(x) ∈ L(C). If
neither ϕ nor ¬ϕ belongs to r, then both r ∪ {ϕ} and r ∪ {¬ϕ} are not finitely
satisfied in M and so neither is r (see the proof of Lemma 2.2.2).
Definition 8.1.4. A type p(x) ∈ Sn (B) is definable over C if the following
holds: for any L-formula ϕ(x, y) there is an L(C)-formula ψ(y) such that for all
b∈B
ϕ(x, b) ∈ p if and only if |= ψ(b).
We say p is definable if it is definable over its domain B.
We write ψ(y) as dp xϕ(x, y) to indicate the dependence on p, ϕ(x, y) and the
choice of the variable tuple x. (So dp has the syntax of a generalised quantifier,
see [59].) Thus, we have

ϕ(x, b) ∈ p if and only if |= dp xϕ(x, b).

Note that dp xϕ(x, y) is also meaningful for formulas ϕ with parameters in B.

Example. In strongly minimal theories all types p ∈ S(A) are definable. To
see this fix ϕ0 ∈ p of minimal Morley rank k and minimal degree and consider
a formula ψ(x, y) without parameters. The discussion on page 97 shows that
ψ(x, a) ∈ p if and only if MR(ϕ0 (x) ∧ ¬ψ(x, a)) < k. By Corollary 6.4.4 this is
an A-definable property of a.
Lemma 8.1.5. A definable type p ∈ S(M ) has a unique extension q ∈ S(B)
definable over M for any set B ⊃ M , namely

{ϕ(x, b) | ϕ(x, y) ∈ L, b ∈ B, C |= dp xϕ(x, b)},

and q is the only heir of p.

Proof. The fact that the dp xϕ(x, y) define a type is a first-order property ex-
pressible in M and is hence true in any elementary extension of M . This
proves existence. On the other hand, if q is a definable extension of p, then
CHAPTER 8. STABLE THEORIES 129

dq xϕ(x, y) and dp xϕ(x, y) agree on M and hence in all elementary extensions

proving uniqueness. Clearly, q is an heir of p. If q 0 ∈ S(B) is different from
q, then for some ϕ(x, b) ∈ q 0 we have 6|= dp xϕ(x, b). But there is no m with
ϕ(x, m) ∧ ¬dp xϕ(x, m) ∈ p, so q 0 is not an heir of p.
Lemma 8.1.6. A global type which is a coheir of its restriction to a model M
is invariant over M .
Proof. Let q ∈ S(C) be finitely satisfiable in M and α ∈ Aut(C/M ). Consider a
formula ϕ(x, c). Since c and α(c) have the same type over M , we have ϕ(x, c) ∧
¬ϕ(x, α(c)) is not satisfiable in M . So ϕ(x, c) ∈ q implies ϕ(x, α(c)) ∈ q.
We conclude by exhibiting coheirs in strongly minimal theories.
Proposition 8.1.7. Let T be strongly minimal, M a model and B an exten-
sion of M . Then tp(a/B) is an heir of tp(a/M ) if and only if MR(a/B) =
MR(a/M ).
Note that in strongly minimal theories MR(a/B) = MR(a/M ) is equivalent
to a and B being geometrically independent over M (see Exercise 6.4.1). This
is a symmetric notion, which implies that in strongly minimal theories heirs and
coheirs coincide. We will later see in Corollary 8.3.7 (see also Corollary 8.5.11)
that this is actually true for all stable theories. Note also that this implies that
in strongly minimal theories types over models have a unique extension of the
same Morley rank, i.e., they have Morley degree 1. This is true in all totally
transcendental theories (Corollary 8.5.12, see also Corollary 8.5.4.)
Proof. Let k be the Morley rank of p = tp(a/M ). Choose a formula ϕ0 ∈ p of
same rank and degree as p. We saw in the Example on page 128 that the unique
heir q of p on B is given by
{ψ(x) L(B)-formula | MR(ϕ0 (x) ∧ ¬ψ(x, a)) < k}.
On the other hand this set of formulas must be contained in all extensions of p
to B having rank k. So q is also the unique extension of p of rank k.
Corollary 8.1.8 (Hrushovski–Chatzidakis). Let K, L, H be algebraically closed
extensions of an algebraically closed field M . If H algebraically independent
from KL over M , then
(KH)alg (LH)alg ∩ (KL)alg = KL.

Proof. We work in the monster model C. Let c be an element of the left hand
side. So there are tuples a ∈ (KH)alg and b ∈ (LH)alg such that c ∈ dcl(a, b),
witnessed by, say, |= ϕ(a, b, c). Furthermore, there are tuples a0 ∈ K and h1 ∈ H
such that a is algebraic over a0 h1 witnessed by, say, |= ϕ1 (a, a0 , h1 ). Similarly,
we find |= ϕ2 (b, b0 , h2 ) for b. By independence and the heir property there are
h01 , h02 ∈ M such that
|= ∃x, y ϕ(x, y, c) ∧ ϕ1 (x, a0 , h01 ) ∧ ϕ2 (y, b0 , h02 ).
Since K and L are algebraically closed, this implies c ∈ KL.
CHAPTER 8. STABLE THEORIES 130

Exercise 8.1.1. Let X be a compact topological space and F an ultrafilter on

I. Then every family (xi )i∈I has a unique F-limit, which is the unique x such
that {i ∈ I | xi ∈ N } belongs to F for every neighbourhood N of x.
Let A be an extension of the model M and p ∈ M . Show that one can
construct all coheir extensions of p ∈ S(M ) to A as follows: choose an ultrafilter
F on M such that p is the F-limit of (tp(m/M ))m∈M , then the F-limit of
(tp(m/A))m∈M in S(A) is a coheir of p.
Exercise 8.1.2. Use Example 7.2.10 and Lemma 8.1.6 to give an alternative
proof of Corollary 7.3.3.
Exercise 8.1.3. Show that a formula which is satisfiable in every model ex-
tending A does not divide over A.
Exercise 8.1.4. Let T be a complete theory, M an ω-saturated model.
1. Let ψ(x) be a formula over M with Morley rank, and ϕ a formula over
arbitrary parameters with the same Morley rank and ϕ(C) ⊆ ψ(C). Show
that ϕ is realised in M .

2. Let B an extension of M and MR(a/B) = MR(a/M ) < ∞. Show that

tp(a/B) is a coheir of tp(a/M ).
It will follow from Corollaries 8.3.7 and 8.5.11 that in totally transcendental
theories this is true for arbitrary M . In fact this holds for arbitrary theories,
see Exercise 8.5.5.
Exercise 8.1.5 (Hrushovski–Pillay). Let p(x) and q(y) be global types, and
suppose that p(x) is A-invariant. We define a global type p(x) ⊗ q(y) by setting
(p ⊗ q)  B = tp(ab/B) for any B ⊃ A where b realises q(y)  B and a realises
p  Bb. Show that p(x) ⊗ q(y) is well defined, and A-invariant if both p(x) and
q(y) are.

8.2 Stability
By analogy with simple theories we here define stable theories via (several equiv-
alent) properties of their formulas and note that this definition fits well with
the definition of κ-stability given in Section 5.2.
In this section, let T be a complete (possibly uncountable) theory. For a
formula ϕ(x, y) let Sϕ (B) denote the set of all ϕ-types over B; these are maximal
consistent sets of formulas of the form ϕ(x, b) or ¬ϕ(x, b) where b ∈ B. Recall
that the variables x and y may have different sorts representing nx and ny -tuples
of elements, respectively.
Definition 8.2.1. Let ϕ(x, y) be a formula in the language of T .
1. The formula ϕ is stable if there is an infinite cardinal λ such that | Sϕ (B)| ≤
λ whenever |B| ≤ λ. The theory T is stable if all its formulas are stable.
CHAPTER 8. STABLE THEORIES 131

2. The formula ϕ has the order property if there are elements a0 , a1 , . . . and
b0 , b1 , . . . such that for all i, j ∈ ω

|= ϕ(ai , bj ) if and only if i < j.

3. The formula ϕ(x, y) has the binary tree property if there is a binary tree
(bs | s ∈ <ω 2) of parameters such that for all σ ∈ ω 2, the set

{ϕσ(n) (x, bσn ) | n < ω}

is consistent. (We use the notation ϕ0 = ¬ϕ and ϕ1 = ϕ.)

It is important to note that T is stable if and only if it is κ-stable for some κ,
see Exercise 8.2.7.
Remark 8.2.2. The notion of ϕ(x, y) having the order property is symmetrical
in x and y. This means that if ϕ(x, y) has the order property, then there are
elements a0 , a1 , . . . and b0 , b1 , . . . such that |= ϕ(ai , bj ) if and only if j < i.
Proof. Apply Lemma 7.1.1 to I = (ai bi )i<ω and J = (ω, >).
Theorem 8.2.3. For a formula ϕ(x, y) the following are equivalent:
a) ϕ is stable.
b) | Sϕ (B)| ≤ |B| for any infinite set B.
c) ϕ does not have the order property.
d) ϕ does not have the binary tree property.
Proof. a) ⇒ d): Let µ be minimal such that 2µ > λ. Then the tree I = <µ 2 has
cardinality at most λ. If ϕ(x, y) has the binary tree property, by compactness
there are parameters bs , (s ∈ I), such that for all σ ∈ µ 2, qσ = {ϕσ(α) (x, bσα ) |
α < µ} is consistent. Complete every qσ to a ϕ-type pσ over B = {bs | s ∈ I}.
Since the pσ are pairwise different, we have |B| ≤ λ < 2µ ≤ | Sϕ (B)|.
d) ⇒ c): Choose a linear ordering of I = ≤ω 2 such that for all σ ∈ ω 2 and
n<ω
σ < σ  n ⇔ σ(n) = 1.
If ϕ(x, y) has the order property, then by Lemma 7.1.1 one can find ai and bi
indexed by I such that

|= ϕ(ai , bj ) if and only if i < j.

Now the tree ϕ(x, bs ), s ∈ <ω 2, shows that ϕ has the binary tree property.
c) ⇒ b): Let B be an infinite set of parameters and | Sϕ (B)| > |B|. For
any a the ϕ-type of a over B is given by Sa = {b ∈ B n ||= ϕ(a, b)}. Since
|B| = |B n | we may assume for simplicity that n = 1 and so Sa ⊆ B. Applying
the Erdős–Makkai Theorem C.2.1 to B and S = {Sa | a ∈ C}, we obtain a
sequence (bi | i < ω) of elements of B and a sequence (ai )i<ω such that either
CHAPTER 8. STABLE THEORIES 132

bi ∈ Saj ⇔ j < i or bi ∈ Saj ⇔ i < j for all i, j. In the first case ϕ has the
order property by definition. In the second case ϕ(x, y) has the order property
by Remark 8.2.2.
b) ⇒ a): Clear.
If ϕ has the binary tree property witnessed, say, by (bs | s ∈ <ω 2), then the
family ^
ϕs = ϕs(n) (x, bsn ), s ∈ <ω 2,
n<|s|

is a binary tree of consistent formulas. This shows that totally transcendental

theories are stable. We will see below (Corollary 8.3.6, also Exercise 8.2.11))
that stable theories are simple.
Remark 8.2.4. By Example 8.6.6 and Exercise 8.2.7 the theory of any R-
module is stable (but not necessarily totally transcendental) providing a rich
class of examples for stable theories. Note that the theory of the random graph
is simple by Corollary 7.3.14 but not stable (see Exercise 8.2.3).
Exercise 8.2.1. The theory T is unstable if and only if there is an L-formula
ψ(x, y) and elements a0 , a1 , . . . , ordered by ψ; i.e., such that

|= ψ(ai , aj ) ⇐⇒ i < j.

ψ may contain parameters.

Exercise 8.2.2. A formula ϕ(x, y) is said to have the independence property
(IP) if there are ai , i ∈ ω, such that for each A ⊆ ω the set {ϕ(x, ai ) | i ∈
A} ∪ {¬ϕ(x, ai ) | i 6∈ A} is consistent. Show that T is unstable if it contains a
formula with the independence property.
Exercise 8.2.3. Show that the theory of the random graph is not stable.
Exercise 8.2.4. A formula ϕ(x, y) is said to have the strict order property
(SOP) if there is a sequence (ai )i<ω such that

|= ∀y(ϕ(ai , y) → ϕ(aj , y)) ⇔ i ≤ j.

The theory T has the strict order property if there is a formula in T with the
strict order property. Show that T has the SOP if and only if there is a partial
ordering with infinite chains definable in T eq . (For the definition of T eq see
p. 138.)
Exercise 8.2.5. Show that a theory with SOP is not simple.
Exercise 8.2.6. (Shelah) If T is unstable, there is a formula having the IP or
a formula having the SOP.
Exercise 8.2.7. The following are equivalent:
a) T is stable.
CHAPTER 8. STABLE THEORIES 133

b) T is λ-stable for all λ such that λ|T | = λ.

c) T is λ-stable for some λ.
It follows from this and Lemma 5.2.2 that T is stable if and only if all ϕ(x, y)
for a single variable x are stable.
Exercise 8.2.8. Show that for any infinite λ there is a linear order of cardinality
greater than λ with a dense subset of size λ.
Exercise 8.2.9. Show that for all tuples of variables x, y the set of stable formu-
las ϕ(x, y) is closed under Boolean combinations, i.e., conjunction (disjunction)
and negation. Use this to show that the theory Tree, defined on page 61, is
stable.
Exercise 8.2.10. Fix an L-formula ϕ(x, y). Let Φ denote the class of Boolean
combinations of formulas of the form ϕ(x, b). Define the ϕ-rank Rϕ as the
smallest function from formulas ψ(x) to {−∞} ∪ On ∪ {∞} (here On is the class
of ordinals) such that

Rϕ (ψ) ≥ 0 if ψ is consistent;
Rϕ (ψ) ≥ β + 1 if there are infinitely many δi ∈ Φ which are pairwise
inconsistent and such that Rϕ (ψ ∧ δi ) ≥ β for all i.

Prove:
1. Rϕ (ψ) < ∞ if and only if ψ(x) ∧ ϕ(x, y) is stable.
2. If Rϕ (ψ) < ∞, then Rϕ (ψ) < ω.
Exercise 8.2.11. If ϕ has the tree property, it is unstable.
This shows that stable theories are simple. We will give a different proof in
Corollary 8.3.6.

8.3 Definable types

Definability of types turns out to be a crucial feature of stable theories. We
show here that in stable theories the extensions of a type over a model given by
its definition agree with the non-forking extensions (and with heirs and coheirs).
We continue to assume that T is a complete theory.
Theorem 8.3.1. The formula ϕ(x, y) is stable if and only if all ϕ-types are
definable.
Proof. Let A be a set of parameters of size ≥ |T |. If all ϕ-types over A are
definable, there exists no more ϕ-types over A than there are defining formulas,
i.e., at most |A| many. So ϕ is stable.
For the converse assume that ϕ(x, y) is stable. Define for any formula θ(x)
the degree Dϕ (θ) to be the largest n for which there is a finite tree (bs | s ∈ <n 2)
CHAPTER 8. STABLE THEORIES 134

of parameters such that for every σ ∈ n 2 the set {θ(x)} ∪ {ϕσ(i) (x, bσi ) | i < n}
is consistent. This is well defined since ϕ does not have the binary tree property.
Now, let p be a ϕ-type over B. Let θ be a conjunction of formulas in p with n =
Dϕ (θ) minimal. Then ϕ(x, b) belongs to p if and only if n = Dϕ (θ(x) ∧ ϕ(x, b)).
This shows that p is definable.

Corollary 8.3.2. The theory T is stable if and only if all types are definable.
Observe that the proof of Theorem 8.3.1 applies also to a proper class of
parameters. From this we obtain the following important corollary.
Corollary 8.3.3 (Separation of variables). Let T be stable and let F be a 0-
definable class. Then any definable subclass of Fn is definable using parameters
from F.
Proof. Let ψ(a, C) be a definable subclass of Fn . The type q = tp(a/F) is
definable over a subset of F by Corollary 8.3.2. Thus,

ψ(a, C) = {f ∈ Fn ||= dq xψ(x, f )}.

If the property in the conclusion of Corollary 8.3.3 holds for a 0-definable

class F (in a not necessarily stable theory T ), then F is called stably embedded.
For equivalent definitions see Exercise 10.1.5.
At first glance, the next lemma looks mysterious. In essence it states that
in stable theories heirs and coheirs coincide. We need it in the proof of Corol-
lary 8.5.3.
Lemma 8.3.4 (Harrington). Let T be stable and let p(x) and q(y) be global
types. Then for every formula ϕ(x, y) with parameters

dp xϕ(x, y) ∈ q(y) ⇔ dq yϕ(x, y) ∈ p(x).

Proof. Let p, q and ϕ be definable over A. We recursively define sequences ai

and bi , i ∈ ω: if a0 , . . . , an−1 and b0 , . . . , bn−1 have been defined, let bn be a
realisation of q  Aa0 , . . . , an−1 and an a realisation of p  Ab0 , . . . , bn . Then we
have for i < j

|= ϕ(ai , bj ) ⇔ |= dq yϕ(ai , y) ⇔ dq yϕ(x, y) ∈ p(x)

and for j ≤ i

|= ϕ(ai , bj ) ⇔ |= dp xϕ(x, bj ) ⇔ dp xϕ(x, y) ∈ q(y).

Since ϕ does not have the order property, the claim follows.
Lemma 8.3.5. Let p ∈ S(C) be a global type.
1. If p is definable over A, then p does not divide over A.
CHAPTER 8. STABLE THEORIES 135

2. If T is stable and p does not divide over the model M , then p is definable
over M .
Note that for global types dividing and forking coincide (Exercise 7.1.3).
Proof. 1): Consider a formula ϕ(x, m) ∈ p and an infinite sequence of indis-
cernibles m = m0 , m1 , . . . over A. If p is definable over A, all ϕ(x, mi ) belong
to p. So ϕ(x, m) does not divide over A by Lemma 7.1.4.
2): Now assume that T is stable and p does not divide over the model M .
We will show that p is an heir of p  M . By Corollary 8.3.2 and Lemma 8.1.5
this implies that p is definable over M . So assume that ϕ(x, b) ∈ p, we want to
show that ϕ(x, b0 ) ∈ p for some b0 ∈ M .
Let I = (bi )i<ω be a Morley sequence of a global coheir extension of tp(b/M )
over M starting with b0 = b (see Example 7.2.10 and Lemma 8.1.6). Since
tp(a/M b) does not divide over M , Lemma 7.1.5 implies that we may assume that
I is indiscernible over M a. So we have |= ϕ(a, bi ) for all i. By Corollary 8.3.2,
the type q = tp(a/M {bi | i < ω}) is definable. Assume that the parameters of
dq xϕ(x, y) are in M {b0 , . . . , bn−1 }. Since tp(bn /M {b0 , . . . , bn−1 }) is a coheir of
tp(b/M ), and since |= dq xϕ(x, bn ), there is a b0 ∈ M with |= dq xϕ(x, b0 ). This
implies |= ϕ(a, b0 ) and so ϕ(x, b0 ) ∈ tp(a/M ) = p  M .
Corollary 8.3.6. Stable theories are simple.
Proof. Let p be a type over a model M . Then p is definable over some A ⊆ M
of cardinality ≤ |T |. Let p0 be the global extension of p given by the defini-
tion over A. By Lemma 8.3.5(1), p0 and hence also p does not divide over A.
Proposition 7.2.5 implies that T is simple.
This implies in particular that forking and dividing coincide in stable theories
(see Proposition 7.2.15).
Corollary 8.3.7. Let T be a stable theory, p a type over a model M and A
an extension of M . Then p has a unique extension q ∈ S(A) with the following
equivalent properties:
a) q does not fork over M .
b) q is definable over M .
c) q is an heir of p.
d) q is a coheir of p.
Proof. By Lemma 8.3.5, q does not fork over M if and only if it is definable
over M . Since p is definable, we know by Lemma 8.1.5 that there is a unique
extension q which is definable over M , and which is also the unique heir of p.
To prove the equivalence with d) we may assume that A = M ∪ {a} for
a finite tuple a. Fix a realisation b of q. Then q = tp(b/M a) is a coheir of
p = tp(b/M ) if and only if tp(a/M b) is an heir and hence, by the first part of
the proof, a non-forking extension of tp(a/M ). Now forking symmetry and the
first part of the proof imply the desired.
CHAPTER 8. STABLE THEORIES 136

Exercise 8.3.1. Find a theory T and a type p over the empty set such that no
definition of p defines a global type. (A definition which defines a global type is
called a good definition of p, see Theorem 8.5.1).
Exercise 8.3.2. Let T be an arbitrary complete theory and M be a model.
Consider the following four properties of a global type p:

(D) p is definable over M .

(C) p is a coheir of p  M .
(I) p is M -invariant.

(H) p is an heir of p  M .
Use the example T = DLO and M = Q to show that D→I, C→I, D→H are the
only logical relations between these notions.
Exercise 8.3.3. Let p(x) ∈ S(C) be definable over B. Then, for any n, the
map
r(y1 , . . . , yn ) 7→ {ϕ(x, y1 , . . . , yn ) | dp xϕ ∈ r}
defines a continuous section πn : Sn (B) → Sn+1 (B). Show that this defines a
bijection between all types definable over B and all “coherent” families (πn ) of
continuous sections Sn (B) → Sn+1 (B).
Exercise 8.3.4. Let ϕ(x) be a formula without parameters and let M be
a model of T . Show that ϕ(M ) is stably embedded in M (i.e., every M -
definable relation of ϕ(M ) is definable over ϕ(M )) if and only if for all n,
every p(x1 , . . . , xn ) ∈ Sn (ϕ(M )) which contains ϕ(x1 ) ∧ . . . ∧ ϕ(xn ) has a unique
extension p0 ∈ Sn (M ). If ϕ is (absolutely) stably embedded and p is definable,
show that p0 is definable over ϕ(M ).
Exercise 8.3.5. Call a formula ψ(x) in one variable (though possibly repre-
senting a tuple) stable if ψ(x) ∧ ϕ(x, y) is stable for all ϕ(x, y) according to
Definition 8.2.1. We call a type stable if it contains a stable formula. Prove:
1. Types with Morley rank are stable.
2. Stable types are definable.
3. Stable formulas are stably embedded.

Exercise 8.3.6. Let T be stable, and p ∈ S(A). Show that p is definable over
C if p is finitely satisfiable in C. Furthermore for every ϕ(x, y), dp xϕ(x, y) is a
positive Boolean combination of formulas ϕ(c, y), c ∈ C.
Exercise 8.3.7. If T is stable, then for any formula ϕ(x, y), there is a formula
∆(y, z) such that for every set A and every type p(x) over A there is a tuple b
in A such that {a ∈ A | ϕ(x, a) ∈ p(x)} = {a ∈ A ||= ∆(a, b)}.
CHAPTER 8. STABLE THEORIES 137

Exercise 8.3.8. We call q a weak heir of p ∈ S(M ) if the heir property holds
for all ϕ(x, y) without parameters. Show that in stable theories, weak heirs are
in fact heirs.
Exercise 8.3.9. In Corollary 8.3.7 prove the equivalence of c) and d) directly
from Lemma 8.3.4.

Exercise 8.3.10. Show that in a stable theory a formula does not fork over A
if and only if it is realised in every model which contains A.

8.4 Elimination of imaginaries and T eq

This section is an excursion outside the realm of stable theories: for a model M of
an arbitrary complete theory T and any 0-definable equivalence relation E(x, y)
on n-tuples, we now consider the equivalence classes of M n /E as elements of a
new sort of so-called imaginary elements. Adding these imaginaries makes many
arguments more convenient. For certain theories, these imaginaries are already
coded in the original structure. However, if this is not already the case, then
adding imaginaries leads to a new theory T eq which does have this property so
that we do not run into an infinite regression.
Definition 8.4.1. A finite tuple d ⊆ C is called a canonical parameter for a
definable class D in Cn if d is fixed by the same automorphisms of C which leave
D invariant. A canonical base for a type p ∈ S(C) is a set B which is pointwise
fixed by the same automorphisms which leave p invariant.
Lemma 6.1.10 implies that D is definable over d, and by Corollary 6.1.12(1)
d is determined by D up to interdefinability. We write d = pDq, or d = pϕ(x)q
if D = ϕ(C). Note that the empty tuple is a canonical parameter for every
0-definable class.

Definition 8.4.2. The theory T eliminates imaginaries if any class e/E of a

0-definable equivalence relation E on Cn has a canonical parameter d ⊆ C.
Theorem 8.4.3. If T eliminates imaginaries, then the following hold:
1. Every definable class D ⊆ Cn has a canonical parameter c.

2. Every definable type p ∈ S(C) has a canonical base.

Proof. Write D = ϕ(C, e). Define the equivalence relation E by

y1 Ey2 ⇐⇒ ∀x ϕ(x, y1 ) ↔ ϕ(x, y2 )

and let d be a canonical parameter of e/E. Then d is a canonical parameter of

D.
If dp is a definition of p, the set B = {pdp xϕ(x, y)q | ϕ(x, y) L-formula} is
a canonical base of p.
CHAPTER 8. STABLE THEORIES 138

Lemma 8.4.4. Assume that T eliminates imaginaries. Let A be a set of pa-

rameters and D a definable class. Then the following are equivalent:
a) D is acl(A)-definable.
b) D has only finitely many conjugates over A.

c) D is the union of equivalence classes of an A-definable equivalence relation

with finitely many classes (a finite equivalence relation).
Proof. Let d be a canonical parameter of D. Then D is definable over acl(A) if
and only if d belongs to acl(A). On the other hand D has as many conjugates
over A as d. So a) and b) are equivalent.
For the equivalence of b) and c), first notice that any class of an A-definable
finite equivalence relation has only finitely many conjugates over A, which yields
c) ⇒ b). For the converse, let D = D1 , . . . , Dn be the conjugates of D over A.
Consider the finite equivalence relation E(c, c0 ) defined by

c ∈ Di if and only if c0 ∈ Di for all i.

Clearly D is a union of E-classes. Also E is definable and since it is invariant

under all A-automorphisms of C, it is in fact definable over A.
Note that elimination of imaginaries was only used for b) ⇒ a).
The previous results show why it is convenient to work in a theory eliminat-
ing imaginaries. It is easy to see that a theory eliminates imaginaries if every 0-
definable equivalence relation arises from fibres of a 0-definable function. While
not all theories have this property (e.g., the theory of an equivalence relation
with infinitely many infinite classes), we now show how to extend any complete
theory T to a theory T eq (in a corresponding language Leq ) which does.
Let Ei (x1 , x2 ), (i ∈ I), be a list of all 0-definable equivalence relations on
ni -tuples. For any model M of T we consider the many-sorted structure

M eq = (M, M ni /Ei )i∈I ,

which carries the home sort M and for every i the natural projection

πi : M ni → M ni /Ei .

The elements of the sorts Si = M ni /Ei are called imaginary elements, the
elements of the home sort are real elements.
The M eq form an elementary class axiomatised by the (complete) theory
T which, in in the appropriate many-sorted language Leq , is axiomatised by
eq

the axioms of T and for each i ∈ I by

.
∀y ∃x πi (x̄) = y (y a variable of sort Si )

and
.
∀x1 , x2 (πi (x1 ) = πi (x̄2 ) ↔ Ei (x1 , x2 )).
CHAPTER 8. STABLE THEORIES 139

The algebraic (definable, respectively) closure of in M eq is denoted by acleq

(dcleq , respectively).
The first two statements of the following proposition explain why we consider
T eq as an inessential expansion of T .
Proposition 8.4.5. 1. Elements of Ceq are definable over C in a uniform
way.
2. The 0-definable relations on the home sort of C eq are exactly the same as
those in C.
3. The theory T eq eliminates imaginaries,
Proof. 1: Every element of sort Si has the form πi (a) for an ni -tuple a from C.
2: We show that every Leq -formula ϕ(x) with free variables from the home-
sort is equivalent to an L-formula ϕ∗ (x) by induction on the complexity of ϕ.
.
If ϕ is atomic, it is either an L-formula or of the form πi (x̄1 ) = πi (x2 ), in which
∗
case we set ϕ = Ei (x̄1 , x2 ). We let ∗ commute with negations, conjunctions
and quantification over home-sort variables. Finally, if y is a variable of sort Si ,
we set
∗
∃yψ(x, y) = ∃x0 ψ(x, πi (x0 ))∗ .
3: We observe first that Ceq is the monster model of T eq , i.e., that every
type p(y) over a set A is realised in Ceq . By part 1 we may assume that A ⊆ C.
If y is of sort Si , the set Σ(x) = p(πi (x)) is finitely satisfiable. By part 2 Σ is
equivalent to a set Σ∗ of L-formulas; this set has a realisation b, which gives us
a realisation πi (b) of p.
It is now clear that πi (e) is a canonical parameter of the class e/Ei . By the
proof of Theorem 8.4.3 this implies that every relation in Ceq that is definable
with parameters from C has a canonical parameter in Ceq . On the other hand,
by part 1, every definable relation in Ceq is definable in C.
Corollary 8.4.6. The theory T eliminates imaginaries if and only if in T eq
every imaginary is interdefinable with a real tuple.
Proof. Since every automorphism of C extends (uniquely) to an automorphism
of Ceq , a real tuple d is a canonical parameter of e/Ei in the sense of T if and
only if it is a canonical parameter in the sense of T eq . But this is equivalent to
d being interdefinable with πi (e).
The proof of the following criterion for elimination of imaginaries shows how
T eq can be used.
Lemma 8.4.7. The following are equivalent:
a) The theory T eliminates imaginaries and has at least two 0-definable ele-
ments.
b) Every 0-definable equivalence relation on Cn is the fibration of a 0-definable
function f : Cn → Cm .
CHAPTER 8. STABLE THEORIES 140

Proof. b) ⇒ a): If E is the fibration of a 0-definable function f : Cn → Cm , we

have pe/Eq = f (e). To see that there are at least two 0-definable elements look
at the following equivalence relation on C2 :
x1 x2 E y1 y2 ⇔ (x1 = x2 ↔ y1 = y2 ).
This has two classes, which are both 0-definable. If E is the fibration of a
0-definable π : C2 → Cm , the two images of π are two different 0-definable m-
tuples.
a) ⇒ b): Let E be a 0-definable equivalence relation on Cn . Every e/E is
interdefinable with an element of some power Cme . So by Exercise 6.1.12, e/E
belongs to a 0-definable D ⊆ Cn /E with a 0-definable injection f : D → Cme .
A compactness argument shows that we can cover Cn /E by finitely many 0-
definable classes D1 , . . . , Dk with 0-definable injections fi : Di → Cmi . We may
assume that the Di are pairwise disjoint, otherwise we replace Di by Di \ (D0 ∪
· · · ∪ Di−1 ). Now, using the two 0-definable elements, we can find, for some
big m, 0-definable injections gi : Cmi → Cm with pairwise disjoint images. The
union of the gi fi is a 0-definable injection from Cn /E into Cm .
Using parts 1 and 2 of the previous proposition, one can see that in general
all properties of T which concern us here are preserved when going from T to
T eq . Here are some examples.
Lemma 8.4.8. 1. The theory T is ℵ1 -categorical if and only if T eq is ℵ1 -
categorical.
2. T is λ-stable if and only if T eq is λ-stable.
3. T is stable if and only if T eq is stable.
Proof. Part 1 is clear.
For Part 2 let A be a set of parameters in T eq of cardinality λ. This set A
is contained in the definable closure of some set B of cardinality λ of the home
sort. For any p ∈ S(B) we may first take the unique extension of p to dcleq (B)
and then its restriction to A. This defines a surjection S(B) → S(A). Notice
that we now have to specify not only the number of variables but also the sorts
for the variables in the types.
If S0 (A) consists of types of elements of the sort Cn /E, Sn (A) denotes the
n-types of the home sort and π the projection Cn → Cn /E, then tp(b/A) 7→
tp(π(b̄)/A) defines a surjection Sn (A) → S0 (A). This shows that T eq is stable if
T is.
Of course Part 3 follows from 2. and Exercise 8.2.7. Still we give a direct
proof as an example of how to translate between T and T eq . Let ϕ(y1 , y2 ) be
a formula in T eq with the order property. If y1 and y2 belong to Cn1 /E1 and
Cn2 /E2 , respectively, there are tuples ai of the home sort such that
|= ϕ(π1 (a1 ), π2 (a2 )) if and only if i < j.
By Proposition 8.4.5 the formula ϕ(π1 (x̄1 ), π2 (x2 )) is equivalent to some L-
formula ψ(x1 , x̄2 ), which has the order property in T .
CHAPTER 8. STABLE THEORIES 141

For applications the following special cases are often useful.

Definition 8.4.9. 1. The theory T eliminates finite imaginaries if every
finite set of n-tuples has a canonical parameter.
2. T has weak elimination of imaginaries if for every imaginary e there is a
real tuple c such that e ∈ dcleq (c) and c ∈ acl(e).
Lemma 8.4.10. Then theory T eliminates imaginaries if and only if it has
weak elimination of imaginaries and eliminates finite imaginaries
Proof. This follows from the observation that T has weak elimination of imag-
inaries if and only if every imaginary e is interdefinable with the canonical
parameter of a finite set of real n-tuples. Indeed, if e ∈ dcleq (c) and c ∈ acl(e),
and {c1 , . . . , cm } are the conjugates of c over e, then e is interdefinable with
p{c1 , . . . , cm }q. If conversely e is interdefinable with p{c1 , . . . , cm }q, then e ∈
dcleq (c1 . . . cm ) and c1 . . . cm ∈ acl(e).
Lemma 8.4.11 (Lascar–Pillay). Let T be strongly minimal and acl(∅) infinite.
Then T has weak elimination of imaginaries.
Proof. Let e = c/E be an imaginary. It suffices to show that c/E contains
an element algebraic over e or, more generally, that every non-empty definable
X ⊆ Cn contains an element of A = acl(pXq). We proceed by induction on n.
For n = 1 there are two cases: if X is finite, it is a subset of A; if X is infinite,
almost all elements of acl(∅) belong to X. If n > 1, consider the projection Y
of X to the first coordinate. Such a Y contains an element a of acl(pY q), which
is a subset of A. By induction the fibre Xa contains an element b of acl(pXa q),
which is also a subset of A. So (a, b) is in X ∩ A.
Corollary 8.4.12. The theory ACFp of algebraically closed fields of character-
istic p eliminates imaginaries.
Proof. By the preceding lemmas it suffices to show that every theory of fields
eliminates finite imaginaries. Let S = {c0 , . . . , ck−1 } be a set of n-tuples ci =
(ci,j )j<n . Consider the polynomial
Y X

p(X, Y0 , . . . , Yn−1 ) = X− ci,j Yj .
i<k j<n

An automorphism leaves p fixed if and only if it permutes S. So the coefficients

of p serve as a canonical parameter of S.
Lemma 8.4.13. A totally transcendental theory in which every global type has
a canonical base in C has weak elimination of imaginaries.
Proof. Let e = c/E be an imaginary and α the Morley rank of the class c/E.
Let p be a global type of Morley rank α which contains E(x, c). By assumption
p has a canonical base d ⊆ C. Since there are only finitely many such p, d is
algebraic over e. Also e is definable from d since for an automorphism α fixing
p, c/E and αc/E cannot be disjoint, so they must be equal. Clearly we may
assume that d is a finite tuple (see also Exercises 8.4.1 and 8.4.7).
CHAPTER 8. STABLE THEORIES 142

Corollary 8.4.14. DCF0 eliminates imaginaries.

Proof. By quantifier elimination every global type p(x) is axiomatised by its
quantifier-free part pqf (x), which is equivalent to a union of qi (x, dx, . . . ,
di x), i = 0, 1, . . ., where the qi (x0 , . . . , xi ) are quantifier-free pure field-theoretic
types. If Ci is the canonical base of qi in the sense of ACF0 , then C0 ∪ C1 ∪ · · ·
is a canonical base of p.
Exercise 8.4.1. Let D be a definable class. Assume that there is a set D
which is fixed by the same automorphisms which leave D invariant. Show that
D contains a canonical parameter of D.
Exercise 8.4.2. A theory T has weak elimination of imaginaries if and only if
for every definable class D there is a smallest algebraically closed set over which
D is definable.
Exercise 8.4.3. Use Exercise 8.4.2 to prove that the theories Infset and DLO
(not easy) have weak elimination of imaginaries. Show also that DLO has elim-
ination of imaginaries, but Infset does not.
Exercise 8.4.4. Show that all extensions of p ∈ S(A) to acl(A) are conjugate
over A. More generally this remains true for every normal extension B of A.
These are sets which are invariant under all α ∈ Aut(C/A). Note that normal
extensions must be subsets of acl(A).
Exercise 8.4.5. An algebraic type over A has a good definition (see Exer-
cise 8.3.1 or p. 143) over B ⊆ A if and only if it is realised in dcl(B).
Exercise 8.4.6. Let d be a canonical parameter of D. Then d is 0-definable in
the L ∪ {P }-structure (C, D).
Exercise 8.4.7. Let T be totally transcendental and p a global type.
1. Show that p has a finite canonical base in Ceq .
2. If p has a canonical base D ⊆ C, then it has a finite base d ⊆ C.
Exercise 8.4.8. Show that Lemma 8.4.13 is true for stable theories. (Hint: In
the proof of 8.4.13 replace p by a suitable E(x, y)-type.)

Exercise 8.4.9. Define the strong type of a over A as stp(a/A) = tp(a/

acleq (A)). Show that stp(a/A) is axiomatised by

{E(x, a) | E(x, y) A-definable finite equivalence relation}.

Exercise 8.4.10 (Poizat). Let T be a complete theory with elimination of

imaginaries. Consider the group G = Aut(acl(∅)) of elementary permutations
of acl(∅). This G is a topological group if we use the stabilisers of finite sets
as a basis of neighbourhoods of 1. Show that there is a Galois correspondence
between the closed subgroups H of G and the definably closed subsets A of
acl(∅).
CHAPTER 8. STABLE THEORIES 143

8.5 Properties of forking in stable theories

Except in Theorem 8.5.10 we assume throughout this section that T is stable.
We now collect the crucial properties of forking in stable theories. As with
simple theories, we will see in Theorem 8.5.10 that these properties characterise
stable theories and forking. Since we already know that stable theories are
simple, some of these properties are immediate. For completeness and reference
we restate them in the context of stable theories.
Let p ∈ S(B) be defined by L(B)-formulas dp xϕ. We call the definition good
if it defines a global type (or, equivalently, if it defines a type over some model
containing B).
Theorem 8.5.1. Let T be stable. A type p ∈ S(B) does not fork over A ⊆ B if
and only if p has a good definition over acleq (A).

Proof. If p does not fork over A, p has a global extension p0 which does not
fork over A. Let M be any model which contains A. By Lemma 8.3.5, p0 is
definable over M , so the canonical base of p0 belongs to M eq . By Exercise 6.1.2
the canonical base belongs to acleq (A).
If conversely p has a good definition over acleq (A), p does not fork over
eq
acl (A) and therefore does not fork over A.
Definition 8.5.2. A type is stationary if and only if it has a unique non-forking
extension to any superset.
Corollary 8.5.3 (Uniqueness). If T is stable, any type over acleq (A) is sta-
tionary.

If T eliminates imaginaries, this just says that any type over an algebraically
closed set is stationary.
Proof. Let A = acleq (A). Let p0 and p00 two global non-forking extensions of
p ∈ S(A). Consider any formula ϕ(x, b), and let q(y) be a global non-forking
extension of tp(b/A). By Theorem 8.5.1, p0 , p00 and q are definable over A. Now
we apply Harrington’s Lemma 8.3.4:

ϕ(x, b) ∈ p0 ⇔ dp0 xϕ(x, y) ∈ q ⇔ dq yϕ(y, x) ∈ p

⇔ dp00 xϕ(x, y) ∈ q ⇔ ϕ(x, b) ∈ p00 .

Corollary 8.5.4. In a stable theory, types over models are stationary.

Proof. This is immediate by the above proof since we can replace acleq (A) every-
where by M . It follows also formally from Corollary 8.5.3 since M eq = dcleq (M )
is an elementary substructure of Ceq and so algebraically closed.
CHAPTER 8. STABLE THEORIES 144

For the remainder of this section we also assume that T eliminates imagi-
naries. In view of T eq (see Section 8.4) this is a harmless assumption.
As stable theories are simple we may first collect some of the properties of
forking established in Section 7.2.
We keep using
a^| B
C
to express that tp(a/BC) does not fork over C.
Theorem 8.5.5. If T is stable, forking independence has the following proper-
ties:
1. (Monotonicity and Transitivity) Let A ⊆ B ⊆ C and q ∈ S(C). Then q
does not fork over A if and only if q does not fork over B and q  B does
not fork over A.
| A b =⇒ b ^
2. (Symmetry) a ^ | Aa
3. (Finite Character) If p ∈ S(B) forks over A, there is a finite subset B0 ⊆ B
such that p  AB0 forks over A.
4. (Local Character) For p ∈ S(A) there is some A0 ⊆ A of cardinality at
most |T | such that p does not fork over A0
5. (Existence) Every type p ∈ S(A) has a non-forking extension to any set
containing A.
6. (Algebraic Closure)
(a) p ∈ S(acl(A)) does not fork over A.
(b) If tp(a/Aa) does not fork over A, then a is algebraic over A.
Proof. This is contained in 7.2.17, 7.2.16, 7.1.9, 7.2.5, 7.2.7 and 7.1.3.
The following properties do not hold in arbitrary simple theories.
Theorem 8.5.6. Assume T is stable.
1. (Conjugacy) If A ⊆ M and M is strongly κ-homogeneous for some κ >
max(|T |, |A|), then all non-forking extensions of p ∈ S(A) to M are con-
jugate over A.
2. (Boundedness) Any p ∈ S(A) has at most 2|T | non-forking extensions for
every B ⊃ A.
Proof. For Part 1 let q1 and q2 be non-forking extensions of p to M . Any A-
automorphism of M which takes q1  acl(A) to q2  acl(A) (see Exercise 8.4.4)
takes q1 to q2 . Since types over algebraically closed sets are stationary by
Corollary 8.5.3, the claim now follows.
To prove Part 2 let A0 be a subset of A of cardinality at most |T | such that
p does not fork over A0 . Then p has at most as many non-forking extensions as
p  A0 has extensions to acl(A0 ).
CHAPTER 8. STABLE THEORIES 145

Corollary 8.5.7. Let T be stable and p ∈ S(A). Then p is stationary if and

only if it has a good definition over A.
Proof. Assume first that p is stationary and let q be the global non-forking
extension. So q is definable and invariant under all automorphisms over A, hence
definable over A by Lemma 6.1.10. This shows that p has a good definition over
A. For the converse assume that p has a good definition over A. So p has a
non-forking global extension p0 , definable over A by 8.5.1. Since all global non-
forking extensions of p are conjugate over A, and p0 is fixed by all automorphisms
over A, p0 is the only global non-forking extension of p.
Let p ∈ S(A) be a stationary type. The canonical base Cb(p) of p is the
canonical base of the non-forking global extension of p. We call p based on B if
p is parallel to some stationary type q defined over B, i.e., if p and q have the
same global non-forking extension (see Exercise 9.1.4). Note that parallel types
necessarily have the same free variables.
Lemma 8.5.8. A stationary type p ∈ S(A) is based on B if and only if
Cb(p) ⊆ dcl(B). So p does not fork over B ⊆ A if and only if Cb(p) ⊆ acl(B).
Proof. Let r be the global non-forking extension of p and q = r  B. Assume
that p is based on B. Then q is stationary and r the unique non-forking extension
of q. By Corollary 8.5.7, q has a good definition over B, which also defines r.
So r is definable over B, which means Cb(p) ⊆ dcl(B).
If, conversely, r is definable over B, we know by Theorem 8.5.1 that r does
not fork over B and that q is stationary by Corollary 8.5.7.
The last statement follows from the easy fact that p does not fork over B if
and only if p is based on acl(B).
For A ⊆ B let N(B/A) be the set of all types over B that do not fork over
A. By Remark 7.1.8, N(B/A) is closed in S(B). For future reference we record
the following useful fact.
Theorem 8.5.9. (Open mapping) The restriction map π : N(B/A) −→ S(A)
is open.
Proof. It is easy to see that we may replace B by C. If π(q) = π(q 0 ) for some
q, q 0 ∈ N(C/A), then q and q 0 are conjugate. So if O is a (relative) open subset
of N(C/A), then
[
O0 = π −1 (π(O)) = {α(O) | α ∈ Aut(C/A)}

is again open. So
S(A) \ π(O) = π(N(C/A) \ O0 )
is closed since it is the image of a closed set.
Theorem 8.5.10 (Characterisation of Forking). Let T be a complete theory
and n > 0. Then T is stable if and only if there is a special class of extensions
of n-types, which we denote by p < q, with the following properties.
CHAPTER 8. STABLE THEORIES 146

a) (Invariance) < is invariant under Aut(C),

b) (Local character) There is a cardinal κ such that for q ∈ Sn (C) there is
C0 ⊆ C of cardinality at most κ such that q  C0 < q.
c) (Weak Boundedness) For all p ∈ Sn (A) there is a cardinal µ such that p
has, for any B ⊃ A, at most µ extensions q ∈ Sn (B) with p < q.
If < satisfies in addition
d) (Existence) For all p ∈ Sn (A) and A ⊆ B, there is q ∈ Sn (B) such that
p < q,
e) (Transitivity) p < q < r implies p < r,
f ) (Weak Monotonicity) p < r and p ⊆ q ⊆ r implies p < q,
then < coincides with the non-forking relation.
Proof. In a stable theory, non-forking extensions satisfy properties a), b) and
c) (and d) , e), f)) by Theorems 8.5.5 and 8.5.6.
Assume conversely that properties a), b) and c) hold. Then a) and c) allow
us to find a sufficiently large cardinal µ0 so that for all A0 of cardinality at most
κ all n-types over A0 have at most µ0 <-extensions to any superset.
Let A be a set of parameters. Then the number of n-types over A is bounded
by the product of the number of subsets A0 of A of cardinality at most κ, times
a bound for the number of types p over A0 , times a bound for the number of
<-extensions of p ∈ Sn (A0 ) to A. So we have
| Sn (A)| ≤ |A|κ · 2max(κ,|T |) · µ0
and it follows that T is λ-stable if λκ = λ and λ ≥ max(2|T | , µ0 ), hence stable
by Exercise 8.2.7.
Now let < have the properties a) to f). Consider a type p ∈ Sn (A) with an
extension q ∈ Sn (B).
Assume first that p < q. Let µ be the cardinal given by Weak Bounded-
ness applied to p. By Exercise 7.1.5 there is an extension M of B such that
every r ∈ Sn (M ) which forks over A has more than µ conjugates over A. By
Existence and Transitivity q has an extension r to M such that p < r.
By Invariance we have p < r0 for all conjugates r0 . So r has no more than µ
conjugates, which implies that r does not fork over A and that q is a non-forking
extension of p.
Now assume that q is a non-forking extension of p. Choose an extension
M of B which is sufficiently saturated in the sense of Theorem 8.5.6(1). Let
r ∈ Sn (M ) be a non-forking extension of q and r0 ∈ Sn (M ) such that p < r0 .
By the above r0 is a non-forking extension of p. So r and r0 are conjugate over
A. This implies p < r and p < q by Weak Monotonicity.
Corollary 8.5.11. Let T be totally transcendental, p ∈ S(A) and q an extension
of p to some superset of A. Then q is a non-forking extension if and only if
MR(p) = MR(q). Hence p is stationary if and only if it has Morley degree 1.
CHAPTER 8. STABLE THEORIES 147

Proof. In a totally transcendental theory, extensions having the same Morley

rank satisfy the conditions of Theorem 8.5.10 (see Section 6.2).
The same is true for types with Morley rank in stable theories (see Exercise
8.5.5). It follows in particular that in totally transcendental theories for any
type p ∈ S(A) there is a finite set A0 ⊆ A such that p does not fork over A0 .
Stable theories with this property are called superstable: see Section 8.6.
Corollary 8.5.12. In a totally transcendental theory, types over models have
Morley degree 1.
Proof. This follows from Corollaries 8.5.4 and 8.5.11.

Corollary 8.5.13. If T is strongly minimal, we have A ^ | B C if and only if

A and C are algebraically independent over B in the pregeometry sense, i.e., if
dim(a/B) = dim(a/BC) for all finite tuples a ∈ A.
Proof. This follows from Theorem 6.4.2 and Corollary 8.5.11.
Corollary 8.5.14. Let K ⊆ F1 , F2 be differential fields contained in a model
| K F2 if and only if F1 and F2 are algebraically independent
of DCF0 . Then F1 ^
over K.
Proof. By Exercises 3.3.2 and 7.2.7, F1 ^ | K F2 implies algebraic independence.
For the converse we may assume that K is algebraically closed. So let F1 and F2
be algebraically independent over K. By the existence of non-forking extensions
choose a copy F 0 of F1 satisfying the same type over K and forking independent
of F2 over K. Then F 0 is algebraically independent of F2 over K. Since K is
algebraically closed, F 0 and F1 satisfy the same type over F2 in the sense of
field theory. Since F 0 , F1 and F2 are d-closed and F 0 and F1 are isomorphic
as d-fields, we conclude that F 0 and F1 have the same type over F2 . Thus,
F1 ^ | K F2 .

Exercise 8.5.1. Show that in ACFp the type of a finite tuple a over a field K
is stationary if and only if K(a) and K sep are linearly disjoint over K.
We continue assuming that T eliminates imaginaries.
Exercise 8.5.2 (Finite Equivalence Relation Theorem). Let p ∈ S(B) and
q ∈ S(B) be two different types which do not fork over A ⊆ B. Then there is
an A-definable finite equivalence relation E with p(x) ∪ q(y) ` ¬E(x, y).
If p and q have different domains, the same conclusion holds if one assumes that
p and q do not have a common non-forking extension.

Exercise 8.5.3. If a and b are independent realisations of the same type over
acl(A), then tp(a/Ab) is stationary. Hence the canonical base of tp(b/ acl(A))
is contained in dcl(bA).
Exercise 8.5.4. Prove the following.
CHAPTER 8. STABLE THEORIES 148

1. Let p ∈ S(A) be stable and q an extension of p. Then q does not fork over
A if and only if q has a good definition over acl(A).
2. Stable types over algebraically closed sets are stationary.
Exercise 8.5.5.
1. Let T be stable, p ∈ S(A) a type with Morley rank and q an extension of
p to some superset of A. Then q is a non-forking extension if and only if
MR(p) = MR(q). It follows that a type with Morley rank is stationary if
and only if it has Morley degree 1.
2. Show that the same is true for an arbitrary theory T .
Exercise 8.5.6. Assume that T is stable. For any p ∈ S(A) there is some
A0 ⊆ A of cardinality at most |T | such that p is the unique non-forking extension
of p  A0 to A. If p has Morley rank, A0 can be chosen as a finite set.
Exercise 8.5.7 (Forking multiplicity). (T stable) Define the multiplicity of a
type p as the number mult(p) of its global non-forking extensions. Show:
1. If p is algebraic, mult(p) equals deg(p), the number of realisations of p.
(See page 79 and Remark 5.6.3.)
2. If T is countable, then mult(p) is either finite or 2ℵ0 .
3. If p has Morley rank, show that mult(p) = MD(p).
Exercise 8.5.8. Let G be a totally transcendental group. Show:
1. If a and b are independent elements, then MR(a · b) ≥ MR(a). Equality
holds if and only if a · b and b are independent.
2. Assume that G is ω-saturated. Then a ∈ G is generic, i.e., MR(a) =
MR(G), if and only if for all b ∈ G

| b ⇒a·b^
a^ | b.

It can be shown that all ω-saturated stable groups contain generic elements,
i.e., elements satisfying property 2, see Poizat [46], or Wagner [61].
Exercise 8.5.9 (Group configuration). Let G be a totally transcendental
group, and let a1 , a2 , a3 ∈ G be independent generic elements, i.e., elements
of maximal rank in Th(G). Put b1 = a1 · a2 , b2 = a1 · a2 · a3 and b3 = a2 · a3 .
We consider these six elements as the points of a geometry with “lines” A0 =
{a1 , b1 , a2 }, A1 = {a2 , b3 , a3 }, A2 = {a1 , b2 , b3 } and A3 = {b1 , b2 , a3 }. It is easy
to see that every permutation of the four lines gives rise to an automorphism of
this geometry.
CHAPTER 8. STABLE THEORIES 149
a1 a3
Q 

A Q 

A Q Q 

b
A  2Q

 Q
b1 b3
A

A

A

a2

Show:

1. Each point on a line is algebraic over the other two points on the line.
2. Any three non-collinear points are independent.
Any family of points a1 , a2 , a3 , b1 , b2 , b3 with these properties1 is called a
group configuration. Hrushovski proved that whenever a totally transcenden-
tal structure contains a group configuration, there is a group definable in this
structure whose Morley rank equals the Morley rank of any of the points. For
more details see Bouscaren [11], Wagner [61] or Pillay [44].
Exercise 8.5.10. Let T be an arbitrary complete theory, not necessarily stable.
For any set of parameters A the map S(acl(A)) −→ S(A) is open. (For stable
theories, this is just the Open Mapping Theorem.)

8.6 SU-rank and the stability spectrum

We saw that in totally transcendental theories forking is governed by the Morley
rank. The SU-rank, which we define here, generalises this to superstable theories.
We use it to show that the stability spectrum of countable theories is rather
restrictive: there are only four possibilities for the class of cardinals in which a
countable theory is stable.
Definition 8.6.1. Let T be a simple theory. We define SU(p) ≥ α for a type p
by recursion on α:
SU(p) ≥ 0 for all types p;
SU(p) ≥ β + 1 if p has a forking extension q with SU(q) ≥ β;
SU(p) ≥ λ (for a limit ordinal λ) if SU(p) ≥ β for all
β < λ.

and the SU-rank SU(p) of p as the maximal α such that SU(p) ≥ α. If there is
no maximum, we set SU(p) = ∞.
1 It is easy to see that Part 2 can equivalently be replaced by: 2a Any two points on a line

are independent and 2b Any two lines are independent over their intersection.
CHAPTER 8. STABLE THEORIES 150

Lemma 8.6.2. Assume T to be simple. Let p have ordinal valued SU-rank and
let q be an extension of p. Then q is a non-forking extension of p if and only if
q has the same SU-rank as p. If p has SU-rank ∞, then so does any non-forking
extension.
Proof. It is clear that the SU-rank of an extension cannot increase. So it is
enough to show for all α that SU(p) ≥ α implies SU(q) ≥ α whenever q is a
non-forking extension of p ∈ S(A). The interesting case is where α = β + 1
is a successor ordinal. Then p has a forking extension r with SU(r) ≥ β. By
the Diamond Lemma (Exercise 7.2.2) there is an A-conjugate r0 of r with a
non-forking extension s which also extends q. By induction SU(s) ≥ β. But s
is a forking extension of q, so SU(q) ≥ β.

Since every type does not fork over some set of cardinality at most |T |, there
are at most 2|T | different SU-ranks. Since they form an initial segment of the
ordinals, all ordinal ranks are smaller than (2|T | )+ . (Actually one can prove
that they are smaller than |T |+ .) It follows that every type of SU-rank ∞ has
a forking extension of SU-rank ∞.

Definition 8.6.3. A simple theory is supersimple if every type does not fork
over some finite subset of its domain. A stable, supersimple theory is called
superstable.
Note that totally transcendental theories are superstable.

Lemma 8.6.4. The theory T is supersimple if and only if every type has SU-
rank < ∞.
Proof. If SU(p) = ∞, there is an infinite sequence p = p0 ⊆ p1 ⊆ · · · of forking
extensions of SU-rank ∞. The union of the pi forks over every finite subset of
its domain. If p ∈ S(A) forks over every finite subset of A, there is an infinite
sequence A0 ⊆ A1 ⊆ · · · of finite subsets of A such that p  Ai+1 forks over Ai .
This shows that p  ∅ has SU-rank ∞.
Let T be a complete theory. The stability spectrum Spec(T ) of T is the class
of all infinite cardinals in which T is stable.
Theorem 8.6.5. Let T be a countable complete theory. There are four cases:

1. T is totally transcendental. Then Spec(T ) = {κ | κ ≥ ℵ0 }.

2. T is superstable but not totally transcendental. Then Spec(T ) = {κ | κ ≥
2ℵ0 }.
3. T is stable but not superstable. Then Spec(T ) = {κ | κℵ0 = κ}.

4. T is unstable. Then Spec(T ) is empty.

Note that these are really four different possibilities: it follows from Theo-
rem A.3.6 that κℵ0 > κ for all κ with countable cofinality, e.g., for all κ = iω (µ).
CHAPTER 8. STABLE THEORIES 151

Proof. 1: This follows from Theorem 5.2.6.

2: Let T be superstable and |A| = κ. Since every type over A does not fork
over a finite subset of A, an upper bound for the size of S(A) can be computed
as the product of
• the number of finite subsets E of A,

• the number of types p ∈ S(E),

• the number of non-forking extensions of p to A.
So we have | S(A)| ≤ κ·2ℵ0 ·2ℵ0 = max(2ℵ0 , κ). If T is not totally transcendental,
the proof of Theorem 5.2.6 shows that T cannot be stable in cardinals smaller
than 2ℵ0 .
3: If T is stable, then T is κ-stable whenever κℵ0 = κ by Exercise 8.2.7.
If T is not superstable, the proof of Lemma 8.6.4 shows that there is a type
p over the empty set of infinite SU-rank with a forking extension p0 of infinite
SU-rank. Let q be a non-forking global extension of p0 and let κ ≥ ℵ0 . By
Exercise 7.1.5 q has κ many different conjugates qα , (α < κ). Choose A0 of size
κ such that all pα = qα  A0 are different. By Lemma 8.6.2 the pα have again
infinite SU-rank. Continuing in this manner we get a sequence A0 ⊆ A1 ⊆ · · ·
of parameter sets and a tree of types pα0 ,...,αn ∈ S(An+1 ), (n < ω, αi < κ). We
may assume thatSall Ai have cardinality κ. Each path through this tree defines
a type over A = n<ω An . So we have | S(A)| ≥ κℵ0 .
4: Clear from the definition.

The spectrum of uncountable theories is more difficult to describe, see [54,

Chapter III].
Example 8.6.6 (Modules). For any R-module M , the LMod (R)-theory of M
is κ-stable if κ = κ|R|+ℵ0 .

See [47] or [63] for more on the model theory of modules.

Proof. Let B be a subset of some model N of Th(M ), |B| ≤ κ. Let SN (B)
denote the set of all complete types over B which are realised in N . Every type
tp(a/B) is axiomatised by

tp± (a/B) = tp+ (a/B) ∪ tp− (a/B),

where
tp+ (a/B) = {ϕ(x, b) | ϕ pp-formula, b ∈ B, N |= ϕ(a, b)}
and
tp− (a/B) = {¬ϕ(x, b) | ϕ pp-formula, b ∈ B, N |= ¬ϕ(a, b)}.
Clearly, tp− is determined by tp+ .
By Corollary 3.3.8, tp+ (a/B) contains – up to equivalence – at most one
formula ϕ(x, b) for any pp-formula ϕ(x, y). Hence tp+ (a/B) is determined by a
CHAPTER 8. STABLE THEORIES 152

partial map f from the set of pp-formulas to the set of finite tuples in B in the
sense that it is axiomatised by {ϕ(x, f (ϕ)) | ϕ pp-formula}. Hence we have

| SN (B)| ≤ (|B| + ℵ0 )|R|+ℵ0 .

Thus T is κ-stable in every κ with κ = κ|R|+ℵ0 .

Example 8.6.7 (Separably closed fields). The theory SCFp,e of separably closed
fields is stable for all κ with κ = κℵ0 .
Proof. Let L be a model of SCFp,e . Fix a p-basis b1 , . . . , be and consider the
corresponding λ-functions λν . Now let K be a subfield of cardinality κ. By
Theorem 3.3.20 every type tp(a/K) is axiomatised by Boolean combinations of
.
equations t(x) = 0 where the t(x) are L(c1 , . . . , ce , λν )ν∈pe -terms with parame-
ters from K. To compute an upper bound for the number of types over K we
may assume that K contains the p-basis and is closed under the λ-functions. It
is now easy to see that every t(x) is equivalent to a term p(λν̄1 (x), . . . , λν̄k (x))
where p(X1 , . . . , Xk ) ∈ K[X1 , . . . , Xk ] and the λν̄i are iterated λ-functions:

λν̄ = λν 1 ...ν m = λν 1 ◦ . . . ◦ λν m .

So, if λν̄0 , λν̄1 , . . . is a list of all iterated λ-functions, the type of a over K is deter-
mined by the sequence of the quantifier-free L-types of the tuples (λν̄0 (a), . . . , λν̄n (a))
over K. By Example 5.2.3 for each n there are only κ many such types. So we
we can bound the number of types over K by κℵ0 .
Exercise 8.6.1. Show that the types of SU-rank 0 are exactly the algebraic
types. Show also that a type is minimal if and only if it is stationary and has
SU-rank 1.
Exercise 8.6.2. Let T be an arbitrary theory. Define the U-rank (or Lascar
rank ) U(p) of a type p ∈ S(A) as its SU-rank, except for the clause
U(p) ≥ β + 1 if for any κ there is a set B ⊃ A to which p has at
least κ many extensions q with U(q) ≥ β.
Show that in stable theories U-rank and SU-rank coincide.
Exercise 8.6.3. Use Exercise 7.2.5 to show that a simple theory is supersimple
if every 1-type has SU-rank < ∞.
Exercise 8.6.4. Show that in simple theories SU(p) ≤ MR(p).
Exercise 8.6.5 (Lascar inequality). Let T be simple, SU(a/C) = α and SU(b/aC) =
β. Prove2
β + α ≤ SU(ab/C) ≤ β ⊕ α.
If a and b are independent over C, we have SU(ab/C) = β ⊕ α.
2 Ordinal addition was defined in Exercise 6.4.4. The strong sum ⊕ is the smallest function

On × On → On which is strictly monotonous in both arguments.

CHAPTER 8. STABLE THEORIES 153

Exercise 8.6.6. Let ϕ(x, y) be a formula without parameters and k natural

number. Define the rank D(p, ϕ, k) by
D(p, ϕ, k) ≥ β + 1 if p has an extension q with D(q, ϕ, k) ≥ β and
which contains a formula ϕ(x, b) which divides
over the domain of p with respect to k.
Show for simple T :

1. That D(p, ϕ, k) is bounded by a natural number which depends only on ϕ

and k.
2. Let q be an extension of p. Then p is a non-forking extension of p if and
only if D(p, ϕ, k) = D(q, ϕ, k) for all ϕ and k.
Exercise 8.6.7. A countable theory is ω-stable if and only if it is superstable,
small and if every type (over a finite set) has finite multiplicity.
Exercise 8.6.8 (Lachlan). Show that in an ℵ0 -categorical theory there are only
finitely many strong 1-types over a finite set. Conclude that an ℵ0 -categorical
superstable theory is ω-stable.
Note that there are stable ℵ0 -categorical theories which are not ω-stable (see
[26]).
Exercise 8.6.9. Let T be a simple theory. Assume that there is a sequence of
definable equivalence relations E0 ⊆ E1 ⊆ · · · such that every Ei -class contains
infinitely many Ei+1 -classes. Show that T is not supersimple.

Exercise 8.6.10. Use Exercise 8.6.9 to show that a superstable group has no
infinite descending sequence of definable subgroups G = G0 ≥ G1 ≥ G2 ≥ G3 ≥
· · · each of infinite index in the previous one. Conclude from this that SCFp,e
is not superstable if e > 0.
Exercise 8.6.11. Prove that a module M is totally transcendental if and only
if it has the dcc on pp-definable subgroups of M . A module M is superstable
if and only if there is no infinite descending sequence of pp-definable subgroups
each of which is of infinite index in its predecessor.
Chapter 9

Prime extensions

In this chapter we return to questions around the uniqueness of prime extensions.

We will now prove their uniqueness for totally transcendental theories and for
countable stable theories having prime extensions.

9.1 Indiscernibles in stable theories

We assume throughout this section that T is complete, stable and eliminates
imaginaries. Indiscernibles in stable theories are in fact indiscernible for every
ordering of the underlying set. More importantly, we show that they form a
Morley sequence in some appropriate stationary type.
A family I = (ai )i∈I of tuples is totally indiscernible over A, if

C |= ϕ(ai1 , . . . , aik ) ↔ ϕ(aj1 , . . . , ajk )

for all L(A)-formulas ϕ and sequences i1 , . . . , ik , j1 , . . . , jk of pairwise distinct

indices.
Lemma 9.1.1. If T is stable, indiscernibles are totally indiscernible.

Proof. Assume that I = (ai )i∈I is indiscernible over A, but not totally in-
discernible over A. By Lemma 7.1.1 we may assume (I, <) = (Q, <). Be-
cause any permutation on {1, . . . , n} is a product of transpositions of neigh-
bouring elements, there are some L(A)-formula ϕ(x1 , . . . , xn ), rational numbers
r1 < · · · < rn and some j ∈ Q such that

|= ϕ(ar1 , . . . , arj , arj+1 , . . . , arn )

and
|= ¬ϕ(ar1 , . . . , arj+1 , arj , . . . , arn ).
The formula ψ(x, y) = ϕ(ar1 , . . . , x, y, . . . , arn ) orders the elements (ar ), (rj <
r < rj+1 ). By Exercise 8.2.1, this contradicts stability.

154
CHAPTER 9. PRIME EXTENSIONS 155

Let p ∈ S(A) be a stationary type. Recall that a Morley sequence in p is

an A-indiscernible sequence of realisations of p independent over A. Morley
sequences (aα )α<λ are easy to construct as follows: choose aα realising the
unique non-forking extension of p to A ∪ {aβ | β < α} (see Example 7.2.10).
Since any well-ordered Morley sequence arises in this way, Morley sequences in
p are uniquely determined by their order-type up to isomorphism over A.

Theorem 9.1.2. If T is stable, then any infinite sequence of indiscernibles over

A is a Morley sequence for some stationary type defined over some extension
of A.
Proof. Let I = (ai )i∈I be indiscernible over A. Notice that for all formulas
ϕ(x, b) the set
Jϕ = {i ∈ I| |= ϕ(ai , b)}
is finite or cofinite in I: otherwise for all J ⊆ I the set of formulas

{ϕ(ai , y) | i ∈ J} ∪ {¬ϕ(ai , y) | i 6∈ J}

would be consistent. So there would be 2|I| many ϕ-types over I, contradicting

stability of T .
This shows that for every ϕ either Jϕ or I \Jϕ is bounded by some kϕ (which
depends only on ϕ).
The average type of I is a global type defined by

Av(I) = {ϕ(x, b) | b ∈ C, |= ϕ(ai , b) for all but finitely many i ∈ I}.

By the preceding remarks, this is a complete type. Let I0 be an infinite

subset of I. Since ϕ(x, b) ∈ Av(I) if and only if {i ∈ I0 | |= ϕ(ai , b)} contains
more than kϕ many (and hence infinitely many) elements, Av(I) is definable
over I0 . Hence Av(I) does not fork over I0 and its restriction to I0 is stationary
(see Theorem 8.5.1 and Corollary 8.5.7.)
It is easy to see that all ai , i ∈ I \ I0 , realise the type

p = Av(I)  AI0 .

As this is also true for all I00 ⊃ I0 , we see that ai , i ∈ I \ I0 , forms a Morley
sequence for p.
At the beginning of the proof we can now replace I by an infinite set of
indiscernibles I 0 containing I as a coinfinite subset which shows I to be a
Morley sequence for p0 = Av(I 0 )  A(I 0 \ I).
Exercise 9.1.1. If p is stationary and q a non-forking extension of p, then any
Morley sequence of q is also a Morley sequence for p.
Exercise 9.1.2. Let p ∈ S(A) be stationary and I a Morley sequence of p.
a) Let B ⊃ A and I0 ⊆ I such that B ^ | AI I. Then I \I0 is a Morley sequence
0
of the non-forking extension of p to B.
CHAPTER 9. PRIME EXTENSIONS 156

b) The type Av(I) is the non-forking global extension of p.

Exercise 9.1.3. We call indiscernibles I0 and I1 parallel if there is some infinite
set J such that I0 J and I1 J are indiscernible. Show that I0 and I1 are parallel
if and only if they have the same average type.
Exercise 9.1.4. Show that two types are parallel if and only if two (or all) of
their infinite Morley sequences are parallel.
Exercise 9.1.5. Show the converse of Lemma 9.1.1: if all indiscernibles se-
quences are totally indiscernible, then T is stable.

9.2 Totally transcendental theories

Let T be a totally transcendental theory. We will prove the following theorem:
Theorem 9.2.1 (Shelah [53]). Let T be totally transcendental.
1. A model M is a prime extension of A if and only if M is atomic over A
and does not contain an uncountable set of indiscernibles over A.
2. Prime extensions are unique.
We first aim to show that a constructible set does not contain an uncountable
set of indiscernibles.

Lemma 9.2.2. Let I be indiscernible over A and B a countable set. Then I

contains a countable subset I0 such that I \ I0 is indiscernible over ABI0 .
Proof. By Theorem 9.1.2, I is a Morley sequence of some stationary types
over some extension A0 of A. Since T is totally transcendental, we only need
to extend A by finitely many elements (this follows using Exercise 8.5.6 and
Exercise 9.1.1). So we may assume A0 = A. For every finite tuple b in B there
is some finite set I0 such that b ^
| AI I. In this way we find a countable set I0
0
with
BI0 ^| I.
AI0

It now follows from Exercise 9.1.2 that I\I0 is a Morley sequence over ABI0 .

We need a bit of set theory: a club D ⊆ ω1 is a closed and unbounded subset

where closed means that sup(α ∩ D) ∈ D for all α ∈ ω1 .
Theorem 9.2.3 (Fodor). Let D ⊆ ω1 be a club and f : D → ω1 a regressive
function, i.e., f (α) < α for all α ∈ D. Then f is constant on an unbounded
subset of ω1 .

Proof. Suppose not; then for each α ∈ ω1 , the fibre Dα = {x ∈ D | f (x) = α}

is bounded. Construct a sequence α0 < α1 < · · · of elements of D as follows.
CHAPTER 9. PRIME EXTENSIONS 157

Let α0 be arbitrary. If αn is constructed, choose for αn+1 an upper bound of

S
β<αn Dβ . So we have for all γ ∈ D

γ ≥ αn+1 ⇒ f (γ) ≥ αn .
For δ = supn<ω αn , this implies f (δ) ≥ δ. A contradiction.
Recall from Section 5.3 that a set B = {bα }α<µ is a construction over A
if all tp(bα /ABα ) are isolated where as above Bα = {bβ | β < α}). A subset
C ⊆ B is called construction closed if for all bα ∈ C the type tp(bα /ABα ) is
isolated by some formula over A ∪ (Bα ∩ C).
The following lemma holds for arbitrary T .
Lemma 9.2.4. Let B = {bα }α<µ be a construction over A.
1. Any union of construction closed sets is construction closed.
2. Any b ∈ B is contained in a finite construction closed subset of B.
3. If C ⊆ B is construction closed, then B is constructible over AC.
Proof. The first part is clear. For the second part, let b = bα ∈ B. Since the
type of b = b0 is isolated over A, we can do induction on α. As {bβ }β<µ is
a construction over A, tp(bα /ABα ) is isolated by some formula ϕ(x, c) where
c = bβ1 . . . bβn with βi < α. By induction, each bβi is contained in a finite
construction closed set Ci . Thus C1 ∪ · · · ∪ Cn ∪ {bα } is finite and construction
closed.
For Part 3 we assume A = ∅ to simplify notation. We will show that the
type tp(bα /CBα ) is isolated for all α. This is clear if bα ∈ C. So assume bα 6∈ C.
From the assumption it is easy to see that C is isolated over Bα+1 where the
isolating formulas only contain parameters from Bα+1 ∩ C ⊆ Bα .
We thus have
tp(C/Bα ) ` tp(C/Bα bα ).
If ϕ(x) isolates the type tp(bα /Bα ), then ϕ(x) also isolates the type
tp(bα /CBα ).
Lemma 9.2.5. If B is constructible over A, then B does not contain an un-
countable set of indiscernibles over A.
Proof. We assume A = ∅. Let I = {cα }α<ω1 be indiscernible. By Lemma 9.2.4
we can build a continuous sequence Cα of countable construction closed subsets
of B such that cα ∈ Cα+1 . By Lemma 9.2.2 there is a club D consisting of limit
ordinals such that for all δ ∈ D the set {cα | α ≥ δ} is indiscernible over Cδ .
Each cδ is isolated over Cδ by a formula with parameters from Cδ0 with δ 0 < δ.
By Fodor’s Theorem there is some δ0 such that for some cofinal set of δs from
D the parameters can be chosen in Cδ0 . Assume that δ1 < δ2 are such elements.
Then cδ1 and cδ2 have the same type over Cδ0 . But this is impossible since
tp(cδ2 /Cδ0 ) ` tp(cδ2 /Cδ0 cδ1 ).
CHAPTER 9. PRIME EXTENSIONS 158

Let M be a model and A ⊆ M . We call a subset B of M normal in M over

A if for every element b ∈ B all realisations of tp(b/A) in M are contained in B.
Lemma 9.2.6. Let T be a (not necessarily totally transcendental) theory which
has prime extensions. If M is atomic over A and B is a normal subset of M
containing A, then M is atomic over B.

Proof. Let c be a tuple from M , so tp(c/A) is isolated. Since the isolated types
are dense over B by Theorem 4.5.7, there is some d ∈ M atomic over B and
realising the type tp(c/A). Let tp(d/B) be isolated by ψ(x, d0 ) for some tuple
d0 ∈ B. Since c and d satisfy the same type over A there is some c0 ∈ M such
that tp(cc0 /A) = tp(dd0 /A). Then c satisfies ψ(x, c0 ) and as B is normal, we
have c0 ∈ B. It follows easily that ψ(x, c0 ) is complete over B as well.
Proof of Theorem 9.2.1(1). If M is a prime extension of A, then M is atomic
over A by Corollary 5.3.7 and since M can be embedded over A into some con-
structible prime extension, by Lemma 9.2.5 M does not contain an uncountable
set of indiscernibles over A.
For the converse assume again A = ∅, i.e., suppose M is atomic over ∅ with-
out uncountable set of indiscernibles. In order to prove that M can be embedded
into any model N we enumerate all types over ∅ realised inSM as (pµ )µ<ν and
recursively extend the empty map to the normal sets Cµ = α<µ pα (M ). That
this is possible follows from the following.
Claim. Let M be atomic over B, p ∈ S(B) and B ⊆ C ⊆ M normal over
B. Then any elementary map C −→ N can be extended to C ∪ p(M ).
We proof the claim by induction on the Morley rank of p and note that the
claim is clear if p is algebraic.
Assume inductively that the claim is proved for all types of Morley rank less
than α (over arbitrary sets B). Then any given elementary map f : C −→ N
with B ⊆ C ⊆ M and C normal over B can be extended to C ∪ {a ∈ M |
MR(a/B) < α}.
Let now p ∈ S(B) with MR(p) = α. Let {ci }i<µ be a maximal set of
realisations of p in M independent over B. By Exercise 9.2.1, {ci }i<µ splits into
a finite number of indiscernible sequences, which implies that µ is countable,
and we can assume that µ ≤ ω. Let Bi = B ∪ {c0 , . . . , ci−1 } and CiS= C ∪ {a ∈
M | MR(a/Bi ) < α}, so B = B0 . By maximality we have p(M ) ⊆ i<µ Ci . As
M is atomic over B, it is also atomic over Bi and since Ci is normal over Bi even
atomic over Ci . If f has been extended to Ci , we may extend f to Ci Bi+1 by
the atomicity. By the induction hypothesis applied to Bi+1 and Ci Bi+1 there
is an extension to Ci+1 .
The proof of 9.2.1(1) can easily be symmetrised to yield 9.2.1(2).
Example (Shelah [55]). Let L contain a binary relation symbol Eα for every
ordinal α < ω1 and let T be the theory stating that each Eα is an equivalence
relation such that E0 consists of only one class and for any α < β < ω1 each Eα -
equivalence class is the union of infinitely many Eβ -equivalence classes. Then
T is complete, stable (but not totally transcendental) and admits quantifier
CHAPTER 9. PRIME EXTENSIONS 159

elimination. Every consistent L(A)-formula ϕ(x) can be completed over A.

Therefore there exists a prime model M which is constructible over the empty
set.
In M any chain (Kα )α<ω1 of Eα -equivalence classes has non-empty inter-
section: otherwise there would be a countable subset A of M and some limit
ordinal η < ω1 such that:
(i) A is construction closed (with respect to a fixed construction);
T
(ii) A ∩ α<η Kα = ∅;
(iii) A ∩ Kα 6= ∅ for all α < η.

By (i) M/A is atomic. Let c ∈ Kη ; then tp(c/A) is isolated by some formula

ϕ(x; a); by (ii) there exists some α < η such that a ∩ Kα = ∅. By (iii) there
is some d ∈ A ∩ Kα ; since a ∩ Kα = ∅, it would implyT |= ϕ(d; a), but this is
impossible as ϕ isolates tp(c/A). Therefore we have (Kα )α<ω1 6= ∅.
Now let a ∈ M and let N be the set of all b from M for which there is some
ordinal α < ω1 with |= ¬aEα b. Then N is also a prime model, but M and N
are not isomorphic.
Exercise 9.2.1. Let T be totally transcendental, and I be an independent set
of realisations of p ∈ S(A). Then I can be decomposed into a finite number
I1 ,. . . ,In of indiscernible sets over A.

9.3 Countable stable theories

We assume throughout this section that T is countable and stable. For such T
we will show that prime extensions, if they exist, are unique. The main point
is Shelah’s result that in this situation subsets of constructible sets are again
constructible.
The proof of Theorem 9.2.1(1) showed that atomic extensions of A with-
out uncountable sets of indiscernibles are constructible and hence prime. The
uniqueness of such prime extensions then also follows directly from the following
theorem which holds for arbitrary theories.

Theorem 9.3.1 (Ressayre). Constructible prime extensions are unique.

Proof. It suffices to prove the theorem for constructible prime extensions M and
M 0 over the empty set. Let f0 : E0 → E00 be a maximal elementary map between
construction closed subsets E0 ⊆ M and E00 ⊆ M 0 . If E0 6= M , there is some
proper construction closed finite extension E1 of E0 . Since E1 is atomic over
E0 by Lemma 9.2.4(3) and Corollary 5.3.6 there is an extension f1 : E1 → E10 of
f0 . Then E10 need not be construction closed, but there is a construction closed
finite extension E20 of E10 . Similarly there exists a (not necessarily construction
closed) set E2 ⊆ M and some extension f1 to an isomorphism f2 : E2 → E20 .
Continuing in this way we obtain an ascending chain of elementary isomorphisms
CHAPTER 9. PRIME EXTENSIONS 160

0 0
S S 0
fi : Ei →
S Ei . Then E∞ := Ei and E∞ := Ei are construction closed and
0
f∞ := fi is an elementary isomorphism from E∞ to E∞ , contradicting the
maximality of f0 .
Theorem 9.3.2 (Shelah [55]). If T is stable and countable, then any subset of
a set constructible over A is again constructible over A.
We immediately obtain the following corollary.
Corollary 9.3.3. If T is a countable stable theory with prime extensions (see
Exercise 5.3.2), then all prime extensions are unique.
For the proof of the theorem we need the following lemma.
Lemma 9.3.4. Let T be countable and stable; if A and B are independent over
C and B 0 is countable, then there is a countable subset C 0 of A with A and BB 0
independent over CC 0 .
Proof. Using the properties of forking, we find a countable subset C 0 ⊆ A with
ABC ^ | BCC 0 B 0 ; then A ^
| BCC 0 B 0 , and since A ^ | CC 0 BB 0 .
| CC 0 B we have A ^

Proof of Theorem 9.3.2. Let B be constructible over A and D a subset of B.

We may assume that B is infinite since finite sets are constructible. If E is an
arbitrary construction closed subset of B and E 0 ⊆ B a countable extension of
E (i.e., E 0 \ E is countable), then there is some countable construction closed
extension E 00 of E 0 . Similarly, for any E with D ^
| A(D∩E) E and for every count-
able extension E of E, there is a countable extension E 00 with D ^
0
| A(D∩E 00 ) E 00
(Lemma 9.3.4).
Applying these closure procedures alternatingly countably many times, one
sees that for any construction closed subset E of B with
D |
^ E
A(D∩E)

and for any countable extension E 0 of E there exists a construction closed count-
able extension E 00 of E 0 such that
D |
^ E 00 .
A(D∩E 00 )

In this way we
S obtain a continuous chain (Cα )α<ξ of construction closed sets
with C0 = ∅, α<ξ Cα = B, countable differences Cα+1 \ Cα and
D |
^ Cα .
A(D∩Cα )

For each α we can choose an ω-enumeration of D ∩ (Cα+1 \ Cα ). These

enumerations can be composed to an enumeration of D. In order to show that
this enumeration is a construction of D it suffices to show that every initial
segment d of D ∩ (Cα+1 \ Cα ) is atomic over A(D ∩ Cα ). This follows from the
Open Mapping Theorem (8.5.9) as d is atomic over ACα .
Chapter 10

The fine structure of

ℵ1-categorical theories

10.1 Internal types

By the results in Sections 5.8 and 6.3 we know that models of ℵ1 -categorical
theories are (minimal) prime extensions of strongly minimal sets. We will see in
the next section that in this case the prime extensions M of ϕ(M ) are obtained
in a particularly simple way, namely as an iterated fibration where each fibre is
the epimorphic image of some ϕ(M )k . Unless stated otherwise, we assume in
this section that T is totally transcendental.
We need the concept of an internal type. We fix a 0-definable infinite sub-
class F = ϕ0 (C) of C.
Definition 10.1.1. A partial type π(x) over the empty set is called F-internal
if for some set B, the class π(C) is contained in dcl(FB).

Lemma 10.1.2. π is F-internal if and only if there is some finite conjunction

ϕ of formulas in π which is F-internal.
Proof. The complement of dcl(FB) can be defined by a partial type σ(x). So
π(C) ⊆ dcl(FB) if and only if π ∪ σ is inconsistent and this is witnessed by a
finite part of π.
Example. Let G be a group. Let

M = (G, A)

be a two-sorted structure where A is a copy of G without the group structure.

Instead, the structure M contains the map

π: G × A → A

161
CHAPTER 10. THE FINE STRUCTURE OF ℵ1 -CATEGORICAL THEORIES162

defined as π(g, a) = ga. Then M is prime over G and clearly the type p of an
element in A is G-internal, in fact, p(M ) ⊆ dcl(G, a) for any a ∈ A. We will see
in Corollary 10.1.7 that this is the typical picture for internal types.
Lemma 10.1.3. A type p ∈ S(A) is F-internal if and only if there is some set
of parameters B and some realisation e of p such that e ∈ dcl(FB) and e ^
| A B.

Proof. If p(C) ⊆ dcl(FB), we just choose e as a realisation of p independent of

B over A.
Conversely, given a realisation e of p and some b ∈ B with e ^ | A b and
e ∈ dcl(Fb), we choose a Morley sequence b0 , b1 , . . . of tp(b/ acleq (A)) of length
|T |+ . By Exercise 7.2.1 for each e0 ∈ p(C) there is some i such that e0 ^ | A bi .
If p is stationary, then eb and e0 bi realise the same type over A and hence
e0 ∈ dcl(Fbi ). So p(C) ⊆ dcl(FB) for B = {b0 , b1 , . . .}. If p is not stationary,
then by the previous argument applied to each extension of p to acleq (A) these
are all F-internal. By taking unions we obtain a set B such that the realisations
of all extensions of p to acleq (A) are contained in dcl(FB).
Lemma 10.1.4. A consistent formula ϕ is F-internal if and only if there is a
definable surjection from some Fn onto ϕ(C).
Proof. If there is a B-definable surjection Fn → ϕ(C), ϕ(C) is contained in
dcl (FB). For the converse assume that ϕ(C) ⊆ dcl (FB). Then, by Exer-
cise 6.1.12, for every e ∈ ϕ(C) there is a B-definable class De ⊆ Fne and a
B-definable map fe : De → C such that e is in the image of fe . A compactness ar-
gument shows that there is a finite number of definable classes D1 , . . . , Dm ⊆ Fn
and definable maps fi : Di → C such that ϕ(C) is contained in the union of the
fi (Di ). Fix a sequence of distinct elements a1 , . . . , am ∈ F and an element b of
ϕ(C). Define f : Fn+1 → ϕ(C) by setting f (x, y) = fi (x) if y = ai and x ∈ Di
and f (x, y) = b otherwise. Then f is a surjection from Fn+1 onto ϕ(C).

Lemma 10.1.5. Let T be an arbitrary theory and F a stably embedded 0-

definable class. If a and b have the same type over F, they are conjugate under
an element of Aut(C/F).
Proof. We construct the automorphism as the union of a long ascending se-
quence of elementary maps α : A ∪ F → B ∪ F which are the identity on F.
Assume that α is constructed and consider an element c ∈ C. Since F is stably
embedded, the type of cA over F is definable over some subset C of F. Choose
some d ∈ C with tp(dB/C) = tp(cA/C). We can then extend α to an elemen-
tary map α0 : {c} ∪ A ∪ F → {d} ∪ B ∪ F. To see this assume that |= ϕ(c, a, f ),
where a ∈ A and f ∈ F. Then ϕ(x, y, f ) belongs to the type of cA over F, so
ϕ(x, y, f ) belongs also to the type of dB over F which shows |= ϕ(d, α(a), f ).

A groupoid is a (small) category in which all morphisms are invertible, i.e.,

isomorphisms. A groupoid is connected if there are morphisms between any two
objects. A definable groupoid G is one whose objects are given by a definable
family (Oi )i∈I of classes and whose morphisms by a definable family (Mi,j )i,j∈I
CHAPTER 10. THE FINE STRUCTURE OF ℵ1 -CATEGORICAL THEORIES163

of bijections Oi → Oj . We use the notation HomG (Oi , Oj ) and AutG (Oi ) for
Mi,j and Mi,i .
We denote by Feq ⊆ Ceq the collection of those 0-definable equivalence classes
having representatives in F.
Theorem 10.1.6 (Hrushovski’s Binding Groupoid). Let T be totally transcen-
dental, E and F be 0-definable and assume that E is F-internal and non-empty.
Then E is an object of a 0-definable connected groupoid G with the following
properties:
a) E is not the only object in G. The objects other than E are subsets of Feq .

b) AutG (E) = Aut(E/F).

The group AutG (E) is Zilber’s binding group (see also [46]).
Proof. By Lemma 10.1.4 there is a definable surjection from some power Fn to
E. This induces a bijection f : O → E for some definable class O of Feq . Let
f = fc be defined from a parameter c and O = Od from a parameter d. By
Lemma 8.3.3 we can find d in F(M0 ) for an atomic model M0 of T . Since,
by Lemma 5.3.4, the isolated types are dense over any parameter set, we may
assume that the type of c over F ∩ M0 is isolated, say by a formula ϕ(x, a). It is
easy to see that ϕ(x, a) isolates a type over F. By extending d if necessary we
may assume that d = a. Now let ψ(y) isolate the type of d.
As objects of G we take E and the Oe where e realises ψ(y) and we let
HomG (Oe , E) be the set of all fc0 with c0 realising ϕ(x, e). We claim that
HomG (Oe , E) is a right coset of Aut(E/F), namely, if c0 realises ϕ(x, e), we
have
HomG (Oe , E) = Aut(E/F) ◦ fc0 .
This follows easily from the fact that, by Lemma 10.1.5, the elements of HomG (Oe , E)
are of the form fα(c0 ) for some automorphism α ∈ Aut(C/F) and from the for-
mula
fα(c0 ) = α ◦ fc0 .
We can now set

HomG (E, Oe ) = {f −1 | f ∈ HomG (Oe , E)}

HomG (E, E) = {f ◦ g −1 | f, g ∈ HomG (Oe , E)}
HomG (Oe , Oe0 ) = {f −1 ◦ g | f ∈ HomG (Oe0 , E), g ∈ HomG (Oe , E)}.

Recall that a group G acts regularly on a set A if for all a, b ∈ A there exists
a unique g ∈ G with ga = b.
Corollary 10.1.7 (Binding group). Let T be a totally transcendental theory, E
and F be 0-definable and assume that E is F-internal. Then the following holds.
CHAPTER 10. THE FINE STRUCTURE OF ℵ1 -CATEGORICAL THEORIES164

1. There is a definable group G ⊆ Feq , the binding group, and a definable

class A on which G acts regularly and such that E ⊆ dcl(Fa) for all a ∈ A.
The group G, the class A, the group operation and the action of G on A
are definable with parameters from F.
2. Aut(E/F) is a 0-definable permutation group, Aut(E/F) acts regularly on
A and is definably isomorphic to G.
Proof. Let G be the groupoid of Theorem 10.1.6. Fix any object Oe different
from E. Set G = AutG (Oe ) and A = HomG (Oe , E). Now replace the definable
bijections in G and A by their canonical parameters in order to obtain elements
of Feq and Ceq respectively.
Remark 10.1.8. Note that in the last corollary Aut(E/F) is infinite if E 6⊆
acleq (F).
Exercise 10.1.1. Let C be F-internal. Show that every model M is a minimal
extension of F(M ).
Exercise 10.1.2. Let T , E and F be as in Theorem 10.1.6.
1. Show that there is a finite subset A of E such that E is contained in
dcl(F ∪ A).
2. Show that the converse of Remark 10.1.8 is also true: Aut(E/F) is finite
if E is contained in acl(F).
Exercise 10.1.3. Prove that every element of dcleq (F) is interdefinable with
an element of Feq .
Exercise 10.1.4. Let T be arbitrary, F 0-definable and C a subset of F. Show
that tp(a/F) is definable over C if and only if tp(a/C) ` tp(a/F).
Exercise 10.1.5 (Chatzidakis–Hrushovski, [17]). Let T be arbitrary and F
0-definable. Show that the following are equivalent:
a) F is stably embedded.
b) Every type tp(a/F) is definable over a subset C of F.
c) For every a there is a subset C of F such that tp(a/C) ` tp(a/F).
d) Every automorphism of F extends to an automorphism of C.

10.2 Analysable types

Throughout this section we assume that T is a stable theory eliminating imagi-
naries.
Definition 10.2.1. Let F be a 0-definable class. A type p ∈ S(∅) is called F-
analysable if for every realisation a of p there is a sequence of tuples a0 , . . . , an =
a in dcl(a) such that tp(ai /a0 . . . ai−1 ) is F-internal for i = 0, . . . , n.
CHAPTER 10. THE FINE STRUCTURE OF ℵ1 -CATEGORICAL THEORIES165

Theorem 10.2.2 (Hrushovski [27]). Let T be ℵ1 -categorical and F a 0-definable

strongly minimal set. Then every type p ∈ S(∅) is F-analysable.
We need some preparation in order to prove this theorem.
Theorem 10.2.3. Let p ∈ S(A) be a stationary type and I an infinite Morley
sequence for p. Then Cb(p) ⊆ dcl(I).
Proof. By Exercise 9.1.2(b) the average type Av(I) is the non-forking extension
of p to the monster model. The proof of Theorem 9.1.2 and Lemma 8.5.8 imply
that Av(I) is based on I.
Definition 10.2.4. 1. We call two types p, q ∈ S(A) almost orthogonal if
any realisation of p is independent over A from any realisation of q.
2. Two types over possibly different domains are orthogonal if all non-forking
extensions to any common domain are orthogonal.
3. A theory T is called unidimensional if all stationary non-algebraic types
are pairwise non-orthogonal.

Note that algebraic types are orthogonal to all types.

Let F be a strongly minimal set. We call a type p ∈ S(A) orthogonal to F if
for every realisation b of p, any c ∈ F and any extension B of A over which F is
defined we have

| B
b^ | c.
=⇒ b ^
A B

If q is a type of Morley rank 1 containing F(x), this is equivalent to p being

orthogonal to q.
Lemma 10.2.5. Let T be ℵ1 -categorical and F a strongly minimal set. Then
no non-algebraic type is orthogonal to F.
Proof. Let F be defined over A and suppose that p ∈ S(A) is orthogonal to F.
Choose a model M containing A, a realisation b of p independent from M over
A, and a model N prime over M b. Then b is independent from F(N ) over M .
By Theorem 5.8.1, there is some c ∈ F(N ) \ M . Let ϕ(x, y) ∈ L(M ) so that
ϕ(x, b) isolates the type of c over M b. Then ϕ(x, b) cannot be realised in F(M )
and hence must be algebraic. So tp(c/M b) and by symmetry tp(b/M c) fork
over M , contradicting the choice of b.
Proof of Theorem 10.2.2. By induction on α = MR(p). If α = 0, p is alge-
braic and hence trivially internal (in any definable class). If α > 0, we apply
Lemma 10.2.5 and find a realisation b of p, some c ∈ F and some set of param-
eters B such that b ^ | B and b ^6 | B c. By finite character of forking we may
assume that B is finite.
CHAPTER 10. THE FINE STRUCTURE OF ℵ1 -CATEGORICAL THEORIES166

Let D be the canonical base of tp(cB/ acl(b)). As cB ^ | D b, but cB ^

6 | b,
6 | D. Let (ci Bi ) be an infinite Morley sequence for tp(cB/D). By
we have b ^
Theorem 10.2.3, we have

D ⊆ dcl(c0 B0 c1 B1 . . .).

It follows from b ^| B (and D ⊆ acl(b)) that we must have D ^ | B and hence

D^ | B0 B1 . . .. This shows that the type of any tuple d ∈ D is F-internal by
Lemma 10.1.3. We choose a finite tuple d ∈ D with b ^ 6 | d. Then we have
MR(b/d) < α. We may absorb the parameter d into the language and apply
the induction hypothesis to T (d) = T ∪ {ϕ(d) | ϕ ∈ tp(d)} to find a sequence
b1 , . . . , bn = b such that bi ∈ dcl(db) and the types tp(bi /db1 . . . bi−1 ) are F-
internal. Setting b0 = d, we would be done if we knew that d ∈ dcl(b). For this
we replace d by the canonical parameter d0 of the finite set {d1 , . . . , dk } of conju-
gates of d over b. We have thus achieved d0 ∈ dcl(b). Because d0 ∈ dcl(d1 . . . dk )
the type of d0 is F-internal and since d ∈ acl(d0 ), we have MR(b/d0 ) ≤ MR(b/d) <
α. We can use this d0 to finish the proof.

A complete theory is called almost strongly minimal if there is a strongly

minimal formula ϕ (possibly containing parameters) such that C is in the alge-
braic closure of ϕ(C) ∪ A for some set A ⊆ C.
Corollary 10.2.6 (Zilber). Let T be an ℵ1 -categorical theory. If there is no
infinite group definable in T eq , then T is almost strongly minimal.

Proof. Let T be an ℵ1 -categorical theory and let F be a 0-definable strongly

minimal set, possibly after adding parameters. If T is not almost strongly
minimal, we can use Theorem 10.2.2 to find a definable class E which is F-
internal but not contained in acleq (F). Then Aut(E/F) is an infinite definable
group by Corollary 10.1.7 and Remark 10.1.8.

Theorem 10.2.7 (Baldwin [3]). Any ℵ1 -categorical theory has finite Morley
rank.
To prove Theorem 10.2.7 we need the following definition which allows us
to extend additivity of Morley rank beyond strongly minimal sets (see also
Proposition 6.4.9 and Exercise 6.4.3).

Definition 10.2.8. Let f : B → A be a definable surjection. We say that the

fibres of f have definable Morley rank if there is a finite bound for the Morley
rank of the fibres f −1 (a) and if for every definable B0 ⊆ B and every k 0 the class
{a ∈ A | MR(f −1 (a) ∩ B0 ) = k 0 } is definable.
Remark 10.2.9. If B is a power of a strongly minimal set, the fibres of f have
definable Morley rank by Corollary 6.4.4.
For the next statement remember that the Morley rank of the empty set is
defined as −∞.
CHAPTER 10. THE FINE STRUCTURE OF ℵ1 -CATEGORICAL THEORIES167

Lemma 10.2.10. If the fibres of f : B → A have definable Morley rank and

MR(A) is finite, we have

MR(B) = max(MR(Ak ) + k),

k<ω

where Ak = {a ∈ A | MR(f −1 (a)) = k}.

Proof. We leave it as an exercise (Exercise 10.2.1) to show that MR B ≥ MR Ak +
k for all k. For the converse we may assume that all fibres have Morley rank k
and that A has Morley degree 1 and Morley rank β. We show MR B ≤ β + k
by induction on β.
Let Bi be an infinite family of disjoint definable subsets of B. We want to
show that one of the Bi has smaller Morley rank than β + k. Consider any
a ∈ A. Then for some i the fibre f −1 (a) ∩ Bi must have rank smaller than k. So
the intersection of all Aik = {a | MR(f −1 (a) ∩ Bi ) = k} is empty, which implies
that one of the Aik has smaller rank that β. Induction yields MR Bi < β + k.
Proof of Theorem 10.2.7. It is enough to prove that every element a has finite
Morley rank over ∅. Each a has an analysing sequence a0 , . . . , an = a where
all types tp(ai /a0 . . . ai−1 ) are F-internal. We prove by induction on n that the
tuple a0 . . . an has finite Morley rank. By Lemma 6.4.1 this implies that an has
finite Morley rank.
By the induction hypothesis, a0 . . . an−1 is contained in a 0-definable set A
of finite Morley rank. Since tp(an /a0 . . . an−1 ) is F-internal, an is contained
in an (a0 . . . an−1 )-definable set which is an image of some power of F by a
definable map. So we may assume that a0 . . . an belongs to a 0-definable set
B which projects onto A by the restriction map π : B → A and such that the
fibres π −1 (a) are definable images of some power of F. By Corollary 10.2.9 the
fibres of π have definable Morley rank. If the rank of the fibres is bounded by
k, Lemma 10.2.10 bounds the rank of B by MR A + k.
We end this section with a different characterisation of ℵ1 -categorical theo-
ries due to Erimbetov [18].
Theorem 10.2.11. A countable theory T is ℵ1 -categorical if and only if it is
ω-stable and unidimensional.
Proof. Assume first that T is ℵ1 -categorical. Let F be strongly minimal, defined
over A, p and q be two stationary types over A. By Lemma 10.2.5 there is an
extension A ⊆ B, realisations a, b, c1 , c2 of p, q and F such that a ^ | A B,
b^ | A B, a ^ 6 | B c2 . That means that c1 ∈ acl(aB) \ acl(B) and c2 ∈
6 | B c1 , b ^
acl(bB) \ acl(B). So c1 and c2 have the same type over B and we may assume
that c1 = c2 . But then c1 ^ 6 | B c1 implies a ^
6 | B b and p and q are not orthogonal.
For the converse assume that T is ω-stable and unidimensional. The proof
of the Baldwin–Lachlan Theorem (5.8.1 shows that it is enough to prove that
there are no Vaughtian pairs M  N for strongly minimal formulas ϕ(y) defined
over M . Let a be any element in N \ M and p(x) = tp(a/M ). By assumption
there is an extension M 0 of M and an element c ∈ ϕ(C)\M 0 such that a ^ | M M0
CHAPTER 10. THE FINE STRUCTURE OF ℵ1 -CATEGORICAL THEORIES168

and c ∈ acl(aM 0 ). Let δ(a, m0 , y) be a formula which isolates the type of c over
aM 0 . Then the following sentences are true in M 0

dp x∃yδ(x, m0 , y)
dp x∀y(δ(x, m0 , y) → ϕ(y))
∀y dp x¬δ(x, m0 , y).

So we find an m ∈ M for which the corresponding sentences are true in M .

This implies that there is a b ∈ B such that N |= δ(a, m, b) and that all such b
lie in ϕ(N ) \ M . So M  N is not a Vaughtian pair for ϕ.
Exercise 10.2.1. Let f : B → A definable and assume that all fibres have
Morley rank at least α. Then MR B ≥ α + MR A.
Exercise 10.2.2. Two types p ∈ Sm (A) and q ∈ Sn (A) are weakly orthogonal ,
if p(x) ∪ q(y) axiomatises a complete type in Sm+n (A). Show
1. p and q are weakly orthogonal if and only if for any realisation a of p, q
has a unique extension to Aa.
2. In a simple theory two stationary types p and q are weakly orthogonal if
and only if they are almost orthogonal.

Exercise 10.2.3. Assume T simple and p and p0 two stationary parallel types.
Then p is orthogonal to a type q if and only if p0 is.
Exercise 10.2.4. In a simple theory call a stationary, non-algebraic type p ∈
S(A) regular if it is orthogonal to every forking extension. Prove the following

1. If T is stable, then also every type parallel to p is regular.

2. cl(B) = {c ∈ p(C) | B ^
6 | A c} defines a pregeometry on p(C).
3. Dependence is transitive if the middle element realises a regular type:
6 | A c, c ^
if b ^ 6 | A d and c realises p, then b ^
6 | A d. It follows that non-
orthogonality is an equivalence relation on the class of regular types.

10.3 Locally modular strongly minimal sets

In this section, we let T denote a complete stable theory and ϕ(x) a strongly
minimal formula without parameters.
We call ϕ modular if its pregeometry is modular in the sense of Defini-
tion C.1.9, i.e., if for all relatively algebraically closed A, B in ϕ(C)

dim(A ∪ B) + dim(A ∩ B) = dim(A) + dim(B). (10.1)

We say T is locally modular if (10.1) holds whenever A ∩ B contains an element

not in acl(∅).
CHAPTER 10. THE FINE STRUCTURE OF ℵ1 -CATEGORICAL THEORIES169

It is easy to see that ϕ is modular if and only if any two relatively alge-
braically closed subsets A and B of ϕ(C) are independent over their intersection:

| B;
A ^
A∩B

see Lemma C.1.10. In fact this holds for arbitrary sets B, not necessarily con-
tained in ϕ(C).
Lemma 10.3.1. If ϕ is modular, then

| B
A ^
A∩B

for algebraically closed B and any A which is relatively algebraically closed in

ϕ(C).
Proof. Let C be the intersection of B and ϕ(C). It is enough to show that A
is independent from B over C. For this we may assume that B is the algebraic
closure of a finite set and the elements of A are algebraically independent over
C. We have to show that the elements of A remain algebraically independent
over B. Choose a B-independent sequence A0 , A1 , . . . of sets realising the same
type as A over B. For any i the intersection of acl(A0 . . . Ai ) and acl(Ai+1 ) is
contained in B. So by local modularity A0 . . . Ai and Ai+1 are independent over
C. This implies that the elements of A0 ∪ A1 ∪ · · · are algebraically independent
over C. By Exercise 9.1.2 for some i the elements of Ai ∪ Ai+1 ∪ · · · are alge-
braically independent over B. Hence also the elements of A are algebraically
independent over B.
Definition 10.3.2. A formula ψ(x) without parameters is 1-based if

A |
^ B
acleq (A) ∩ acleq (B)

for all B and all subsets A of ψ(C).

For a set B and a tuple a, the strong type stp(a/B) = tp(a/ acleq (B)) is
stationary (see Exercises 8.4.9 and 8.5.4). We denote by Cb(a/B) the canonical
basis of stp(a/B). Note that Cb(a/B) is a subset of Ceq .
Lemma 10.3.3. The formula ψ is 1-based if and only if

Cb(a/B) ⊆ acleq (a)

for all sets B and finite tuples a in ψ(C).

Proof. Let C be the intersection acleq (a) ∩ acleq (B). If T is 1-based, the strong
type of a over B does not fork over C. So by Lemma 8.5.8 Cb(a/B) is con-
tained in C ⊆ acleq (a). If conversely Cb(a/B) is contained in acleq (a), it is also
contained in C and a ^ | Cb(a/B) B implies a ^| C B.
CHAPTER 10. THE FINE STRUCTURE OF ℵ1 -CATEGORICAL THEORIES170

Corollary 10.3.4. 1. 1-basedness is preserved under adding and removing

parameters, i.e., ψ is 1-based if and only if ψ is 1-based in CA for any set
A of parameters.
2. If ψ is 1-based and if every element of ψ 0 (C) is algebraic over ψ(C), then
ψ 0 (C) is 1-based.

Proof. 1. If ψ is 1-based, then CbA (a/B) = Cb(a/AB) ⊆ acleq (a) ⊆ acleq A (a).
If conversely ψ is 1-based in CA and a ∈ ψ(C) and B are given, we may assume
that a, B are independent from A. We have then Cb(a/B) = Cb(a/AB) =
CbA (a/B) ⊆ acleq A (a). Since Cb(a/B) is also contained in acleq (B), we con-
clude Cb(a/B) ⊆ acleq (a).
2. First note that if c is algebraic over a, then, for any set B, we have that c
and B are independent over Cb(a/B), hence Cb(c/B) ⊆ acleq Cb(a/B). Now let
a0 be a finite tuple from ψ 0 (C) and B any set. Choose a tuple a1 from ϕ(C) over
which a0 is algebraic. If a1 and a0 are interalgebraic, we are done. Otherwise
choose a2 which realises the type of a1 over a0 and is independent from a1 over
a0 . Then Cb(a0 /B) ⊆ acleq Cb(a1 /B) ∩ acleq Cb(a2 /B) ⊆ acleq (a1 ) ∩ acleq (a2 ) ⊆
acleq (a0 ).
Theorem 10.3.5. Let T be totally transcendental and ϕ a strongly minimal
formula without parameters. Then the following are equivalent.
a) ϕ is locally modular.
b) ϕ is 1-based.
c) Every family of plane curves in ϕ has dimension at most 1. This means that
for all B and elements a, b of ϕ(C), if tp(ab/B) has Morley rank 1, then
Cb(ab/B) has Morley rank at most 1 over the empty set.

Proof. a) ⇒ b): If ϕ is locally modular, ϕ becomes modular if we add a name

for any element x ∈ ϕ(C)\acleq (∅) to the language. If B and a ∈ ϕ(C) are given,
we choose x independent from aB (over the empty set). It follows from Lemma
10.3.1 that a and Bx are independent over acleq (ax) ∩ acleq (Bx). This implies
Cb(a/Bx) ⊆ acleq (ax). By the choice of x we have Cb(a/Bx) = Cb(a/B) ⊆
acleq (B). Since x and B are independent over a, we have acleq (ax) ∩ acleq (B) ⊆
acleq (a). This implies Cb(a/B) ⊆ acleq (a).
b) ⇒ c): Write d = Cb(ab/B). Then MR(ab/d) = 1. If the Morley rank
of d is not zero, ab and d are dependent over the empty set. By the definition
of 1-basedness and Lemma 10.3.3, we have d ∈ acleq (ab), and so MR(abd) ≤ 2.
Since MR(ab/d) = 1, we have MR(d) ≤ 1 using Proposition 6.4.9.
c) ⇒ a): Let x be a non-algebraic element of ϕ(C). By Lemma C.1.11
we have to show the following: for all elements a, b and sets B in ϕ(C) with
MR(ab/x) = 2 and MR(ab/Bx) = 1, there is some c ∈ acl(Bx) such that
MR(ab/cx) = 1. We may assume that a 6∈ acl(Bx). Consider the imaginary
element d = Cb(ab/Bx). Since d is contained in acleq (Bx), a is not algebraic
over d. Also, since x 6∈ acl(ab) and d is algebraic over ab by assumption, x is
CHAPTER 10. THE FINE STRUCTURE OF ℵ1 -CATEGORICAL THEORIES171

not algebraic over d. So a and x have the same type over d and we can find an
element c such that xc and ab have the same type over d. Since b ∈ acl(ad) and
d ∈ acleq (ab) this implies c ∈ acl(xd) ⊆ acl(Bx) and d ∈ acleq (xc). So we have
MR(ab/cx) = 1 as required.
By a theorem of Zilber any ω-categorical strongly minimal theory is locally
modular (see [64]). On the other hand (see Exercise 4.3.1) we have the following,
which holds more generally for stable theories, see [43].
Proposition 10.3.6. A totally transcendental theory T which contains an in-
finite definable field is not 1-based.
Proof. Let K be an infinite definable set with a definable field structure. We
may assume that everything is definable over the empty set. Let α be the Morley
rank of K. We call an element x of K generic if MR(x) = α. We note first that
if p = (x, y) is an element of the line ga,b = {(x, y) | ax + b = y} which is not
algebraic over a, b, then a, b ∈ dcl Cb(p/a, b). This follows from the fact that
two lines intersect in at most one point. So it is enough to find such p and ga,b
with (a, b) not algebraic over p.
For this we choose four independent generic elements a, b, a0 , b0 and let p =
(x, y) be the intersection of ga,b and ga0 ,b0 . Since x and b0 are interdefinable
over a, b, a0 , we can conclude that x, a, b, a0 are generic and independent. So p
is not algebraic over a, b. Since y and b are interdefinable over x, a we have that
x, y, a, a0 are generic and independent. This implies that a, b is not algebraic
over p.
The converse of the previous proposition for strongly minimal theories was
known as Zilber’s conjecture. This conjecture, namely that for any non-locally
modular strongly minimal theory T an infinite field is definable in T eq , was
refuted by Hrushovski in [28]. A variant of his construction of a new strongly
minimal set will be given in the next section. However, Hrushovski and Zilber
proved in their fundamental work [30] that the conjecture holds for so-called
Zariski structures.

10.4 Hrushovski’s examples

To end, we present a modification of Hrushovski’s ab initio example of a new
strongly minimal set [28]. This counterexample to Zilber’s conjecture has been
the starting point of a whole new industry constructing new uncountably cate-
gorical groups [7], fields [6], [8], and geometries [5], [58]. The dimension function
defined below also reappears in Zilber’s work around Shanuel’s Conjecture [65].
Following Baldwin [5] (see also [58]), we construct an almost strongly mini-
mal projective plane as a modified Fraı̈ssé limit: instead of considering structures
with all their substructures we restrict the amalgamation to so-called strong sub-
structures and embeddings. This will allow us to keep control over the algebraic
closure of sets so that the resulting structure is uncountably categorical.
CHAPTER 10. THE FINE STRUCTURE OF ℵ1 -CATEGORICAL THEORIES172

Recall that a projective plane is a point-line geometry such that any two
lines meet in a unique point, any two points determine a unique line through
them and there are four points no three of which are collinear. For convenience,
we consider a projective plane as a bipartite graph whose vertices are the points
and lines of the projective plane and in which the incidence between a point
and a line is represented by an edge, making this graph naturally bipartite. In
these terms a projective plane is a bipartite graph with the property that the
distance between any two vertices is at most 3, the smallest cycles in the graph
have length 6, and any element has at least 3 neighbours.
We fix a language L = {P, E} for bipartite graphs where P is a predicate
denoting the colouring and E denotes the edges. For a finite graph A, let e(A)
denote the number of edges in A and |A| the number of vertices. We define
δ(A) = 2|A| − e(A) and put δ(A/B) = δ(AB) − δ(B). Then δ satisfies the
submodular law of dimension functions for pregeometries (see page 204):

δ(AB) + δ(A ∩ B) ≤ δ(A) + δ(B)

or equivalently
δ(A/B) ≤ δ(A/A ∩ B).
We call a finite set B ⊆ M strong in M , B ≤ M , if δ(A/B) ≥ 0 for all finite
A ⊆ M . It follows easily from submodularity that this is a transitive relation,
see Exercise 10.4.1.
Let K be the class of graphs M , bipartite with respect to P , not containing
any 4-cycles, and such that δ(A) ≥ 4 for any finite subgraph A of M with
|A| ≥ 3. Note that this implies δ(A) ≥ 2 for all finite A ∈ K. Any finite subset
A of M is contained in a finite strong subset F of M : we can choose F to be
any finite extension of A with δ(F ) minimal.
Given graphs A ⊆ M, N we denote by M ⊗A N the trivial amalgamation
of M and N over A, obtained as the graph whose set of vertices is the disjoint
union (M \ A) ∪ (N \ A) ∪ A with incidence and predicate P induced by M and
N.

Lemma 10.4.1. If A ≤ N and δ(F ) ≥ 4 for all finite subgraphs F of M and

N with |F | ≥ 3, the same is true for M ⊗A N . If M is finite, then M is strong
in M ⊗A N .
Proof. This follows from the fact that every finite F ⊆ M ⊗A N has the form
M 0 ⊗A0 N 0 where M, A0 , N 0 are the intersection of F with M, A, N respectively,
and from the formula δ(F ) = δ(M 0 ) + δ(N 0 /A0 ).
Definition 10.4.2. We call a proper strong extension F over A minimal if it
cannot be split into two proper strong extensions A ≤ C ≤ F . We call a minimal
extension i-minimal if δ(F/A) = i. We use this terminology also for pairs (A, B)
of disjoint sets, which we call i-minimal – or we say that B is i-minimal over
A – if AB is an i-minimal extension of A. A 0-minimal pair (A, B) is called a
simple pair if B is not 0-minimal over any proper subset of A.
CHAPTER 10. THE FINE STRUCTURE OF ℵ1 -CATEGORICAL THEORIES173

The following is easy to see.

Remark 10.4.3. Let B be 0-minimal over A and A0 the set of elements of A
which are connected to an element of B. Then (A0 , B) is simple and A ∪ B =
A ⊗A0 A0 B.
Lemma 10.4.4. A proper strong extension A ≤ F is minimal if and only if
δ(C/A) > δ(F/A) for all C properly between A and F .
Proof. Let C be a set properly between A and F with c = δ(C/A) ≤ δ(F/A)
minimal. Then F is a strong extension of C.
Corollary 10.4.5. A minimal extension F of A is i-minimal for i = 0, 1 or 2.
The 1-minimal and 2-minimal extensions are of the form F = A ∪ {b} where b
is connected to at most one element of A.
Proof. Let F be an i-minimal extension of A. Assume that B = F \ A has more
than one element and i > 0. Since δ(b/A) ≤ 2 for any b ∈ B, by the previous
lemma we must have i = 1 and no element of B is connected with A. Thus
1 = δ(B), which is impossible for B ∈ K.
We next fix a function µ from simple pairs (A, B) into the natural numbers
satisfying the following properties:

1. µ(A, B) depends only on the isomorphism type of (A, B).

2. µ(A, B) ≥ δ(A).

We will be only interested in simple pairs (A, B) where AB belongs to K. Note

that this implies A 6= ∅ and hence µ(A, B) ≥ 1.
For any graph N and any simple pair (A, B) with A ⊆ N we define χN (A, B)
to be the maximal number of pairwise disjoint graphs B 0 ⊆ N such that B and
B 0 are isomorphic over A.
Let now Kµ be the subclass of K consisting of those N ∈ K satisfying
χN (A, B) ≤ µ(A, B) for every simple pair (A, B) with A ⊆ N . Clearly Kµ
depends only on the values µ(A, B), where AB belongs to K.
Lemma 10.4.6. Let N ∈ Kµ contain two finite subgraphs A ≤ F . If δ(F/A) =
0, then N contains only finitely many copies of F over A.
Proof. It suffices to consider the case that F = A ∪ B for B simple over A. As-
sume that B has infinitely many copies over A in N . Consider a finite extension
C of A which is strong in N . There is a copy of B which is not contained in C.
It follows from minimality that B is disjoint from C. So we can construct an
infinite sequence of disjoint copies, contradicting χN (A, B) ≤ µ(A, B).
We need the following lemma:
Lemma 10.4.7. Let M be in Kµ , A a finite subgraph of M and (A, B) a simple
pair. If N = M ⊗A AB ∈ / Kµ is witnessed by χN (A0 , B 0 ) > µ(A0 , B 0 ), there are
two possibilities for (A0 , B 0 ):
CHAPTER 10. THE FINE STRUCTURE OF ℵ1 -CATEGORICAL THEORIES174

1. A0 = A and B 0 is an isomorphic copy of B over A.

2. a) A0 is contained in A ∪ B, but not a subset of A.
b) B contains an isomorphic copy of B 0 over A0 .
Proof. First consider the case A0 ⊆ M . Since M ∈ Kµ there is some copy B 00 of
B 0 over A0 which intersects B. If B 00 6⊆ B, then A0 ≤ A0 ∪ (M ∩ B 00 ) ≤ A0 ∪ B 00
contradicting the minimality of B 00 over A0 . So B 00 ⊆ B. Since (A0 , B 00 ) is
simple, every element of A0 is connected with some element of B 00 and so A0
must be a subset of A. If B 00 were a proper subset of B, 0-minimality of (A, B)
would imply that 0 < δ(B 00 /A) ≤ δ(B 00 /A0 ) which is not possible. So B 00 = B,
which implies A = A0 by simplicity of (A, B).
Next consider the case A0 6⊆ M , so A0 ∩ B 6= ∅. Then since B 0 is simple
over A0 , no copy of B 0 over A0 is contained in M \ A. Now suppose that there
are k disjoint copies B10 , . . . , Bk0 of B 0 over A0 contained in M and that the
0 0
disjoint copies Bk+1 , . . . , Bk+l intersect both M and B. Since each Bi0 contains
vertices which are connected to vertices of A0 ∩ B, it follows immediately that
δ(A0 /M ) ≤ δ(A0 /M ∩ A0 ) − k ≤ δ(A0 ) − k. Note that δ(M ∪ A0 ) ≤ δ(A0 ) since
M ∪ A0 is strong in A0 .
Since the Bi0 are 0-minimal over A0 , we have for each i = k + 1, . . . , k + l:
 
δ Bi0 /M ∪ A0 ∪ Bk+1 0 0
∪ · · · ∪ Bi−1 < 0.

This implies
k+l
[
δ( Bi0 /M ∪ A0 ) ≤ −l.
i=k+1

Hence
 k+l
[ 
0≤δ Bi0 ∪ A0 /M ≤ δ(A0 /M ) − l ≤ δ(A0 ) − (k + l).
i=k+1

Thus at most δ(A0 ) many disjoint copies of B 0 over A0 are not contained in B,
leaving at least one copy of B 0 over A0 inside B. Since each element of A0 is
connected to some element of this copy, we see that A must be contained in
A ∪ B, finishing the proof.
As in Section 4.4 we say that M ∈ Kµ is Kµ -saturated if for all finite A ≤ M
and finite strong extensions C of A with C ∈ Kµ there is a strong embedding
of C into M fixing A elementwise. Since the empty graph belongs to Kµ and
is strongly embedded in every A ∈ Kµ , this implies that every finite A ∈ Kµ is
strongly embeddable in M .
Theorem 10.4.8. The class Kµfin of finite elements of Kµ is closed under sub-
structures, and has the joint embedding and the amalgamation property with
respect to strong embeddings. There exists a countable Kµ -saturated structure
Mµ , which is unique up to isomorphism. This structure Mµ is a projective plane
with infinitely many points per line and infinitely lines through any point.
CHAPTER 10. THE FINE STRUCTURE OF ℵ1 -CATEGORICAL THEORIES175

Proof. Clearly, Kµfin is closed under substructures. We will show that Kµfin has
the joint embedding and the amalgamation property with respect to strong em-
beddings. Then exactly as in the proof of Theorem 4.4.4 we obtain a countable
Kµ -saturated structure Mµ , which is unique up to isomorphism. In particular,
in Mµ any partial isomorphism f : A → A0 with A, A0 ≤ Mµ extends to an
automorphism of Mµ . Since the empty graph is in Kµ and strong in A ∈ Kµ , it
suffices to prove the amalgamation property. Let C0 , C1 , C2 ∈ Kµfin , C0 ≤ C1 , C2 .
We have to find some D ∈ Kµfin which contains C1 and C2 as strong subgraphs.
We prove the amalgamation property by induction on the cardinality of C2 \ C0 .
Case 1: C2 is not a minimal extension of C0 . Then there is a set C20 ≤ C2
which lies properly between C0 and C2 . By the induction hypothesis we may
amalgamate C1 with C20 over C0 to get D0 , and then amalgamate D0 with C2
over C20 to obtain D.
Case 2: C2 is a minimal extension of C0 . We will show that either D =
C1 ⊗C0 C1 belongs to Kµ or C1 strongly contains a copy of C2 over C0 .
By Corollary 10.4.5 there are three cases:
Case 2.i): C2 is a 0-minimal extension of C0 . We assume D = C1 ⊗C0 C2 6∈ Kµ
and show that C2 contains a copy of C20 of C2 . Since δ(C20 /C0 ) = 0 this then
implies that C20 is strong in C1 .
That D 6∈ Kµ can have two reasons. First there might be a 4-cycle in D.
This cycle must consist of a0 , a00 in C0 , b1 ∈ C1 \ C0 and b2 ∈ C2 \ C0 such that
b1 and b2 are connected with both a0 and a00 . But then minimality implies that
C2 = C0 ∪ {b2 } and C0 ∪ {b1 } is a copy of C2 over C0 . Note that b1 and b2 must
have the same colour.
The second reason might be that χD (A0 , B 0 ) > µ(A0 , B 0 ) for a simple pair
(A , B 0 ). Let A be the set of elements in C0 which are connected to a vertex in
0

B = C2 \ C0 . Then (A, B) is a simple pair and we have D = C1 ⊗A AB. We can

now apply Lemma 10.4.7. The second case of the Lemma cannot occur since
then every copy B 00 of B 0 over A0 in D must intersect C2 and, since C2 is strong in
D2 , must be contained in C2 . This would imply that χC2 (A0 , B 0 ) = χD (A0 , B 0 ).
So the first case applies and we have A0 = A and B 0 is a copy of B over A.
All other copies B 00 of B over A are contained in C1 by simplicity. Since B 00
is minimal over A, either B 00 must be a subset of C0 or a subset of C1 ⊆ C0 .
Since C2 is in Kµ , there is a B 00 contained in C2 \ C0 . Then C0 ∪ B 00 is over C0
isomorphic to C1 = C0 ⊗A AB.
Case 2.ii): C2 is a 1–minimal extension of C0 . Then C2 = C0 ∪ {b} and b is
connected with a single a ∈ C0 . We show that D = C1 ⊗C0 C2 is in Kµ . Clearly,
D does not contain more cycles than C1 . So consider a simple pair (A, B) in
D. Since A ≤ B and δ(B/A) = 0, b cannot be contained in B. If b ∈ A, then
1 = χD (A, B) ≤ µ(A, B) as b is connected to a unique element of D. So we
have D ∈ Kµ .
Case 2.iii): C2 is a 2-minimal extension of C0 . Then C2 = C0 ∪ {b} where
b is not connected with C0 . The same argument as in the last case shows that
D = C1 ⊗C0 C2 is in Kµ .
Using the fact that any partial isomorphism f : A → A0 with A, A0 ≤ Mµ
CHAPTER 10. THE FINE STRUCTURE OF ℵ1 -CATEGORICAL THEORIES176

extends to an automorphism of Mµ it is easy to see that Mµ is a projective

plane: since any two vertices of the same colour form a strong substructure
of Mµ , for any two pairs of such vertices with the same colouring there is an
automorphism taking one pair to the other. Since there are pairs of vertices
of the same partition at distance 2 in the graph, the same is true for any such
pair. Thus any two points lie on a common line and any two lines intersect in
a point. Uniqueness is immediate since there are no 4-cycles. Similarly, for any
n ∈ ω the graph consisting of a vertex x0 and neighbours x1 , . . . xn of x0 (in
either colouring) lies in Kµ and is a strong extension of x0 . It follows that in
Mµ every vertex has infinitely many neighbours. Translated into the language
of point-line geometries this says that there are infinitely many points per line
and infinitely many lines through any point. This of course already implies
the existence of four points in Mµ no three of which are collinear. But this
also follows from the fact that the corresponding graph, an 8-cycle of pairwise
distinct elements is contained in Kµ .
We now turn to the model-theoretic properties of Mµ .

Theorem 10.4.9. Let Tµ be the theory (in the language of bipartite graphs)
axiomatising the class of models M such that:
1. Every vertex of M has infinitely many neighbours.
2. M ∈ Kµ ;

3. M ⊗A AB 6∈ Kµ for each simple pair (A, B) with A ⊆ M .

Then Tµ = Th(Mµ ).
Proof. Note first that this forms an elementary class whose theory Tµ is con-
tained in Th(Mµ ): clearly, Part 1 is a first-order property, which holds in
Mµ by Theorem 10.4.8. For each simple pair (A, B) we can express that
χM (A, B) ≤ µ(A, B), so Part 2 is first-order expressible and holds in Mµ by con-
struction. For Part 3 notice that if D = M ⊗A AB ∈ / Kµ , then by Lemma 10.4.7
to express the existence of a simple pair (A0 , B 0 ) with χD (A0 , B 0 ) > µ(A0 , B 0 )
one can restrict to pairs which are contained in A ∪ B. So this can be expressed
in a first-order way. To see that this is true in Mµ we argue as follows. As-
sume D ∈ Kµ . Then for every finite C ≤ Mµ which contains A, the graph
C ≤ C ⊗A AB belongs to Kµ and so Mµ contains a copy of B over C. So we
can construct in Mµ an infinite sequence of disjoint copies of B over A. This is
not possible.
Let M be a model of Tµ . We have to show that M is elementarily equivalent
to Mµ . Choose an ω-saturated M 0 ≡ M . By (one direction of) the next claim
M 0 is Kµ -saturated. As in Exercise 4.4.1 M 0 and Mµ are partially isomorphic
and therefore elementarily equivalent.
Claim. The structure M is an ω-saturated model of Tµ if and only if it is
Kµ -saturated.
CHAPTER 10. THE FINE STRUCTURE OF ℵ1 -CATEGORICAL THEORIES177

Proof of Claim. Let M |= Tµ be ω-saturated. To show that M is Kµ -

saturated, let A ≤ M and A ≤ F ∈ Kµfin . By induction we may assume that F
is a minimal extension of A. There are three cases:
Case 1: F is a 0-minimal extension of A. By condition 3 M ⊗A F does not
belong to Kµ . Then by Claim 2 of the proof of Theorem 10.4.8 M contains a
strong copy of F over A.
Case 2: F is a 1-minimal extension of A. Then F = A ∪ {b} where b is
connected to exactly one vertex a ∈ A. Since a is connected to infinitely many
b0 ∈ M , there is such a b0 that is not algebraic over A. Then F 0 = A ∪ {b0 } is
isomorphic to F and strong in M , since by Lemma 10.4.6 F 0 is not contained
in any C ⊆ M with δ(C/A) = 0.
Case 3: F is a 2-minimal extension of A. Then F = A ∪ {b} where b
is not connected to A. By the previous case there are b0 and b00 such that
A ≤ A ∪ {b0 } ≤ A ∪ {b0 , b00 } ≤ M , b0 is connected with exactly one vertex from A
and b00 is connected with b0 but not with A. Then F 0 = A ∪ {b00 } is isomorphic
to F and F ≤ A ∪ {b0 , b00 } implies F 0 ≤ M .
Conversely, suppose M is Kµ -saturated. Since M is partially isomorphic to
Mµ , it is a model of Tµ . Choose an ω-saturated M 0 ≡ M . Then by the above
M 0 is Kµ -saturated. So M 0 and M are partially isomorphic, which implies that
M is ω-saturated by Exercise 4.3.13.
Definition 10.4.10 (Coordinatisation). Let Π = (P, L, I) be a projective
plane, let ` ∈ L be a line and let a1 , a2 , a3 ∈ P be non-collinear points out-
side `. Let D` denote the set of points on `. Then every element of Π is in the
definable closure of D` ∪ {a1 , a2 , a3 }: if the point x ∈ P lies for example, not
on the line (a1 , a2 ), let x1 and x2 , respectively, denote the intersections of the
lines (x, a1 ) and (x, a2 ) with `. Then x ∈ dcl(a1 , a2 , x1 , x2 ). A similar argument
shows that also every line is definable from a1 , a2 , a3 and two elements of `. This
process is called coordinatisation.

Let cl(B) = clM (B) be the smallest strong subgraph of M containing B (see
Exercise 10.4.1). We also define

d(A) = min{δ(B) | A ⊆ B ⊆ M } = δ(cl(A)).

Similarly, we put d(A/B) = d(AB) − d(B).

We will show that for any vertex a ∈ Mµ the set Da of neighbours of a is
strongly minimal. To this end we start with the following easy lemma.
Lemma 10.4.11. Let M and M 0 be models of Tµ . Then tuples a ∈ M and
a0 ∈ M 0 have the same type if and only if the map a 7→ a0 extends to an
isomorphism of cl(a) to cl(a0 ). In particular, d(a) depends only on the type
of a.
Lemma 10.4.12. Let M be a model of Tµ and A a finite subset of M . Then a
is algebraic over A if and only if d(a/A) = 0.
CHAPTER 10. THE FINE STRUCTURE OF ℵ1 -CATEGORICAL THEORIES178

Proof. Clearly cl(A) is algebraic over A. If d(a/A) = 0, there is an extension B

of cl(A) with δ(B/ cl(A)) = 0. By Lemma 10.4.6 B is algebraic over cl(A).
For the converse we may assume that M is ω-saturated. If d(a/A) > 0,
we decompose the extension cl(A) ≤ cl(Aa) into a series of minimal extensions
cl(A) = F0 ≤ · · · ≤ Fn = cl(Aa). One extension Fk ≤ Fk+1 must be i-minimal
for i = 1, 2. By the proof of Theorem 10.4.9, Fk+1 has infinitely many conjugates
over Fk . So cl(Aa) and therefore also a are not algebraic over A.
Proposition 10.4.13. For any model M of Tµ and any a ∈ M , the set Da is
strongly minimal.
Proof. Let A be a strong finite subset of M which contains a and let b be an
element of Da . Then d(b/A) is 0 or 1. If d(b/A) = 0, then b is algebraic over A
by the previous lemma. If d(b/A) = 1, then a is the only element of A connected
with b. Thus Ab is also strong in M . So by Lemma 10.4.11 the type of b over
A is uniquely determined. The claim now follows from Lemma 5.7.3.
Together with coordinatisation, the strong minimality of Da now implies
that Tµ is almost strongly minimal.

Theorem 10.4.14. The theory Tµ is almost strongly minimal, not 1-based and
of Morley rank 2. For finite sets A, F we have MR(F/A) = d(F/A).
Proof. By coordinatisation there is a line a and finite set A of parameters such
that every element is definable from A and two points of Da . Since Da is
strongly minimal, this implies that every element has rank at most 2.
To see that Tµ is not 1-based, let ` be a line of M . Since {`, p} is a strong
subset, the type of (`, p) is uniquely determined. It follows that p is not algebraic
over `, which implies ` ∈ Cb(p/`). Also ` is not algebraic over p, which implies
that Tµ is not 1-based.
To compute the Morley rank of F over A we may assume that A ≤ F ≤ Mµ .
By Proposition 6.4.9 (see also Exercise 6.4.2) we know that Morley rank is
additive in Tµ . This shows that we may assume that F is a minimal extension
of A. If δ(F/A) = 0, F is algebraic over A by Lemma 10.4.12. If δ(F/A) = 1,
we have F = A ∪ {b}, where b is connected with some element of a ∈ A. Then
MR(F/A) = 1 by the proof of Proposition 10.4.13. If δ(F/A) = 2, the proof of
Theorem 10.4.9 shows that F has a 0-minimal extension F 0 which can be reached
from A by two 1-minimal extensions. So MR(F/A) = MR(F 0 /A) = 2.
To show that this indeed yields a counterexample to Zilber’s Conjecture, we
finish by showing that
Proposition 10.4.15. There is no infinite group definable in Tµeq .

Proof. Assume that there is group G of Morley rank n is definable in Tµeq . To

simplify notation we assume that G is definable without parameters. We apply
now Exercise 8.5.9 to obtain a group configuration:
CHAPTER 10. THE FINE STRUCTURE OF ℵ1 -CATEGORICAL THEORIES179
a1 a3
Q 

A Q 

A Q Q 

b
A  2Q

 Q
b1 b3
A

A

A

a2
The ai and bi have Morley rank n; each element of a triple forming a line in
the diagram is algebraic over the two other elements of this triple; any three
non-collinear elements are independent. We will show that such a configuration
cannot exist.
Choose finite closed sets Ai and Bi in M such that ai ∈ dcleq (Ai ) and
bi ∈ dcleq (Bi ). We may assume that Ai is independent over ai from the rest of
the diagram and similarly for Bi . We may also assume that all ranks MR(Ai /ai ),
MR(Bj /bj ) are the same, say k.
By additivity of Morley rank (Proposition 6.4.9 (see also Exercise 6.4.2)) we
have then that the Morley rank of all six points A1 . . . B3 together is 3n + 6k.
Consider the four “lines” E0 = cl(A1 , B1 , A2 ), E1 = cl(A2 , B3 , A3 ), E2 =
cl(A1 , B2 , B3 ) and E3 = cl(B1 , B2 , A3 ) corresponding to the four lines of the
configuration. Since the union E of the Ei has Morley rank 3n + 6k, we have
δ(E) ≥ 3n + 6k. On the other hand, we can bound δ(E) by the inequality of
Exercise 10.4.2. First we note that δ(Ei ) = 2n + 3k. Then for different i, j we
have δ(Ei ∩ Ej ) = n + k, since we have e.g., A2 ⊆ E1 ∩ E2 ⊆ acl(A2 ). Similarly
the intersection of three of the Ei is contained in acl(∅) = ∅. We then have

δ(E) ≤ 4(2n + 3k) − 6(n + k) = 2n + 6k,

which is only possible if n = 0, so G is finite.

We now have the promised counterexample to Zilber’s Conjecture.

Corollary 10.4.16. Let Ta be the induced theory of Tµ on the strongly minimal

set Da (after adding the parameter a to the language). Then Ta is strongly
minimal, not locally modular and does not interpret an infinite field.
Proof. By Corollary 10.3.4 since Tµ is almost strongly minimal over Da and not
1-based, the strongly minimal set Da itself cannot be locally modular. Hence
the induced theory Ta is strongly minimal and not 1-based. If Ta did interpret
an infinite field, then so would Tµ , which it doesn’t.
Exercise 10.4.1. Let A and B be finite subgraphs of M and N an arbitrary
subgraph. Prove:

A≤M ⇒ A∩N ≤N
A≤B≤M ⇒ A≤M
A, B ≤ M ⇒ A ∩ B ≤ M.
CHAPTER 10. THE FINE STRUCTURE OF ℵ1 -CATEGORICAL THEORIES180

Assume that δ(B) ≥ 0 for all B ⊆ M . Then for each A there is a smallest finite
subgraph cl(A) which contains A and is strong in M .
Exercise 10.4.2. Prove the following generalisation of the submodular law:
X \ 
|∆|
δ(A1 ∪ · · · ∪ Ak ) ≤ (−1) δ Ai .
∅6=∆⊆{1,...,k} i∈∆

Exercise 10.4.3. Show that for any a ∈ Aut(Mµ ) acts 2-transitively on Da : for
any two pairs of elements x1 6= x2 and y1 6= y2 in Da , there is some g ∈ Aut(Bµ )
such that g(x1 ) = y1 and g(x2 ) = y2 .

The following exercise shows directly that Tµ is not ω-categorical.

Exercise 10.4.4. Let M = Mµ , and let A = {x0 , . . . , x12 } be a 12-cycle of pair-
wise different elements in M . (Such a set exists in every projective plane.) Show
that acl(A) is infinite. (Hint: For any k ≥ 3, a 2k-cycle (x0 , x1 , . . . , x2k−1 , x2k = x0 )
is a 0-minimal extension of A if every xi has a unique neighbour in A. Show that such
an extension of A is in Kµ by noting that for any simple pair (C, D) any element of
D has at least 2 neighbours in D.)
Appendix A

Set theory

In this appendix we collect some facts from set theory presented from the naive
point of view and refer the reader to [31] for more details. In order to talk about
classes as well as sets we begin with a brief axiomatic treatment.

A.1 Sets and classes

In modern mathematics, the underlying axioms are mostly taken to be ZFC,
i.e., the Zermelo–Fraenkel axioms (ZF) including the axiom of choice (AC) (see
e.g., [31, p. 3]). For the monster model we may work in Bernays–Gödel set
theory (BG) which is formulated in a two-sorted language, one type of objects
being sets and the other type of objects being classes, with the element-relation
defined between sets and sets and between sets and classes only. Since the
axioms are less commonly known, we give them here following [31, p. 70]. We
use lower case letters as variables for sets and capital letters for classes. BG has
the following axioms.
1. (a) Extensionality: Sets containing the same elements are equal.
(b) Empty set: The empty set exists.
(c) Pairing: For any sets a and b, {a, b} is a set. This means that there
is a set which has exactly the elements a and b.
S
(d) Union: For every set a, the union a = {z|∃y z ∈ y ∈ a} is a set.
(e) Power Set: For every set a, the power set P(a) = {y|y ⊆ a} is a set.
(f) Infinity: There is an infinite set. This can be expressed by saying
that there is a set which contains the empty set and is closed under
the successor operation x ∪ {x}.
2. (a) Class extensionality: Classes containing the same elements are equal.
(b) Comprehension: If ϕ(x, y1 , . . . , ym , Y1 , . . . , Yn ) is a formula in
which only set-variables are quantified, and if b1 , . . . , bm , B1 , . . . , Bn

181
APPENDIX A. SET THEORY 182

are sets and classes, respectively, then

{x|ϕ(x, b1 , . . . , bm , B1 , . . . , Bn )}

is a class.
(c) Replacement: If a class F is a function, i.e., if for every set b there
is a unique set c = F (b) such that (b, c) = {{b}, {b, c}} belongs to F ,
then for every set a the image {F (z)|z ∈ a} is a set.
3. Regularity: Every nonempty set has an ∈-minimal element.
For BGC we add:
5. Global Choice: There is a function F such that F (a) ∈ a for every
nonempty set a.
The set-part of a model of BG is a model of ZF. Conversely, a model M of
ZF becomes a model of BG by taking the definable subsets of M as classes. This
shows that BG is a conservative extension of ZF: any set-theoretical statement
provable in BG is also provable in ZF. Similarly, BGC is a conservative extension
of ZFC, see [20]. For a historical discussion see also [9].

A.2 Ordinals
Definition A.2.1. A well-ordering of a class X is a linear ordering of X such
that any non-empty subclass of X contains a smallest element or, equivalently,
such that X does not contain infinite properly descending chains. If X is a
proper class rather than a set, we also ask that for all x ∈ X the set {y|y < x}
of predecessors is a set.
The well-ordering of X is equivalent the following principle of transfinite
induction.
Let E be a subclass of X. Assume that whenever all elements less
than x are in E, then x itself belongs to E. Then E = X.
Well-orderings can be used to define functions by recursion.
Theorem A.2.2 (Recursion Theorem). Let G be a function which takes func-
tions defined on proper initial segments of a well-ordered set X as arguments.
Then there is a unique function F defined on X satisfying the recursion formula

F (x) = G F  {y|y < x})

Proof. It is easy to see by induction that for all x there is a unique function fx
0
S on {y|y ≤ x} and satisfying the recursion formula for all x ≤ x. Put
defined
F = x∈X fx .
Definition A.2.3 (v. Neumann). An ordinal is a well-ordered set in which
every element equals its set of predecessors.
APPENDIX A. SET THEORY 183

Note that the well-ordering of an ordinal is given by ∈, so we can identify an

ordinal with its set of elements. We denote the class of all ordinals by On.
Elements of ordinals are again ordinals, so we have

α = {β ∈ On|β < α}.

Proposition A.2.4. 1. Every well-ordered set (x, <) is isomorphic to a unique

ordinal α.
2. On is a proper class, well-ordered by ∈.
We call α = otp(x, <) the order type of (x, <). For ordinals we write α < β for
α∈β
Proof. 1) Define F on x recursively by F (y) = {F (z)|z < y}. The image of F is
an ordinal which is isomorphic to (x, <) via F . Note that F is the only possible
isomorphism between (x, <) and an ordinal.
2) Consider two different ordinals α and β. We have to show that either
α ∈ β or β ∈ α. If not, x = α ∩ β would be a proper initial segment of α and β
and therefore itself an element of α and β, which is impossible. The class On is
proper because otherwise it would itself be an ordinal.
The proof shows also that every well-ordered proper class is isomorphic to
On.
For any ordinal α its successor is defined as α ∪ {α}: it is the smallest
ordinal greater than α. Starting from the smallest ordinal 0 = ∅, its successor
is 1 = {0}; then 2 = {0, 1} and so on, yielding the natural numbers. The order
type of the natural numbers is denoted by ω = {0, 1, . . .}; the next ordinal is
ω + 1 = {0, 1, . . . , ω}, et cetera.
By definition, a successor ordinal β contains a maximal element α (so β is
the successor of α) and we write β = α + 1. For natural numbers n, we put

α + n = α + 1 + ··· + 1.
| {z }
n times

Ordinals greater than 0 which are not successor ordinals are called limit ordinals.
Whenever {αi |i ∈ I} is a non-empty set of ordinals without biggest element,
supi∈I αi is a limit ordinal. Any ordinal can be uniquely written as

λ + n,

with λ = 0 or a limit ordinal.

We finish with a quick proof of the Well-ordering Theorem, which like Zorn’s
Lemma (see below) is equivalent to the Axiom of Choice, see [31, 5.1].
Proposition A.2.5 (Well-ordering Theorem). Every set has a well-ordering.
APPENDIX A. SET THEORY 184

Proof. Let a be a set. Fix a set b which does not belong to a and define a
function F : On → a ∪ {b} by the following recursion:
(
some element in a \ {F (β)|β < α} if this set is not empty.
F (α) =
b otherwise.

Then γ = {α|F (α) 6= b} is an ordinal and F defines a bijection between γ and

a.
Zorn’s Lemma states that every partially ordered set in which every ordered
subset has an upper bound contains a maximal element. We omit its proof.

A.3 Cardinals
Two sets are said to have the same cardinality if there is a bijection between
them. By the well-ordering theorem any set x has the same cardinality as some
ordinal. We call the smallest such ordinal the cardinality |x| of x.
Ordinals occurring in this way are called cardinals. They are characterised
by the property that all smaller ordinals have smaller cardinality. All natural
numbers and ω are cardinals. The cardinality of a finite set is a natural number,
a set of cardinality ω is called countable.
Proposition A.3.1. The class of all cardinals is a closed and unbounded sub-
class of On.
Proof. Being closed in On means that the supremum supi∈I κi of a set of cardi-
nals is again a cardinal. This is easy to check. For the second part assume that
there is a largest cardinal κ. Then every ordinal above κ would be the order
type of a suitable well-ordering of κ. Since the well-orderings on κ form a set
this would imply that On is a set.
The isomorphism between On and the class of all infinite cardinals is denoted
by α 7→ ℵα , which can be recursively defined by

ω
 if α = 0
ℵα = ℵ+ β if α = β + 1

supβ<α ℵβ if α is a limit ordinal


where κ+ denotes the smallest cardinal greater than κ, the successor cardinal
of κ. Positive cardinals which are not successor cardinals are limit cardinals.
Sums, products, and powers of cardinals are defined by disjoint union, Carte-
sian power and sets of maps, respectively. Thus

|x| + |y| = |x ∪ y| (A.1)

|x| · |y| = |x × y| (A.2)
|x||y| = |y x| (A.3)
APPENDIX A. SET THEORY 185

where we assume in (A.1) that x and y are disjoint. In (A.3), the set y x denotes
the set of all functions from y to x.
It is easy to see that these operations satisfy the
same rules as the corre-
µ
sponding operations on the natural numbers, e.g., κλ = κλ·µ . The following
theorem shows that addition and multiplication are actually trivial for infinite
cardinals.
Theorem A.3.2 (Cantor’s Theorem). 1. If κ is infinite, then κ · κ = κ.
2. 2κ > κ.
Proof. The proof of part 2 is a generalisation of the well-known proof that there
are uncountably many reals: if (fα )α<κ is a sequence of functions from κ to 2,
find a function g : κ → 2 such that g(α) 6= fα (α) for all α < κ. We will prove
part 1 in Lemma A.3.7 below.
Corollary A.3.3. 1. If λ is infinite, then κ + λ = max(κ, λ).
2. If κ > 0 and λ are infinite, then κ · λ = max(κ, λ).
3. If κ is infinite, then κκ = 2κ .
Proof. Let µ = max(κ, λ). Then µ ≤ κ + λ ≤ µ + µ ≤ 2 · µ ≤ µ · µ = µ, and if
κ > 0, then µ ≤ κ · λ ≤ µ · µ = µ.
Finally,
κ
2κ ≤ κκ ≤ (2κ ) = 2κ·κ = 2κ .

Corollary A.3.4. The set [

<ω n
x= x
n<ω

of all finite sequences of elements of a non-empty set x has cardinality

max(|x|, ℵ0 ).
<ω
Note that with this notation 2 is the set of finite sequences in 0 and 1.
Proof. Let κ be the cardinality of all finite sequences in x. Clearly, |x| ≤ κ and
ℵ0 ≤ κ. On the other hand
X  
κ= |x|n ≤ sup |x|n · ℵ0 = max(|x|, ℵ0 ),
n∈N
n∈N

because

1,
 if |x| = 1
n
sup |x| = ℵ0 , if 2 ≤ |x| ≤ ℵ0
n∈N 
|x|, if ℵ0 ≤ |x|.

APPENDIX A. SET THEORY 186

The Continuum Hypothesis (CH) states that there is no cardinal strictly

between ω and the cardinality of the continuum R, i.e.,

ℵ1 = 2ℵ0 .

The Generalised Continuum Hypothesis (GCH) states more generally that

κ+ = 2κ for all infinite κ.

Both CH and GCH are independent of ZFC (assuming these axioms are consis-
tent, see e.g., [31] 14.32).
For every cardinal µ the beth function is defined as

µ,
 if α = 0,
iα (µ) = 2iβ (µ) , if α = β + 1,

supβ<α iβ (µ), if α is a limit ordinal.


For any linear order (X, <) we can easily construct a well-ordered cofinal
subset, i.e., a subset Y such that for any x ∈ X there is some y ∈ Y with x ≤ y.
Definition A.3.5. The cofinality cf(X) is the smallest order type of a well
ordered cofinal subset of X.
It is easy to see that cf(X) is a regular cardinal where an infinite cardinal
κ is regular if cf(κ) = κ. Successor cardinals and ω are regular. The existence
of weakly inaccessible cardinals, i.e., uncountable regular limit cardinals, cannot
be proven in ZFC.
The following is a generalisation of Theorem A.3.2(2), and has a similar
proof, see [31, 3.11].
Theorem A.3.6. If κ is an infinite cardinal, we have κcf(κ) > κ.
We conclude this section with a lemma which implies Theorem A.3.2(1).
Lemma A.3.7 (The Gödel well-ordering, see [31, 3.5]). There is a bijection
On → On × On which induces a bijection κ → κ × κ for all infinite cardinals κ.
Proof. Define

(α, β) < (α0 , β 0 ) ⇔ max(α, β), α, β <lex max(α0 , β 0 ), α0 , β 0

where <lex is the lexicographical ordering on triples. Since this is a well-ordering,

there is a unique order-preserving bijection γ : On × On → On. We show by
induction that γ maps κ × κ to κ for every infinite cardinal κ, which in turn
implies κ·κ = κ. Since the image of κ×κ is an initial segment, it suffices to show
that the set Xα,β of predecessors of (α, β) has smaller cardinality than κ for every
α, β < κ. We note first that Xα,β is contained in δ × δ with δ = max(α, β) + 1.
Since κ is infinite, we have that the cardinality of δ is smaller than κ. Hence by
induction |Xα,β | ≤ |δ| · |δ| < κ.
Appendix B

Fields

B.1 Ordered fields

Let R be an integral domain. A linear < ordering on R is compatible with the
ring structure if for all x, y, z ∈ R
x<y → x+z <y+z
x<y ∧ 0<z → xz < yz.
A field (K, <) together with a compatible ordering is an ordered field .
Lemma B.1.1. Let R be an integral domain and < a compatible ordering of
R. Then the ordering < can be uniquely extended to an ordering of the quotient
field of R.
a
Proof. Put b > 0 ⇔ ab > 0.
It is easy to see that in an ordered field sums of squares can never be negative.
In particular, 1, 2, . . . are always positive and so the characteristic of an ordered
field is 0. A field K in which −1 is not a sum of squares is called formally real.
Lemma B.1.2. A field has an ordering if and only if it is formally real.
Proof. A field with an ordering is formally real by the previous remark. For
the converse first notice that Σ , the set of all sums of squares in K, is a
semi-positive cone, i.e., a set P such that
Σ ⊆ P (B.1)
P +P ⊆ P (B.2)
P ·P ⊆ P (B.3)
−1 6∈ P (B.4)
The first and third condition easily imply
1
x∈P \0 ⇒ ∈ P.
x

187
APPENDIX B. FIELDS 188

Therefore, condition (B.4) is equivalent to P ∩ (−P ) = 0. It is also easy to

see that for all b, the set P + bP has all the properties of a semi-positive cone,
except possibly (B.4). Condition (B.4) holds if and only if b = 0 or −b 6∈ P . We
now choose P as a maximal semi-positive cone. Then

x≤y ⇔y−x∈P

defines a compatible ordering of K.

Corollary B.1.3 (of the proof). Let K be a field of characteristic different from
2 and let a be an element of K. There is an ordering of K making a negative
if and only if a is not the sum of squares.
Proof. If a 6∈ Σ , then Σ  − a · Σ  is a semi-positive cone.
Definition B.1.4. An ordered field (R, <) is real closed if
a) every positive element is a square,
b) every polynomial of odd degree has a zero.
We call (R, <) a real closure of the subfield (K, <) if R is real closed and
algebraic over K.
The field of real numbers is real closed. Similarly, the field of real algebraic
numbers is a real closure of Q. More generally, any field which is relatively
algebraically closed in a real closed field is itself real closed.
Theorem B.1.5. Every ordered field (K, <) has a real closure, and this is
uniquely determined up to isomorphism over K.
Proof (Sketch). Existence: Let K≥0 be the semi-positive cone of (K, <) and let
L be a field extension of K. It is easy to see that the ordering of K can be
extended to L if and only if

P = {x1 y12 + · · · + xn yn2 | xi ∈ K≥0 , yi ∈ L}

is a semi-positive cone of L, i.e., if −1 6∈ P . Therefore we may apply Zorn’s

Lemma to obtain a maximal algebraic extension R of K with an ordering ex-
tending the ordering of K. We claim that R is real closed.
Let r be a positive element of R and assume√ that r is not a square. Since the
ordering cannot be extended to L = R( r), there are ri ∈ R≥0 and si , ti ∈ R
such that X √
−1 = ri (si r + ti )2 .

ri (s2i r + t2i ), contradicting the fact that the right hand side is
P
Then −1 =
positive. Thus every element of R is a square (and the ordering of R is unique).
Let f ∈ R[X] be a polynomial of minimal odd degree n without zero (in
R). Clearly, f is irreducible. Let α be a zero of f (in the algebraic closure
of R) and put L = R(α). Since L cannot be ordered, −1 is a sum of squares
APPENDIX B. FIELDS 189

P polynomials gi ∈ R[X] of degree less than n such that f

in L. So there are
divides h = 1 + gi2 . The leading coefficients of the gi2 are squares in R and
so cannot cancel out. Hence the degree of h is even and less than 2n. But then
the polynomial hf −1 has odd degree less than n and no zero in R since h does
not have a zero. A contradiction.
Uniqueness: Let R and S be real closures of (K, <). It suffices to show that
R and S are isomorphic over K as fields. By (the easy characteristic 0 case
of) Lemma B.3.13 below it is enough to show that an irreducible polynomial
f ∈ K[X] has a zero in R if and only if it has one in S. This follows from the
next lemma which we prove at the end of the section.
Lemma B.1.6 (J. J. Sylvester). Let f be irreducible1 in K[X] and (R, <) a real
closed extension of (K, <). The number of zeros of f in R equals the signature
of the trace form of the K-algebra K[X]/(f ).
Note that a formally real field may have different orderings leading to non-
isomorphic real closures. However, in a real closed field the ordering is uniquely
determined by the field structure. Hence it makes sense to say that a field is
real closed without specifying its ordering.
The Fundamental Theorem of Algebra holds for arbitrary real closed fields:
√
Theorem B.1.7. Let R be real closed. Then C = R( −1) is algebraically
closed.
√
Proof. Notice that all elements of C are squares: one square root of a + b −1
is given by s√ s√
a2 + b2 + a a2 + b2 − a √
± −1,
2 2
where we choose ± according to the sign of b.
Let F be a finite extension of C. We claim that F = C. We may assume that
F is a Galois extension of R. Let G be a 2-Sylow subgroup of Aut(F/R) and L
the fixed field of G. Then the degree L/R is odd. The minimal polynomial of a
generating element of this extension has the same degree and is irreducible. But
since all irreducible polynomials over R of odd degree are linear, we have L = R.
Therefore G = Aut(F/R) and hence also H = Aut(F/C) are 2-groups. Now
2-groups are soluble (even nilpotent). So if H is non-trivial, it has a subgroup
of index 2 and thus C has a field extension of degree 2; but this is impossible
since every element is a square. Hence H = 1 and F = C.
Corollary B.1.8. If R is real closed, the only monic irreducible polynomials
are:
• Linear polynomials

X − a,
1 It suffices that f is non-constant and all zeros in acl(K) have multiplicity 1.
APPENDIX B. FIELDS 190

(a ∈ R).
• Quadratic polynomials

(X − b)2 + c,

(b, c ∈ R, c > 0).

√
Proof. Since all non-constant polynomials f ∈ R[X] have a zero in R( −1), all
irreducible polynomials must be linear or quadratic. Any monic polynomial of
degree 2 is of the form (X − b)2 + c. It is reducible
√ if and only if it has a zero
x in R if and only if c ≤ 0 (namely x = b ± −c).
Finally we prove Sylvester’s Lemma (B.1.6). Let K be an ordered field, R
a real closure of K and f ∈ K[X] irreducible. Consider the finite dimensional
K-algebra A = K[X]/(f ). The trace TrK (a) of a ∈ A is the trace of left
multiplication by a, considered as a vector space endomorphism. The trace
form is a symmetric bilinear form given by

(a, b)K = TrK (ab).

Let a1 , . . . , an be a basis of A diagonalising ( , )K , i.e., (ai , aj )K = λi δij . The

signature of such a form is defined as the number of positive λi minus the
number of negative λi . Sylvester’s Theorem (from linear algebra) states that
the signature is independent of the diagonalising basis. By tensoring A with R,
we obtain the R-algebra

AR = A ⊗ R ∼
= R[X]/(f ).

The basis a1 , . . . , an is also a diagonalising R-basis for the trace form of AR

with the same λi and hence the same signature as the trace form of A. We now
split f in R[X] into irreducible polynomials g1 , . . . , gm . Since f does not have
multiple zeros, the gi are pairwise distinct. By the Chinese Remainder Theorem
we have
R[X]/(f ) ∼= R[X]/(g1 ) × · · · × R[X]/(gm ).
This shows that the trace form of R[X]/(f ) is the direct sum of the trace
forms of the R[X]/(gi ), and hence its signature is the sum of the corresponding
signatures. Sylvester’s Lemma follows once we show that the signature of the
trace form is equal to 1 for a linear polynomial, while the signature is 0 for an
irreducible polynomial of degree 2.
If g is linear, then R[X]/(g) = R. The trace form√(x, y)R = xy has signature
√
1. If g is irreducible of degree 2, then R[X]/(g) = R( √−1) and TrR (x+y −1) =
2x. The trace form is diagonalised by the basis 1, −1. With respect to this
basis we have λ1 = 2 and λ2 = −2 and so its signature is zero.
APPENDIX B. FIELDS 191

B.2 Differential fields

In this section, all rings considered have characteristic 0. Let R be a commu-
tative ring and S an R-module. (We mainly consider the case that S is a ring
containing R.) An additive map d : R → S is called a derivation if

d(rs) = (dr)s + r(ds).

For any ring S the ring of dual numbers is defined as

S[] = {a + b | a, b ∈ S},

where 2 = 0. The following is an easy observation.

Lemma B.2.1. Let S be a ring containing R and let π : S[] → S; a + b 7→ a.
Any derivation d : R → S defines a homomorphism

td : R → S[]; r 7→ r + (dr)

inverting π. Conversely, any such homomorphism arises from a derivation.

Lemma B.2.2. Let S be a ring containing the polynomial ring R0 = R[x1 , . . . ,
xn ] and let d : R → S be a derivation. For any sequence s1 , . . . , sn of elements
of S there is a unique extension of d to a derivation d0 : R0 → S taking xi to si
for i = 1, . . . , n.
Proof. Extend td : R → S[] via td0 (xi ) = xi +si  to a homomorphism td0 : R0 →
S[].
Lemma B.2.3. Let R be a subring of the field K and let R0 be an interme-
diate field algebraic over R. Then every derivation d : R → K can be uniquely
extended to R0 .
Proof. It suffices to consider the following two cases:
1. R0 is the quotient field of R. If a is a unit, then a + b is a unit in K[]
(with inverse a−1 − ba−2 .) Therefore, td takes non-zero elements of R to units
in K[] and thus can be uniquely extended to R0 .
2. R is a field and R0 = R[a] a simple algebraic extension. Let f (x) be the
minimal polynomial of a over R. Then td can be extended to R0 if and only if
(td f )(x) has a zero of the form a + b in K[]. But (td f )(x) = f (x) + (f d )(x),
hence  
d ∂f d
(td f )(a + b) = f (a + b) + f (a) = (a)b + f (a) . (B.5)
∂x
∂f
Since char(K) = 0, f is separable, so ∂x (a) 6= 0 and d can be extended to R0 by
 −1
0 d ∂f
d (a) = a − f (a) (a) .
∂x
APPENDIX B. FIELDS 192

Let C = {c ∈ K | dc = 0} denote the set of constants of K. Clearly, C is a

subring containing 1. The previous lemma implies that C is a subfield which is
relatively algebraically closed in K.
Corollary B.2.4. Let R be a subring of the field K. Any derivation d : R → K
can be extended to a derivation of K.

Remark B.2.5. Let (K, d) be a differential field and F a field extension of K,

a, b ∈ F n . Assume that the ideal of all f in K[x] with f (a) = 0 is generated by
I0 . Then the following are equivalent:
a) There is an extension of d to F with da = b.

b) For all f ∈ I0 we have

∂f
(a)b + f d (a) = 0.
∂x
Proof. It is clear that a) implies b) because

∂f
d(f (a)) = (a)b + f d (a).
∂x
In order to show the converse we have to extend the homomorphism td : K →
F [] to K[a] in such a way that a is mapped to a+b. This is possible if and only
if (td f )(a + b) = 0 for all f ∈ I0 . By (B.5) this implies ∂f d
∂x (a)b + f (a) = 0.

Let (F, d) be a saturated model of DCF0 and V ⊆ F n an irreducible affine

variety (see [51, Chapter 1] for basic algebraic geometry). The torsor T (V ) ⊆
F 2n is defined by the equations
∂f
f (x) = 0 and (x)y + f d (x) = 0
∂x
for all f in the vanishing ideal of V (see p. 195). Clearly (a, da) ∈ T (V ) for all
a∈V.
Remark B.2.5 states that for any small subfield K over which V is defined,
and any (a0 , b0 ) ∈ T (V ) such that a0 ∈ V is generic over K there is a pair
(a, da) ∈ T (V ) satisfying the same field-type over K as (a, b). This already
proves one direction of the following equivalence:
Remark B.2.6. [41] An algebraically closed differential field (K, d) is a model
of DCF0 if and only if for every irreducible affine variety V defined over K and
every regular section2 s : V → T (V ) of the projection T (V ) → V there is some
a ∈ V (K) with da = s(a).

Proof. We show that an algebraically closed differential field (K, d) with this
property is a model of the axioms of DCF0 .
2 A section of a surjection f : A → B is a map g : B → A with f g = id . A map between
B
affine varieties is called regular if it is given by polynomials.
APPENDIX B. FIELDS 193

Let f and g be given with f irreducible. As in the proof of Theorem 3.3.22

we find a field extension F = K(α, . . . , dn α) in which α, . . . , dn−1 α are alge-
braically independent over K and f (α, . . . , dn α) = 0. Let β be the inverse of
∂f
g(α, . . . , dn−1 α) ∂x n
(α, . . . , dn α). Note that K[α, . . . , dn α, β] is closed under d.
Putting c = (α, . . . , dn α, β), let (c, dc) = s(c) for some tuple s of polynomials
in K[x0 , . . . , xn+1 ]. Then s defines a section V → T (V ) where V ⊆ K n+1 is
the variety defined by f = 0. By assumption there is some (a, b) ∈ V (K) with
(a, b, da, db) = s(a, b). Hence f (a, . . . , dn a) = 0 and g(a, . . . , dn−1 a) 6= 0.
Remark B.2.7 (Linear differential equations). Let K be a model of DCF0 and
A an n × n-matrix with coefficients from K. The solution set of the system of
differential equations
dy = Ay (B.6)
is an n-dimensional C-vector space.

Proof. Choose an n × n-matrix Y over an extension of K whose coefficients are

algebraically independent over K. Lemma B.2.2 shows that d can be extended
to K(Y ) by dY = AY . Since K is existentially closed, there is a regular matrix
over K, which we again denote by Y , such that dY = AY . The columns of Y are
n linearly independent solutions of (B.6). Such a matrix is called a fundamental
system of the differential equation. We show that any solution y is a C-linear
combination of the columns of Y . Let z be a column over K with y = Y z. Then

AY z = d(Y z) = (dY )z + Y dz = AY z + Y dz.

Hence Y dz = 0, so dz = 0, i.e., the elements of z are constants.

Remark B.2.8. Let K be a differential field and K 0 an extension of K with

fields of constants C and C 0 , respectively. Then K and C 0 are linearly disjoint
over C, i.e., any set of C-linearly independent elements of K remains linearly
independent over C 0 (see Section B.3).
Proof. Let a0 , . . . , an be in K and linearly dependent over C 0 . Then the columns
of  
a0 ... an
 da0 . . . dan 
B= .
 
.. 
 .. . 
n n
d a0 . . . d an
are linearly dependent. Let m < n be maximal with
 
a0 ... am
 da0 . . . dam 
Y = .
 
.. 
 .. . 
dm a0 . . . dm am
APPENDIX B. FIELDS 194

regular. We want to conclude that a0 , . . . , am+1 is linearly dependent over C.

So we may assume n = m + 1. Then the last row of B is a K-linear combination
of the first m + 1 rows. Thus for some m × m matrix A all columns of Y and
 
an
z =  ... 
 

dm an

are solutions of the differential equation dy = Ay. The proof of Remark B.2.7
shows that z is a C-linear combination of the columns of Y .

B.3 Separable and regular field extensions

Definition B.3.1. Two rings R, S contained in a common field extension are
said to be linearly disjoint over a common subfield k if any set of k-linearly
independent elements of R remains linearly independent over S. Note that then
R is also linearly disjoint over K from the quotient field of S.
Remark B.3.2. It is easy to see that this is equivalent to saying that the ring
generated by R and S is canonically isomorphic to the tensor product R ⊗k S.
So linear disjointness is a symmetric notion.
If S is a normal algebraic extension of K, so invariant under automorphisms
of K alg /K, then the linear disjointness of R and S over K does not depend on
the choice of a common extension of R and S (see Exercise 8.4.4 for a similar
phenomenon).
Recall that an algebraic field extension L over K is separable if every element
of L is a zero of a separable polynomial f ∈ K[X] and Galois if L is normal
and separable over K.
Lemma B.3.3. If F and L are field extensions of K with L Galois over K,
then F and L are linearly disjoint over K if and only if F ∩ L = K.
Proof. One direction is clear (see also the remark below). For the other direction
assume F ∩ L = K. We may assume that L/K is finitely generated. Since L/K
is separable, there is a primitive element, say L = K(a). Let f be the minimal
polynomial of a over F . Since all roots of f belong to L, f is in L[X] and
therefore in K[X]. It follows that [K(a) : K] = [F (a) : F ] and F and L are
linearly disjoint over K.
That h1 , . . . , hn are algebraically independent over L means that the mono-
mials in h1 , . . . , hn are linearly independent over L. This observation implies
that L and H are algebraically independent over K if they are linearly disjoint
over K. The converse holds if L/K is regular :

Definition B.3.4. A field extension L over K is regular if L and K alg are

linearly disjoint over K in some common extension.
APPENDIX B. FIELDS 195

Lemma B.3.5. If L is a regular extension of K and H/K is algebraically

independent from L/K, then L and H are linearly disjoint over K.
The reader may note that in the special case where H = K alg this is just
the definition of regularity.
P
Proof. Let li ∈ L, hi ∈ H, i<n li hi = 0, but not all hi = 0. Since L and H
are independent over K alg , the type tp(L/K alg H) is an heir of tp(L/K alg ) by
Corollaries 6.4.5 or 8.5.13. Thus, there is a non-trivial n-tuple h̄0 ∈ K alg such
that ¯l · h̄0 = 0. Since L/K is regular, there is a non-trivial h̄00 ∈ K such that
¯l · h̄00 = 0. This proves the claim.

Lemma B.3.6. Let K be a field. There is a natural bijection between isomor-

phism types of field extensions K(a1 , . . . , an ) of K with n generators3 and prime
ideals P in K[X1 , . . . , Xn ]. Regular extensions correspond to absolutely prime
ideals, i.e., ideals which generate a prime ideal in K alg [X1 , . . . , Xn ].
Proof. We associate with L = K(a1 , . . . , an ) the vanishing ideal P = {f ∈
K[X1 , . . . , Xn ] | f (a1 , . . . , an ) = 0}. Conversely if P is a prime ideal the quo-
tient field of K[X1 , . . . , Xn ]/P is an extension of K generated by the cosets of
X1 , . . . , Xn .
Let I be the ideal generated by P in K alg [X1 , . . . , Xn ]. Then K[a1 , . . . , an ]⊗K
K and K alg [X1 , . . . , Xn ]/I are isomorphic as K-algebras. Now L/K is regu-
alg

lar if and only if K[a1 , . . . , an ] ⊗K K alg is an integral domain, which means that
I is prime.
Remark B.3.7. It is easy to see that the case tr. deg(a1 , . . . , an ) = n − 1
corresponds exactly to the case where P is a principal ideal, generated by an
irreducible polynomial f ∈ K[X1 , . . . , Xn ]: this polynomial is uniquely deter-
mined up to a constant factor. ([35, VII, Ex.26])
The perfect hull of a field K is the smallest perfect field containing it, so
−n
in characteristic 0 equals K, and in characteristic p is the union of all K p =
n
{a ∈ K alg | ap ∈ K}.
Definition B.3.8. A field extension L over K is separable if, in some common
extension, both L and the perfect hull of K are linearly disjoint over K. This
extends the definition in the algebraic case.

Again this does not depend on the choice of the common extension L and the
perfect hull of K. Note that regular extensions are separable.
Let K be a field of characteristic p > 0. For any subset A we call the field
K p (A) the p-closure of A. This defines a pregeometry on K (see Section C.1)
whose associated dimension is the degree of imperfection. A basis of K in the
called a p-basis. If b = (bi | i ∈ I) is a p-basis of
sense of this pregeometry is Q
K, then the products bν = i∈I bνi i defined for multi-indices ν = (νi | i ∈ I)
where 0 ≤ νi < p and almost all νi equal to zero, form a linear basis of K
3 By quantifier elimination this is just Sn (K) in the theory of K alg .
APPENDIX B. FIELDS 196

over K p . If the degree of imperfection of K is a finite number e, it follows that

[K : K p ] = pe .
Remark B.3.9. A field L is a separable extension of K if and only if L and
−1
K p are linearly disjoint over K. It follows that L/K is separable if and only
if a p-basis of K stays p-independent in L.

Proof. Consider a sequence (ai ) of elements of L which is linearly independent

−1
over K. Then (ai ) is independent over K p by assumption which implies that
p p −1
(ai ) is independent over K. Then (ai ) is independent over K p implying that
−2
(ai ) is independent over K p . In this way one can show inductively that (ai )
−n
is independent over K p for all n.
Lemma B.3.10. If b1 , . . . , bn are p-independent in K, they are algebraically
independent over Fp .

Proof. The b1 , . . . , bn−1 form a p-basis of F = Fp (b1 , . . . , bn−1 ). So K/F is

separable by Remark B.3.9. If an element c ∈ K is algebraic over F , it is
separably algebraic over F . So c is also separably algebraic over K p F , which is
only possible if c belongs to K p F . Since bn does not belong to K p F , it follows
that bn is not algebraic over F .

Lemma B.3.11. 1. Let R be an integral domain of characteristic p and let

b be a p-basis of R in the sense that every element of R is a unique Rp -
linear combination of the bν . Then b is also a p-basis of the quotient field
of R.
2. Let b be p-basis of K and L is a separable algebraic extension of K, then
b is also a p-basis of L.
Proof. 1): Let K be the quotient field of R. Clearly b is p-independent in K.
Since the elements of b are algebraic over K p , it follows that K p [b] is a subfield
of K which contains R = Rp [b]. So K = K p [b].
2): We may assume that L/K is finite. Since Lp and K are linearly disjoint
over K p , we have [Lp K : K] = [Lp : K p ] = [L : K]. It follows that Lp K = L.
Lemma B.3.12. The field L = K(a1 , . . . , an ) is separable over K if and only if
there is a transcendence basis a0 ⊆ {a1 , . . . , an } of L/K such that L is separably
algebraic over K(a0 ).
Proof. This follows from the fact that every irreducible polynomial in K[X] is of
k
the form f (X p ) for some k ≥ 0 and some separable polynomial f ∈ K[X].

Lemma B.3.13 (See [16, Théorème 5.11]). Algebraic field extensions of K are
isomorphic over K if and only if the same polynomials in K[X] have a zero in
these extensions.
APPENDIX B. FIELDS 197

Proof. Consider two algebraic extensions L and L0 of K. A compactness argu-

ment shows that L and L0 are isomorphic over K if and only if they contain, up
to isomorphism over K, the same finitely-generated subextensions. The condi-
tion that the same polynomials in K[X] have zeros in L and L0 is equivalent to
L and L0 having the same simple subextensions. So the lemma is clear if L and
L0 are separable over K which means that in the general case we may assume
that K is infinite.
By symmetry it is enough to show that every finite subextension F/K of
L/K has an isomorphic copy in L0 /K, or in other words that F ⊆ α(L0 ) for
some α ∈ Aut(K alg /K). By our assumption every a ∈ F is contained in some
α(L0 ), so F is the union of all Fα = α(L0 ) ∩ F . There are only finitely many
different Fα . To see this choose a finite normal extension N/K which contains
F . Then Fα depends only on the restriction of α to N . Since F and the Fα are
vector spaces over the infinite field K, we can apply Neumann’s Lemma 3.3.9
and conclude that F equals one of the Fα .
Exercise B.3.1. Show that elementary extensions are regular.

B.4 Pseudo-finite fields and profinite groups

Definition B.4.1. A field K is pseudo algebraically closed (PAC) if every
absolutely irreducible affine variety defined over K has a K-rational point.
See [51, Chapter 1] for basic algebraic geometry.
Lemma B.4.2. Let K be a field. The following are equivalent:
a) K is PAC,
b) Let f (X1 , . . . , Xn , T ) ∈ K[X1 , . . . , Xn , T ] be absolutely irreducible and of
degree greater than 1 in T and let g ∈ K[X1 , . . . , Xn ]. Then there exists
(a, b) ∈ K n × K such that f (a, b) = 0 and g(a) 6= 0.
c) K is existentially closed in every regular field extension.
Proof. a) ⇒ b): Let C be an algebraically closed field containing K. The zero
set of f defines an absolutely irreducible variety X ⊆ Cn × C over K. Then the
set V = {(a, b, c) | f (a, b) = 0, g(a)c = 1} is absolutely irreducible as well and
defined over K.
b) ⇒ c): Let L/K be a finitely generated regular extension. Since regular
extensions are separable, by Lemma B.3.12 we find a transcendence basis a of
L/K such that L is separable algebraic over K(a). Then there is a b separable
algebraic over K(a) such that L = K(a1 , . . . , an , b). Let now c1 , . . . , cm be
elements of L satisfying certain equations over K.4 We need c0i ∈ K satisfying
the same equations. Write ci = hig(a) (a,b)
for polynomials hi (x, y) and g(x) over
K. By Remark B.3.7 the vanishing ideal of (a, b) over K is generated by some
4 As we are working in an infinite field, we don’t need inequalities.
APPENDIX B. FIELDS 198

f (x, y), which is absolutely irreducible since L/K is regular. There are a0 , b0 ∈ K
(a0 ,b0 )
such that f (a0 , b0 ) = 0 and g(a0 ) 6= 0. Hence the c0i = hig(a0) satisfy all equations
satisfied by the ci .
c) ⇒ a): Let V be an absolutely irreducible variety over K. If c ∈ V is an
element of C generic over K, then K(c)/K is regular by Lemma B.3.6.

Corollary B.4.3. “PAC” is an elementary property.

Proof. Being absolutely irreducible is a quantifier-free property of the coeffi-
cients of f (X1 , . . . , Xn , T ) because the theory of algebraically closed fields has
quantifier elimination.

Definition B.4.4. A profinite group is a compact5 topological group with a

neighbourhood basis for 1 consisting of subgroups.
Note that by compactness open subgroups are of finite index.
Profinite groups can be presented as inverse limits

lim i∈I Ai
←

of finite groups, with neighbourhood bases for the identity given by the inverse
images of the Ai , see [62] for more details.

Definition B.4.5. A profinite group is called procyclic if all finite (continuous)

quotients are cyclic.
Lemma B.4.6 (see [62]). Let G be a profinite group. The following are equiv-
alent:

a) G is procyclic;
b) G is an inverse limit of finite cyclic groups;
c) G is (topologically) generated by a single element a, i.e., the group abstractly
generated by a is dense in G.

By a generator for a profinite group we always mean a topological generator,

i.e., an element generating a dense subgroup.
Definition B.4.7. We denote by Z
b the inverse limit of all Z/nZ, with respect
to the canonical projections.

Lemma B.4.8. A profinite group G is procyclic if and only if for every n there
is at most one closed subgroup of index n. If G has exactly one closed subgroup
of index n for each n > 0, then G is isomorphic to Z.
b
5 Note that for us compact spaces are Hausdorff.
APPENDIX B. FIELDS 199

Proof. It suffices to prove the lemma for finite G. So if G has order n, we have
to show that G is cyclic if and only if for each k dividing n there is a unique
subgroup of order k (hence of index n/k). If a is a generator for G and H a
subgroup of G of order k, then an/k is a generator for H, proving uniqueness.
For the other direction, note that by assumption any two elements having the
same order generate the same subgroup. Thus, if G has an element of order k,
then G contains exactly ϕ(k) such elements. The claim now follows from the
equality
X
n= ϕ(k).
k|n

Lemma B.4.9. Let G → H be an epimorphism of procyclic groups. Then any

generator of H lifts to a generator of G.
Proof. First we consider the case where G is finite. We may then assume that
G and H are p-groups. If H = 0, we choose any generator as the preimage. If
H 6= 0, any preimage of a generator of H is a generator of G.
If G is infinite, let h be a generator of H. The finite case implies that
for every open normal subgroup A of G we find an inverse image of h which
generates G/A. The claim follows now by compactness.
For a field K we write G(K) for its absolute Galois group Aut(K alg /K).
Definition B.4.10. A perfect field K is called procyclic if G(K) is procyclic,
and 1-free if G(K) ∼
= Z.
b

By Lemma B.4.8 a perfect field is procyclic (1-free, respectively) if and only

if it has at most one (exactly one, respectively) extension of degree n in K alg
for every n. Thus, being procyclic or 1-free is an elementary property of K.
Definition B.4.11. A perfect 1-free PAC-field is called pseudo-finite.
Remark B.4.12. Being pseudo-finite is an elementary property. A theorem of
Lang and Weil (see [36]) on the number of points on varieties over finite fields
shows that infinite ultraproducts of finite fields are PAC and hence pseudo-finite.
Conversely, any pseudo-finite field is elementarily equivalent to an ultraproduct
of finite fields (see [1] and Exercise 7.5.1).
If L/K is regular, then clearly K is relatively algebraically closed in L. If K
is perfect, the converse holds as well:
Proposition B.4.13. Let K be perfect and L/K a field extension. Then the
following are equivalent:
a) L/K regular;
b) K is relatively algebraically closed in L;
APPENDIX B. FIELDS 200

c) the natural map G(L) → G(K) is surjective.

Proof. Using Remark B.3.2 and the functoriality of tensor products, it is easy
to see that a) implies c) (even without K being perfect). Clearly, c) implies
b). To see that b) implies a) assume that K is relatively algebraically closed
in L. Since every finite extension between K and K alg is generated by a single
element, it suffices to show for a ∈ K alg that the minimal polynomial f of a
over L has coefficients in K. But this follows as the coefficients of f can be
expressed by conjugates of a ∈ K alg ∩ L over K.
If L is perfect and N/L and L/K are regular extensions, then also N/K is
regular.
Corollary B.4.14. Let L/K be regular, L procyclic and K 1-free. Then L
is 1-free. If N/L is an extension of L such that N/K is regular, then N/L is
regular.
Proof. If G(L) is procyclic and π : G(L) → Z
b = G(K) is surjective, then π is
an isomorphism. Let ρ : G(N ) → G(L) be the natural map. If ρπ is surjective,
then so is ρ.
Any profinite group G acts faithfully and with finite orbits on the set S
consisting of all left cosets of finite index subgroups. That point stabilisers are
open subgroups of G just says that the action is continuous with respect to the
discrete topology on S. We use this easy observation to prove the following:
Lemma B.4.15. Any procyclic field has a regular pseudo-finite field extension.
Proof. Let K be a procyclic field. Let S be as in the previous remark for G = Z, b
0 0
let L = K(Xs )s∈S with the action of G on L given by the action of G on S and
L be the fixed field of G. Then L0 is a Galois extension of L with Galois group
G and L0 (and L) are regular extensions of K by Proposition B.4.13. Let σ ∈
Aut(K alg L0 /L) be a common extension of a generator of G(K) and a generator
of Zb = Aut(L0 /L). Extend σ to some σ 0 ∈ Aut(Lalg /L) and consider the fixed
field L00 of σ 0 . Then G(L00 ) is generated by σ 0 and has Z b as a homomorphic
image, so L00 is 1-free. Also L00 is a regular extension of K by Proposition B.4.13.
It is now easy to construct, by a long chain, a regular procyclic extension N of L00
which is existentially closed in all regular procyclic extensions. This extension
N is again regular over K and 1-free.
We use the criterion given in Lemma B.4.2 to show that N is PAC. Let N 0
be a regular extension of N . Let π ∈ G(N 0 ) be a lift of a generator of G(N ) and
alg
N 00 the fixed field of π in N 0 . Then N 00 is a regular procyclic extension of N .
By construction, N is existentially closed in N 00 and hence also in N 0 .
Procyclic fields have the amalgamation property for regular extensions:
Lemma B.4.16. If L1 , L2 are regular procyclic extensions of a field K, there
is a common procyclic regular extension H of L1 and L2 in which L1 and L2
are linearly disjoint over K.
APPENDIX B. FIELDS 201

Proof. Let L1 and L2 be regular procyclic extensions of a field K (so K is also

procyclic). We may assume that the Li are algebraically independent over K in
some common field extension. Let σ be a (topological) generator of G(K). By
Lemma B.4.9, we can lift σ to generators σi of G(Li ). By Lemma B.3.5, L1 and
L2 are linearly disjoint. As Lalg alg
1 L2 is the quotient field of the tensor product
of L1 and L2 over K alg , the σi generate an automorphism of Lalg
alg alg alg
1 L2 which
can be extended to an automorphism τ of (L1 L2 )alg . The fixed field of τ is
procyclic with Galois group generated by τ and a regular extension of L1 and
L2 by Proposition B.4.13.
By Lemma B.4.15 this also implies that pseudo-finite fields have the amal-
gamation property for regular extensions.
Appendix C

Combinatorics

C.1 Pregeometries
In this section, we collect the necessary facts and notions about pregeometries,
existence of bases and hence a well-defined dimension, modularity laws etc. First
recall Definition 5.6.5.
Definition. A pregeometry (X, cl) is a set X with a closure operator cl : P(X) →
P(X) such that for all A ⊆ X and a, b ∈ X
a) (Reflexivity) A ⊆ cl(A)
b) (Finite character) cl(A) is the union of all cl(A0 ), where the A0 range
over all finite subsets of A.

c) (Transitivity) cl(cl(A)) = cl(A)

d) (Exchange) a ∈ cl(Ab) \ cl(A) ⇒ b ∈ cl(Aa)
Remark C.1.1. The following structures are pregeometries:
1. A vector space V with the linear closure operator.

2. For a field K with prime field F , the relative algebraic closure cl(A) =
F (A)alg
3. The p-closure in a field K of characteristic p > 0, i.e., cl(A) = K p (A).

Proof. We prove Exchange for algebraic dependence in a field K as an ex-

ample.1 We may assume that A is a subfield of K. Let a, b be such that
a ∈ A(b)alg . So there is a non-zero polynomial F ∈ A[X, Y ] with F (a, b) = 0.
If we also assume that b ∈ / A(a)alg , it follows that F (a, Y ) = 0. This implies
alg
a∈A .
1 For the p-closure see [10, § 13].

202
APPENDIX C. COMBINATORICS 203

A pregeometry in which points and the empty set are closed, i.e., in which

cl(∅) = ∅ and cl(x) = {x} for all x ∈ X,

is called geometry. For any pregeometry (X, cl), there is an associated geometry
(X 0 , cl0 ) obtained by setting X 0 = X • / ∼, and cl0 (A/ ∼) = cl(A)• / ∼ where ∼
is the equivalence relation on X • = X \ cl(∅) defined by cl(x) = cl(y). Starting
from a vector space V , the geometry obtained in this way is the associated
projective space P(V ). The important properties of a pregeometry are in fact
mostly properties of the associated geometry.
Definition C.1.2. Let (X, cl) be pregeometry. A subset A of X is called

1. independent if a 6∈ cl(A \ {a}) for all a ∈ A;

2. a generating set if X = cl(A);
3. a basis if A is an independent generating set.
Lemma C.1.3. Let (X, cl) be a pregeometry with generating set E. Any inde-
pendent subset of E can be extended to a basis contained in E. In particular
every pregeometry has a basis.
Proof. Let B be an independent set. If x is any element in X \ cl(B), B ∪ {x} is
again independent. To see this consider an arbitrary b ∈ B. Then b 6∈ cl(B\{b}),
whence b 6∈ cl(B \ {b} ∪ {x}) by exchange.
This implies that for a maximal independent subset B of E, we have E ⊆
cl(B) and therefore X = cl(B).
Definition C.1.4. Let (X, cl) be a pregeometry. Any subset S gives rise to
two new pregeometries, the restriction (S, clS ) and the relativisation (X, clS ),
where

clS (A) = cl(A) ∩ S,

clS (A) = cl(A ∪ S).

Remark C.1.5. Let A be a basis of (S, clS ) and B a basis of (X, clS ). Then
the (disjoint) union A ∪ B is a basis of (X, cl).
Proof. Clearly A ∪ B is a generating set. Since B is independent over S, we
have b 6∈ clS (B \ {b}) = cl(A ∪ B \ {b}) for all b ∈ B. Consider an a ∈ A. We
have to show that a 6∈ cl(A0 ∪ B), where A0 = A \ {a}. As a 6∈ cl(A0 ), we let B 0
be a maximal subset of B with a 6∈ cl(A0 ∪ B 0 ). If B 0 6= B this would imply that
a ∈ cl(A0 ∪ B 0 ∪ {b}) for any b ∈ B \ B 0 which would in turn imply b ∈ cl(A ∪ B 0 ),
a contradiction.

We say A is a basis of S and B a basis over or relative to S.

Lemma C.1.6. All bases of a pregeometry have the same cardinality.
APPENDIX C. COMBINATORICS 204

Proof. Let A be independent and B a generating subset of X. We show that

|A| ≤ |B|.

Assume first that A is infinite. Then we extend A to a basis A0 . Choose for

every b ∈ B a finite subset Ab of AS0 with b ∈ cl(Ab ). Since the union of the Ab
is a generating set, we have A0 = b∈B Ab . This implies that B is infinite and
|A| ≤ |A0 | ≤ |B|.
Now assume that A is finite. That |A| ≤ |B| follows immediately from the
following exchange principle: Given any a ∈ A \ B there is some b ∈ B \ A
such that A0 = {b} ∪ A \ {a} is independent. For, since a ∈ cl(B), B cannot be
contained in cl(A \ {a}). Choose b in B but not in cl(A \ {a}). It follows from
the exchange property that A0 is independent.
Definition C.1.7. The dimension dim(X) of a pregeometry (X, cl) is the car-
dinality of a basis. For a subset S of X let dim(S) be the dimension of (S, clS )
and dim(X/S) the dimension of (X, clS ).
By Remark C.1.5 we have

Lemma C.1.8. dim(X) = dim(S) + dim(X/S).

The dimensions in our three standard examples are:
• The dimension of a vector space.

• The transcendence degree of a field.

• The degree of imperfection of a field of finite characteristic (see [10, § 13,
Ex.1]).
Let (X, cl) be a pregeometry. For arbitrary subsets A and B one sees easily
that the submodular law holds:

dim(A ∪ B) + dim(A ∩ B) ≤ dim(A) + dim(B).

One may hope for equality to hold if A and B are closed.

Definition C.1.9. We call a pregeometry (X, cl) modular, if

dim(A ∪ B) + dim(A ∩ B) = dim(A) + dim(B) (C.1)

for all cl-closed A and B.

The main examples are

• trivial pregeometries where cl(A ∪ B) = cl(A) ∪ cl(B) for all A, B.

• vector spaces with the linear closure operator.
APPENDIX C. COMBINATORICS 205

Let K be a field of transcendence degree at least four over its prime field F .
The following argument shows that the pregeometry of algebraic dependence
on K is not modular. Choose x, y, x0 , y 0 ∈ K algebraically independent over
F . From these elements we can compute a, b ∈ K such that ax + b = y and
ax0 +b = y 0 . Since the elements x, x0 , a, b generate the same subfield as x, y, x0 , y 0 ,
they are also algebraically independent. This implies that F (x) and F (x0 ) are
isomorphic over F (a, b)cl , where the superscript cl denotes the relative algebraic
closure in K. This isomorphism maps y to y 0 and therefore we have

F (x, y)cl ∩ F (a, b)cl ⊆ F (x, y)cl ∩ F (x0 , y 0 )cl = F cl .

So K is not modular since

tr. deg F (x, y, a, b) + tr. deg F = 3 + 0

< 2 + 2 = tr. deg F (x, y) + tr. deg F (a, b).

Note that the proof shows that tr. deg(K) ≥ 3 actually suffices.

Let us call two sets A and B (geometrically) independent over C if all subsets
A0 ⊆ A and B0 ⊆ B which are both independent over C are disjoint and their
union is again independent over C. The following is then easy to see.
Lemma C.1.10. For a pregeometry (X, cl) the following are equivalent:
1. (X, cl) is modular.
2. Any two closed A and B are independent over their intersection.
3. For any two closed sets A and B we have dim(A/B) = dim(A/A ∩ B).
Considering the closed 1- and 2-dimensional subsets of a modular pregeom-
etry (X, cl) as points and lines, respectively, these satisfy the Veblen–Young
Axioms of Projective Geometry provided any line contains at least three points:
namely, any two distinct points a, b lie on a unique line ab; and for four dis-
tinct points a, b, c, d, if the lines ab and cd intersect, then so do ac and bd. If
the dimension of X is at least 4, then by the fundamental theorem of projec-
tive geometry, this is indeed isomorphic to the projective geometry of a vector
space over some skew field (see e.g., [13], Thm. 1). The projective planes in
Section 10.4 are examples of projective geometries of dimension 3 which do not
arise from vector spaces over skew fields. Note that by Exercise C.1.4 a subset
A of a modular geometry is closed if and only if for any distinct a, b ∈ A the
line containing them is also contained in A.
Lemma C.1.11. A pregeometry (X, cl) is modular if and only if for all a, b, B
with dim(ab) = 2, dim(ab/B) = 1, there is c ∈ cl(B) such that dim(ab/c) = 1.
Proof. If (X, cl) is modular and a, b, B are as in the lemma, then ab and B are
dependent, but independent over the intersection of cl(ab) and cl(B). Let c be
an element of the intersection which is not in cl(∅). Then dim(ab/c) = 1.
APPENDIX C. COMBINATORICS 206

Assume that the property of the lemma holds. We show that the third
condition of Lemma C.1.10 is satisfied. For this we may assume that n =
dim(A/A ∩ B) is finite and proceed by induction on n. The cases n = 0, 1
are trivial. So assume n ≥ 2. Let a1 , . . . , an be a basis of A over A ∩ B.
We have to show that dim(a1 , . . . , an /B) = n. By induction we know that
dim(a1 . . . an−2 /B) = n − 2. So it is enough to show that dim(an−1 an /B 0 ) = 2
where B 0 be the closure of {a1 , . . . , an−2 } ∪ B. If not, by our assumption there
is c ∈ B 0 such that dim(an−1 an /c) = 1.
Definition C.1.12. A pregeometry (X, cl) is locally modular if (C.1) holds for
all closed sets A, B with dim(A ∩ B) > 0.
Remark C.1.13. Clearly, (X, cl) is locally modular if and only if for all x ∈
X \ cl(∅) the relativised pregeometry (X, clx ) is modular.
An affine subspace of a vector space V is a coset of a subvector space. The
affine subspaces of V define a locally modular geometry, which is not modular,
because of the existence of parallel lines. Note that in this example, points have
dimension 1, lines dimension 2, etc.
The arguments on page 205 show also that a field of transcendence degree
at least 4 is not locally modular. Just replace F by a subfield of transcendence
degree 1.
Exercise C.1.1. Consider a pregeometry (X, cl). Let A ^|cl C B be the rela-
0
| C B the relation cl(AC) ∩
tion of A and B being independent over C and A ^
cl(BC) = cl(C). Show the following:
1. ^|cl has the following properties as listed in Theorem 7.3.13: Mono-
tonicity, Transitivity, Symmetry, Finite Character and Local
Character.
2. ^ |0 has Weak Monotonicity (see Theorem 8.5.10), Symmetry, Fi-
nite Character and Local Character. Monotonicity holds only
if ^|0 = ^
|cl , i.e., if X is modular.
Exercise C.1.2. Prove that trivial pregeometries are modular.
Exercise C.1.3 (P. Kowalski). Let K be a field of characteristic p > 0 with
degree of imperfection at least 4. Prove that K with p-dependence is not locally
modular.
Exercise C.1.4. (X, cl) is modular if and only if for all c ∈ cl(A ∪ B) there are
a ∈ cl(A) and b ∈ cl(B) such that c ∈ cl(a, b).
Exercise C.1.5. The set of all closed subsets of a pregeometry forms a lattice,
where the infimum is intersection and the supremum of X and Y is X t Y =
cl(X ∪ Y ). Show that a pregeometry is modular if and only if the lattice of
closed sets is modular, i.e., if for all closed A, B, C

A ⊆ C ⇒ A t (B ∩ C) = (A t B) ∩ C.
APPENDIX C. COMBINATORICS 207

Exercise C.1.6. Let (X, cl) be a pregeometry of uncountable dimension. Sup-

pose that for all closed B of countable dimension the automorphism group
Aut(X/B) acts transitively on X \ B. Then (X, cl) is locally modular if and
only if for all closed B of countable dimension and all a, b with dim(a, b) = 2
and dim(a, b/B) = 1, and for every σ ∈ Aut(X/B) the two pairs (a, b) and
(σ(a), σ(b)) are not independent over ∅. Conclude that for any finite A, X is
locally modular if and only if XA is locally modular.

C.2 The Erdős–Makkai Theorem

Theorem C.2.1 (Erdős–Makkai). Let B be an infinite set and S a set of subsets
of B with |B| < |S|. Then there are sequences (bi | i < ω) of elements of B and
(Si | i < ω) of elements of S such that either

bi ∈ Sj ⇔ j < i (C.2)

or

bi ∈ Sj ⇔ i < j (C.3)

for all i, j ∈ ω.
Proof. Choose a subset S 0 of S of the same cardinality as B such that any two
finite subsets of B which can be separated by an element of S can be separated
by an element of S 0 . The hypothesis implies that there must be an element S ∗
of S which is not a Boolean2 combination of elements of S 0 .
Assume that for some n, three sequences (b0i | i < n) in S ∗ , (b00i | i < n) in
B \ S ∗ and (Si | i < n) in S 0 have already been constructed. Since S ∗ is not a
Boolean combination of S0 , . . . , Sn−1 , there are b0n ∈ S ∗ and b00n ∈ B \ S ∗ such
that for all i < n
b0n ∈ Si ⇔ b00n ∈ Si .
Choose Sn as any set in S 0 separating {b00 , . . . , b0n } and {b000 , . . . , b00n }.
Now, an application of Ramsey’s theorem shows that we may assume that
either b0n ∈ Si or b0n 6∈ Si for all i < n. In the first case we set bi = b00i and get
(C.2), in the second case we set bi = b0i+1 and get (C.3).

C.3 The Erdős–Rado Theorem

Definition C.3.1. For cardinals κ, λ, µ we write κ → (λ)nµ (read as κ arrows
λ) to express the fact that for any function f : [κ]n → µ there is some A ⊆ κ
with |A| = λ, such that f is constant on [A]n . In other words, every partition
of [κ]n into µ pieces has a homogeneous set of size λ.
2 It actually suffices to consider positive Boolean combinations.
APPENDIX C. COMBINATORICS 208

With this notation Ramsey’s Theorem 5.1.5 states that

ω → (ω)nk for all n, k < ω.

In an analogous manner one can define a cardinal κ to be a Ramsey cardinal

if κ → (κ)<ω n
2 . In other words, if for any n a partition of [κ] into two classes
is given, there is a set of size κ simultaneously homogeneous for all partitions.
A Ramsey cardinal κ satisfies κ → (κ)<ω γ for all γ < κ (see [33] 7.14). More
generally, an uncountable cardinal κ is called weakly compact if it satisfies κ →
(κ)22 . Such cardinals are weakly inaccessible and their existence cannot be
proven from ZFC.

Theorem C.3.2 (Erdős–Rado). i+ + n+1

n (µ) → (µ )µ .

Proof. This follows from µ+ → (µ+ )1µ and the following lemma.
Lemma C.3.3. If κ+ → (µ+ )nµ , then (2κ )+ → (µ+ )n+1
µ .

Proof. We note first that the hypothesis implies µ ≤ κ. Now let B be a set of
cardinality (2κ )+ and f : [B]n+1 → µ be a colouring. If A is a subset of B, we
call a function p : [A]n → µ a type over A. If b ∈ B \ A, the type tp(b/A) is the
function which maps each n-element subset s of A to f (s ∪ {b}). If |A| ≤ κ,
there are at most 2κ many types over A. Thus an argument as in the proof of
Lemma 6.1.2 shows that there is some B0 ⊆ B of cardinality 2κ such that, for
every A ⊆ B0 of cardinality at most κ, every type over A which is realised in B
is already realised in B0 .
Fix an element b ∈ B \ B0 . We can easily construct a sequence (aα )α<κ+ in
B0 such that every aα has the same type over {aβ | β < α} as b. By assumption
{aα | α < κ+ } contains a subset A of cardinality µ+ such that tp(b/A) is
constant on [A]n . Then f is constant on [A]n+1 .
Appendix D

Solutions to exercises

Exercises whose results are used in the book have their solutions marked with
an asterisk.

Chapter 1. The basics

Exercise 1.2.3. We consider formulas which are built using ¬, ∧, ∃, ∀ and we
move the quantifiers outside using

¬∃xϕ ∼ ∀x¬ϕ
¬∀xϕ ∼ ∃x¬ϕ
(ϕ ∧ ∃xψ(x, y)) ∼ ∃z (ϕ ∧ ψ(z, y))
(ϕ ∧ ∀xψ(x, y)) ∼ ∀z (ϕ ∧ ψ(z, y)) .

In the last equivalence we replaced the bounded variable x by a variable z which

does not occur freely in ϕ.
Exercise 1.2.4. That Πi∈I Ai /F is well defined is easy to see. Los’s The-
orem is proved by induction on the complexity of ϕ. The case of atomic for-
mulas is clear by construction. If ϕ is a conjunction, we use the fact that X ∈
F and Y ∈ F ⇔ X ∩ Y ∈ F. If ϕ is a negation, we use X 6∈ F ⇔ I \ X ∈ F.
If X = {i ∈ I | Ai |= ∃y ψ(āi , y)} ∈ F, choose for every i ∈ I some bi ∈ Ai such
that i ∈ X ⇒ Ai |= ψ(āi , bi ). Then by induction Πi∈I Ai /F |= ψ((āi )F , (bi )F )
and we have Πi∈I Ai /F |= ∃y ψ((āi )F , y). The converse is also easy.
Exercise 1.3.1. The theory DLO of dense linear orders without endpoints
is axiomatised in LOrder by
• ∀x ¬x < x

• ∀x, y, z (x < y ∧ y < z → x < z)

.
• ∀x, y (x < y ∨ x = y ∨ y < x)

209
APPENDIX D. SOLUTIONS TO EXERCISES 210

• ∀x, z (x < z → ∃y (x < y ∧ y < z))

• ∀x∃y x < y
• ∀y∃x x < y.
The class of all algebraically closed fields can be axiomatised by the theory ACF:
• Field (field axioms)
.
• For all n > 0: ∀x0 . . . xn−1 ∃y x0 + x1 y + · · · + xn−1 y n−1 + y n = 0.

Exercise 1.3.3. Let A and B be two elementarily equivalent L-structures.

It is easy to see that A and B are isomorphic if L is finite. Let L be arbitrary
and assume towards a contradiction that A and B are not isomorphic. Then
for every bijection f : A → B there is a Zf ∈ L which is not respected by f .
If L0 is the set of all the Zf , then A  L0 and B  L0 are not isomorphic,
contradicting our first observation. One should note that isomorphism follows
also from Exercise 6.1.1 and Lemma 4.3.3.
∗
Exercise 1.3.5. Show by induction on the complexity of ϕ that for all
f ∈ I and all ā in the domain of f we have A |= ϕ(ā) ⇔ B |= ϕ(f (ā)).

Chapter 2. Elementary extensions and compact-

ness
Exercise 2.1.2. Hint for Part 1: We may assume that C = {Ai | i ∈ I}
is a set. If M is a model of T , choose an ultrafilter F on I which contains
Fϕ = {i ∈ I | Ai |= ϕ} for all ϕ ∈ Th(M).
Exercise 2.2.1. Hint: Let T be a finitely satisfiable theory. Consider the
every ∆ ∈ I choose a model A∆ . Find a
set I of all finite subsets of T . For Q
suitable ultrafilter F on I such that ∆∈I A∆ /F is a model of T .
Exercise 2.3.1. 2. If e is a new element of an elementary extension and if
f and g are almost disjoint, then f (e) and g(e) are different.
3. Let Q be a proper elementary extension. Show first that Q contains a
positive infinitesimal element e. Then show that for every r ∈ R there is an
element qr such that Pr (qr ) and Qr (qr + e) are true in Q.
Exercise 2.3.3. For every prime p, ACFp has a model which is the union
of a chain of finite fields.

Chapter 3. Quantifier elimination

∗
Exercise 3.1.1. We imitate the proof of Lemma 3.1.1. That a) implies b)
is clear. For the converse consider an element y1 of Y1 and Hy1 , the set of all
elements of H containing y1 . Part b) implies that the intersection of the sets
APPENDIX D. SOLUTIONS TO EXERCISES 211

in Hy1 is disjoint from Y2 . So a finite intersection hy1 of elements of Hy1 is

disjoint from Y2 . The hyi , y1 ∈ Y1 , cover Y1 . So Y1 is contained in the union H
of finitely many of the hyi . Hence H separates Y1 from Y2 .
Exercise 3.2.3. Hint: For any simple existential formula write down the
equivalence to a quantifier-free formula. Show that this is equivalent to an
∀∃-sentence and that T is axiomatised by these.
Exercise 3.3.1. Like the proof of Theorem 3.3.2 with order-preserving au-
tomorphisms replaced by edge-preserving ones.
∗
Exercise 3.3.2. The cases ACF, RCF are easy. So we concentrate on
DCF0 . Clearly, the algebraic closure of the differential field generated by A is
contained in the model-theoretic algebraic closure. For the converse, let K0
be an algebraically closed differential field and a0 an element not in K0 . If
dim(a0 /K0 ) is infinite, then a0 and all its derivatives have the same type over K0 ,
so a0 is not not model-theoretically algebraic over K0 . If dim(a0 /K0 ) = n > 0,
consider the minimal polynomial f of a0 over K0 . Let K1 be a d-closed extension
of K0 containing a0 . Then f remains irreducible over K1 and there is some
a1 whose minimal polynomial over K1 is f . Now extend K1 to some field K2
containing a1 etc.. In this way we obtain an infinite sequence of distinct elements
having the same type over K0 as a0 , showing that a0 is not algebraic over K0
in the sense of model theory.
Exercise 3.3.3. Let K be algebraically closed, X = {a | K |= γ(a, b)} a
definable subset of K n and suppose that f : X → X is given by an n-tuple of
polynomials f (x, b). We may assume that γ(x, z) is quantifier-free. We want to
show that K satisfies


∀y ∀x ϕ(x, y) → ϕ(f (x, y), y) ∧
. .

∀x, x0 ϕ(x, y) ∧ ϕ(x0 , y) ∧ f (x, y) = f (x0 , y) → x = x0 →
.

∀x0 ϕ(x0 , y) → ∃x(ϕ(x, y) ∧ f (x, y) = x0 ) .

This is obviously true in finite fields (even in all finite LRing -structures) and
logically equivalent to an ∀∃-sentence. So the claim follows from Exercise 2.3.3.
For the second part use Exercise 6.1.14 and proceed as before.

Chapter 4. Countable models

∗
Exercise 4.2.1. The set [ϕ] is a singleton if and only if [ϕ] is non-empty
and cannot be divided into two non-empty clopen subsets [ϕ ∧ ψ] and [ϕ ∧ ¬ψ].
This means that for all ψ either ψ or ¬ψ follows from ϕ modulo T . So [ϕ] is a
singleton if and only if ϕ generates the type

hϕi = {ψ(x) | T ` ∀x (ϕ(x) → ψ(x))},

which of course must be the only element of [ϕ].

APPENDIX D. SOLUTIONS TO EXERCISES 212

This shows that [ϕ] = {p} implies that ϕ isolates p. If, conversely, ϕ isolates
p, this means that hϕi is consistent with T and contains p. Since p is a type,
we have p = hϕi.
Exercise 4.2.2. a): The sets [ϕ] are a basis for the T closed subsets of Sn (T ).
So the closed sets of Sn (T ) are exactly the intersections ϕ∈Σ [ϕ] = {p ∈ Sn (T ) |
Σ ⊆ p}.
b): The set X is the union of a sequence of countable nowhere dense sets Xi .
We may assume that the Xi are closed, i.e., of the form {p ∈ Sn (T ) | Σi ⊆ p}.
That Xi has no interior means that Σi is not isolated. The claim follows now
from Corollary 4.1.3.
Exercise 4.2.3. Let X = {tp(a0 , a2 , . . .) | the ai enumerate a model of T }.
Consider for every formula ϕ(v̄, y) the set Xϕ = {p ∈ Sω | (∃yϕ(v̄, y) →
ϕ(v̄, vi )) ∈ p for some i}. The Xϕ are open and dense and X is the intersection
of the Xϕ .
∗
Exercise 4.2.5. The homeomorphism from Sm (aB) to the fibre above
tp(a/B) is given by tp(c/aB) 7→ tp(ca/B).
∗
Exercise 4.3.9. Assume that A is κ-saturated, B a subset of A of smaller
cardinality than κ and p(x, ȳ) a (n + 1)-type over B. Let b̄ ∈ A be a realisation
of q(ȳ) = p  ȳ and a ∈ A a realisation of p(x, b̄). Then (a, b̄) realises p.
∗
Exercise 4.3.13. If B is ω-saturated and elementarily equivalent to A,
then the set of all isomorphisms between finitely-generated substructures that
are elementary partial maps in the sense of A and B is non-empty, and has the
back-and-forth property.
Now assume that A and B are partially isomorphic via I; they are elemen-
tarily equivalent by Exercise 1.3.5. Consider a finite subset B0 of B and a type
p ∈ S(B0 ). There is an f ∈ I which contains B0 in its image. Choose a realisa-
tion a of f −1 (p) in A and an extension g ∈ I of f which is defined on a. Then
g(a) realises p.
∗
Exercise 4.4.1. If M and M0 are K-saturated, consider the set I of all
isomorphisms between finitely-generated substructures of M and M0 .
Exercise 4.5.1. Let n be such that Sn (T ) uncountable. Prove that there is
a consistent formula ϕ such that both [ϕ] and [¬ϕ] are uncountable. Inductively
we obtain a binary tree of consistent formulas; see proof of Theorem 5.2.6(2).

Chapter 5. ℵ1 -categorical theories

Exercise 5.1.2. To ease notation we replace the partition by a function γ :
[A]n → {1, . . . , k}. Fix a non-principal ultrafilter U on A. For each s ∈ [A]n−1
choose c(s) such that {a ∈ A | γ(s ∪ {a}) = c(s)} belongs to U. Construct
a sequence a0 , a1 , . . . of distinct elements such that γ(s ∪ {an }) = c(s) for all
s ∈ [{a0 , . . . , an−1 }]n−1 . Apply induction to c restricted to [{a0 , a1 , . . .}]n−1 .
APPENDIX D. SOLUTIONS TO EXERCISES 213

Exercise 5.2.1. Let T define a linear ordering of the universe. By Exer-

cise 8.2.8 there is a linear ordering J of bigger cardinality than κ which has
a dense subset of cardinality κ. A compactness argument shows that T has a
model with a subset B which is order-isomorphic to J. Let A be a dense subset
of B of cardinality κ. Then all elements of B have different types over A and
so | S(A)| ≥ |B| > κ.
Exercise 5.2.3. As an example consider formulas ϕ0 (x), ϕ1 (x), . . . without
parameters such that ϕn (x)Qimplies ϕn+1 (x). We have to show that Φ = {ϕn |
n < ω} is realised in M = i<ω Ai /F if each ϕn is realised in M. For each n
there is a B n ∈ F such that, for all i ∈ B n , ϕn is realised in Ai . We may assume
that the B n are descending and have empty intersection since F is non-principal.
Now for every i ∈ B 0 let n be maximal with i ∈ B n . Choose an element ai
which realises ϕn in Ai . For i outside B 0 choose ai ∈ Ai arbitrary. Then the
class of (ai )i<ω realises Φ in M.
∗
Exercise 5.2.5. If T is totally transcendental, each reduct is also totally
transcendental. The converse follows from the observation that a binary tree
contains only countably many formulas.
∗
Exercise 5.2.6. If T is κ-stable, then | Sn (∅)| ≤ κ. Choose for any two
n-types over the empty set a separating formula. Then any formula is logically
equivalent to a finite Boolean combination of these κ-many formulas.
Exercise 5.3.2. a) ⇒ b): Let A be a countable subset of the model M.
A prime extension of A is just a prime model of TA = Th(MA ). So the claim
follows from Theorem 4.5.7.
b) ⇒ c): Assume that A is contained in the model M and that the isolated
types are not dense over A. So there is a consistent L(A)-formula ϕ, which
does not contain a complete L(A)-formula. Add a predicate P for the set A
and consider the L ∪ {P }-structure (M, A). Choose a countable elementary
substructure (M0 , A0 ) which contains the parameters of ϕ. Then ϕ does not
contain a complete L(A0 )-formula.
c) ⇒ a): Like the proof of 5.3.3.
Exercise 5.5.4. Clearly, T (q) is complete λ-stable if and only if T is. It
is also clear that if T (q) has a Vaughtian pair then so does T . For the converse
use a construction as in Theorem 5.5.2 to find a Vaughtian pair M ≺ N such
that q is realised in N.
Exercise 5.5.7. Hint: Use Exercise 5.5.6.
Exercise 5.6.1. If p ∈ S(A) is not algebraic, then all n-types

q(x1 , . . . , xn ) = {xi , i = 1, . . . , n, satisfies p and the xi are pairwise distinct}

are consistent and hence realised in M.

∗
Exercise 5.6.2. If ci ∈ acl(Acn ) for some i < n, let a0 . . . an realise
tp(c0 . . . cn /A) with a0 . . . an−1 ∈
/ acl(AB). Then an ∈/ B. If ci ∈
6 acl(Acn ) for
APPENDIX D. SOLUTIONS TO EXERCISES 214

all i < n, realise tp(cn /A) by some an ∈

/ B and then tp(c0 . . . cn−1 /Aan ) outside
B.
Exercise 5.7.2. Prove that the theory eliminates quantifiers.

Chapter 6. Morley rank

∗
Exercise 6.1.2. Fix any model M which contains A. If b is not algebraic
over A, thenb has a conjugate over A which does not belong to M . This implies
that M has a conjugate M 0 over A which does not contain b.
Note that Exercise 5.6.2 implies that if C is any set without elements alge-
braic over A, there is a conjugate M 0 of M which is disjoint from C.
Exercise 6.1.3. Choose special models Ai of Ti of the same cardinality
and observe that a reduct of a special model is again special.
Exercise 6.1.4. If θ does not exist, the set T 0 of all L-sentences θ such
that ` ϕ1 → θ or ` ¬ϕ2 → θ is consistent. Choose a complete L-theory T which
contains T 0 and apply Exercise 6.1.3 to T1 = {ϕ1 } ∪ T and T2 = {¬ϕ2 } ∪ T 0 .
Exercise 6.1.5. We use the criterion of Exercise 2.1.2(2). Let C be the
class of all reducts of models of T 0 to L. It is easy to see that C is closed under
ultraproducts. If A belongs to C and B is elementarily equivalent to A, consider
an expansion A0 of A to a model of T 0 . Now choose two special elementary
extensions A0 ≺ D0 , B ≺ E of the same cardinality. Then D ∼ = E belongs to C.
Exercise 6.1.6. Let |A| < κ and p = p(xi )i<κ be a κ-type over p. Denote
by pα the restriction of p to the variables (xi )i<α . Construct a realisation
(ai )i<κ of p inductively: if (ai )i<α realises pα , choose aα as a realisation of
pα+1 ((ai )i<α , xα ).
Exercise 6.1.8. 1) Construct an elementary chain (Mα )α<κ of structures
of cardinality 2<κ such that all types over subsets of Mα of cardinality less than
κ are realised in Mα+1 . This is possible because for regular κ a set of cardinality
2<κ has at most 2<κ -many subsets of cardinality less than κ.
Part 2) follows from 1) since 2<κ = κ.
∗
Exercise 6.1.11. If B ⊆ dcl(A), every formula with parameters in B is
equivalent to a formula with parameters in A. So every type over A axiomatises
a type over B. This proves 1) ⇒ 2). For the converse show that b ∈ dcl(A) if
tp(b/A) has a unique extension to Ab.
∗
Exercise 6.1.12. If ϕ(a, b) is a formula witnessing b ∈ dcl(a), let D
equal the class of elements x for which there is a unique y with ϕ(x, y) and let
f : D → E denote the corresponding map, which we may assume to be surjective.
If furthermore a ∈ dcl(b) is witnessed by ψ(y, x) and a function g : E1 → D1 , we
get a 0-definable bijection {(x, y) ∈ E × D1 |f (x) = y and g(y) = x}.
Exercise 6.1.13. Use Exercise 3.3.2.
APPENDIX D. SOLUTIONS TO EXERCISES 215

Exercise 6.1.14. Use Exercise 6.1.13 and compactness.

Exercise 6.1.16. By induction on n. We distinguish two cases. First
assume that for some i < 0, Hn ∩ Hi has finite index in Hn . We can then cover
every coset of Hn by finitely many cosets of Hi . Since G is not a finite union of
cosets of H0 , . . . , Hn−1 , we are done. Now assume that all Hi0 = Hn ∩ Hi have
infinite index in Hn . Assume towards a contradiction that G is a finite union
of cosets of the H0 , . . . , Hn . Since Hn has infinite index in G, there must be a
coset of Hn which is covered by a finite number of cosets of H0 , . . . , Hn−1 . This
implies that Hn is a finite union of cosets of the Hi0 , i < n, which is impossible
by induction.
∗
Exercise 6.1.17. To see that Exercise 6.1.16 implies Exercise 6.1.15 con-
sider the subgroups Hi := Gci , i ≤ n, each of infinite index in G. The finitely
many cosets aj Hi with aj (ci ) ∈ B, i ≤ n, do not cover G, so there is some g ∈ G
such that for all i ≤ n we have g(ci ) ∈ / B.
For the converse, let H1 , . . . , Hn ≤ G be subgroups of infinite index, and
consider the action of G on the disjoint union of the G/Hi by left translation.
By Exercise 6.1.15, for any a1 , . . . , an ∈ G there is some g ∈ G such that for all
i ≤ n we have g(1 · Hi ) ∈
/ ai Hi , proving Exercise 6.1.16.
∗
Exercise 6.2.1. Let ϕ(x, a) be defined from the parameter tuple a ∈ M .
There is an infinite family (ϕi (x, bi )) of pairwise inconsistent formulas of Morley
rank ≥ α which imply ϕ(x, a). Since M is ω–saturated, there is a sequence (ai )
in M such that tp(a, a0 , a1 , . . .) = tp(a, b0 , b1 , . . .).
Exercise 6.2.2. Let I be the set of all i for which ϕ ∧ ψi has rank α. The
hypothesis implies that all k-element subsets of I contain two indices i, j such
that ϕ ∧ ψi 6∼α ϕ ∧ ψj . So |I| ≤ (k − 1) MD ϕ.
Exercise 6.2.3. Let G0 be the intersection of all definable subgroups of
finite index; it is definable by Remark 6.2.8. If N is a finite subgroup which is
normalised by G, the centraliser of N in G is a definable group of finite index
in G0 .
Exercise 6.2.4. Let M be an ω-saturated model and let p be a type over
M of Morley rank α and degree n, witnessed by ϕ(x, m) ∈ p. If n > 1, there is
a formula ψ(x, b) such that ϕ(x, m) ∧ ψ(x, b) and ϕ(x, m) ∧ ¬ψ(x, b) both have
Morley rank α. Choose a ∈ M with tp(a/m) = tp(b/m). Then both formulas
ϕ(x, m) ∧ ψ(x, a) and ϕ(x, m) ∧ ¬ψ(x, a) have rank α and degree less than n
and one of these formulas belongs to p, contradicting the choice of ϕ(x, m).
Exercise 6.2.5. Assume that there is a formula ϕ(x, b) of rank ≥ |T |+ .
Construct a binary tree of formulas (ϕs (x, y s ) | s ∈ <ω 2) below ϕ(x, b) so that
for all k and all α < |T |+ there are parameters as such that MR ϕs (x, as ) ≥ α
for all s with |s| = k. Conclude that MR ϕ(x, b) = ∞,
∗
Exercise 6.2.8. Let a be in acl(A) and a1 , . . . , an the conjugates of a over
A. Then ϕ(x, a) and ϕ(x, a1 ) ∨ · · · ∨ ϕ(x, an ) have the same Morley rank.
APPENDIX D. SOLUTIONS TO EXERCISES 216

Exercise 6.4.1. In a pregeometry a finite set A is independent from B

over C if and only if dim(A/BC) = dim(A/C). Now use Theorem 6.4.2.
∗
Exercise 6.4.2. Assume that abC is independent from B and apply
Proposition 6.4.9.
Exercise 6.4.4. The first two inequalities follow easily from Lemma 6.4.1.
For the third inequality we may assume that A has Morley degree 1. We dis-
tinguish two cases:
a) βgen > 0. Let Di be an infinite family of disjoint definable subclasses of
B defined over C 0 ⊃ C. Choose a ∈ A which has rank α over C 0 . For some i
the rank of f −1 (a) ∩ Di is bounded by some βgen0
< βgen . By induction we have
0
MR(Di ) ≤ β · α + βgen < β · α + βgen .
b) βgen = 0. Then A contains a definable subclass A0 of rank α over which all
fibres are finite. By the second inequality we then have MR(f −1 (A0 )) ≤ α ≤ β·α.
If A0 6= A we have MR(A \ A0 ) = α0 < α and by induction MR(f −1 (A \ A0 )) ≤
β · α0 + β ≤ β · α.
Exercise 6.4.5. The language L contains a binary relation symbol E and
unary predicate Pni for all n and i ≤ n, and T says that E is an equivalence
relation with infinite classes, the Pni are infinite and disjoint and for each n the
union of Pn0 ∪ · · · ∪ Pnn is an E-equivalence class.

Chapter 7. Simple theories

Exercise 7.1.1. Choose any model M 0 which extends A. Use Lemma 7.1.1
to get a realisation I 0 of EM(I/A) which is indiscernible over M 0 and has the
same order type as I. Then I 0 realises tp(I/A).
Exercise W 7.1.3. If p forks over A, there is some ϕ(x, m) ∈ p which implies
a conjunction `<d ϕ` (x, b) of formulas which divide over A. Choose a tuple b0
in M which realises the type of b over Am. The formulas ϕ` (x, b0 ) fork over A
and one of them belongs to p.
∗
Exercise 7.1.4. Let q be A-invariant. Use Lemma 7.1.4 to show that q
does not divide over A: if π(x, b) belongs to q, then all the π(x, bi ) also belong
to q. That q does not fork over A follows from Exercise 7.1.3.
∗
Exercise 7.1.5. p contains a formula ϕ which divides over A. So there
are κ many αi ∈ Aut(M/A) such that the system of all αi (ϕ) is k-inconsistent.
This implies that κ many of the αi (p) must be distinct.
Exercise 7.1.6. The type p(x) forks over the empty set since it implies
the disjunction of cyc(0, x, 3) and cyc(2, x, 1).
∗
Exercise 7.1.7. This is just a variant of Proposition 7.1.6. Let I = (bi |
i < ω) be a sequence of A-indiscernibles containing b such that (ϕ(xbi ) | i < ω)
is k-inconsistent. Since tp(a/Ab) does not divide over A, we can assume that I
is indiscernible over Aa.
APPENDIX D. SOLUTIONS TO EXERCISES 217

Exercise S with pα ∈ S(Aα ). Their

7.2.1. Let (pα )α∈|T |+ be a chain of types
union α∈|T |+ pα does not fork over a subset A0 of α∈|T |+ Aα of cardinality at
S

most |T | by Proposition 7.2.5 (Local Character) and Proposition 7.2.15. Since

A0 ⊆ Aα for some sufficiently large α ∈ |T |+ , from that value of the index on-
wards the chain no longer forks. (Note that this property is just a reformulation
of Local Character for κ = |T |+ .)
For the last sentence note that otherwise the types pα = tp(c/A{bβ | β < α})
would contradict the first part.
∗
Exercise 7.2.2. Let q = tp(b/B) and r = tp(c/C). Find an A-automor-
phism which maps c to b and C to C 0 such that B ^ | Ab C 0 . Then r0 = tp(b/C 0 )
0
and s = tp(b/BC ) are as required: we have B ^ | A b, which together with
B^ | Ab C 0 yields B ^| A C 0 b from which b ^
| C 0 B.
Exercise 7.2.3. By monotonicity and symmetry it suffices to show that aX
and aY \X are independent over A. So we can assume that X and Y are disjoint
and, by finite character, that X and Y are finite. We proceed by induction
on |X ∪ Y |. Let z ∈ X ∪ Y be maximal. By symmetry we may assume that
z ∈ Y . Then az is independent from aX∪Y \{z} . The claim now follows from the
induction hypothesis and transitivity.
∗
Exercise 7.2.4. This follows from the fact that if (bi | i < ω) is in-
discernible over A, there is an A-conjugate B 0 of B such that (bi | i < ω) is
indiscernible over B 0 .
Exercise 7.2.5. Up to symmetry this is only a reformulation of Monotonic-
ity and Transitivity.
Exercise 7.2.6. For ease of notation we restrict to the following special
case: If b1 b2 is independent from C, we have

| b2
b1 ^ ⇐⇒ | b2
b1 ^
C

since by Corollary 7.2.18 both sides of this equivalence are equivalent to the
triple (b1 , b2 , C) being independent. (For the direction from right to left it
suffices in fact to assume that b1 and b2 are independent from C individually.)
∗
| AB C. Now the claim follows
Exercise 7.2.7. The hypothesis implies a ^
from Remark 7.1.3.
Exercise 7.3.1. Let I be theSsequence b = b0 , b1 , . . .. Use Proposition 7.3.6
n
S that {π(x, bi ) | i < 2 } does not fork over A. So
and induction on n to show
we find a realisation of {π(x, bi ) | i < ω} which is independent from I over A.
That one can choose c in such a way that I is indiscernible over Ac follows from
Lemma 5.1.3 and Finite Character.
Exercise 7.3.2. 1. Let (ci | i < ω) be an antichain for θ1 ∧ θ2 . Then
by Ramsey’s theorem there is an infinite A ⊆ ω such that (ci | i ∈ A) is an
antichain for θ1 or for θ2 .
APPENDIX D. SOLUTIONS TO EXERCISES 218

2. If θ∼ (x, y) is not thick, it has, by compactness, antichains (ci | i ∈ I)

indexed by arbitrary linear orders I. If I ∼ is the inverse order, (ci | i ∈ I ∼ ) is
an antichain for θ.
3. Consider θ ∧ θ∼ .
Exercise 7.3.3. Choose an A-conjugate c of b, different from b, and set
B = b and C = c.
|0 A A with Existence and use Monotonicity
Exercise 7.3.5. Show that a ^
and Transitivity.
Exercise 7.3.6. Let p ∈ S(M ) with two different non-forking extensions
to B ⊃ M . Let B0 , B1 , . . . be an M -independent family of conjugates of B.
Then on each Bi there are two different extensions qi0 , qi of p. Now by the
independence theorem for any function  : ω → 2 there is a non-forking extension
(i)
q  of p to i<ω Bi , which extends every each qi .
S

Exercise 7.4.1. If α ∈ Aut(C) has the given property, choose a model M

of size |T | and β ∈ Autf (C) with β  M = α  M . Then αβ −1 ∈ Aut(C/M ) ⊆
Autf (C).
Exercise 7.4.2. If θ is thick and defined over A, the conjunction of the
A-conjugates of θ is thick and defined over A.
Exercise 7.4.3. Extend B and C to models MB and MC such that
MB ^| A MC , b ^
| A MB and c ^
| A MC . Now it suffices to find some d such that
| A MB MC , tp(d/MB ) = tp(b/MB ) and tp(d/MC ) = tp(c/MC ).
d^
Exercise 7.4.4. We have to show that for every thick θ(x, y) the formula
θ(x, a) does does not divide over A. So let a = a0 , a1 , . . . be indiscernible over
A. Then |= θ(ai , aj ) for all i, j. This shows that {θ(x, ai ) | i < ω} is finitely
satisfiable. If T is simple and B is any set, choose a00 independent from B over
A such that ncA (a00 , a). Finally choose a0 such that tp(a0 /Aa) = tp(a00 /Aa) and
a0 ^
| Aa B.
Exercise 7.4.5. If I is an infinite sequence of indiscernibles over A, then
I is indiscernible over some model which contains A.
Exercise 7.4.6. Use the Erdős–Rado Theorem C.3.2.
Exercise 7.4.7. Let R be bounded and A-invariant and a0 , a1 , . . . indis-
cernible over A. Show that R(ai , aj ) for all i < j.
Exercise 7.4.8. This follows from Exercise 7.4.7(c).
Exercise 7.5.1. If K is a subfield of Falg
p , consider the set I = {i | Fpi ⊆
K}. If F is an ultrafilter
Q on I which contains {j ∈ I | i|j} for all i ∈ I, then K
is the absolute part of i∈I Fpi /F.
Exercise 7.5.2. Use Corollary 7.5.3 and Remark B.4.12.
APPENDIX D. SOLUTIONS TO EXERCISES 219

Chapter 8. Stable theories

Exercise 8.1.1. If q is a coheir of p, the sets ϕ(M ), ϕ ∈ q are non-empty
and closed under finite intersections. So there is an ultrafilter F on M which
contains all ϕ(M ).
Exercise 8.1.2. Consider a Morley sequence of a global coheir extension
of tp(a/M ) = tp(b/M ) over M .
Exercise 8.1.3. Use Exercise 7.1.1.
Exercise 8.1.4. We may assume that ψ has Morley degree 1 and do
induction on MR ψ = α. If α = β + 1, choose an infinite disjoint family of
M -definable classes ψi (C) ⊆ ψ(C) of rank β = MR(ψ ∧ ¬ϕ) < MR ψ. Then
MR(ψi ∧ϕ) = β for some i and ψi ∧ϕ is realised in M by induction. If α is a limit
ordinal, choose some definable ψ 0 (C) ⊆ ψ(C) with MR(ψ ∧ ¬ϕ) < MR(ψ 0 ) < α
and apply the induction hypothesis to ψ 0 and ψ 0 ∧ ϕ.
The second part follows from the first.
Exercise 8.1.5. If b0 is another realisation of q(y)  B and a0 realises
p(x)  Bb0 there is a B-automorphism α taking b0 to b. If p(x) is A-invariant,
α(a0 ) realises tp(a/Bb) and so p(x) ⊗ q(y) is well defined. The same proof shows
p(x) ⊗ q(y) to be A-invariant if p, q both are A-invariant.
∗
Exercise 8.2.1. If ϕ(x, y) has the order property witnessed by a0 , a1 , . . .
and b0 , b1 , . . ., then the sequence a0 b0 , a1 b1 , . . . is ordered by the formula ϕ0 (xy, x0 y 0 ) =
ϕ(x, y 0 ). The converse is obvious.
Exercise 8.2.3. The formula xRy has the binary tree property.
Exercise 8.2.5. If ϕ(x, y) has SOP, the formula ψ(x, y1 , y2 ) = ϕ(y1 , x) ∧
¬ϕ(y2 , x) has the tree property with respect to k = 2.
Exercise 8.2.6. Hint: if T is unstable, there are a formula ϕ(x, y) and
indiscernibles (ai bi | i ∈ Q) with |= ϕ(ai , bj ) ⇔ i < j. If ϕ does not have
the independence property, there are finite disjoint subsets J, K of Q such that
ΦJ,K (y) = {ϕ(ai , y) | i ∈ J} ∪ {¬ϕ(ai , y) | i ∈ K} is inconsistent. Not all of J
can be less than all elements of K. Choose J and K minimising the number
of inversions F = {(j, k) ∈ J × K | k < j}. Choose (j, k) ∈ F so that the
interval (k, j) does not contain any elements of J ∪ K. Write J = J0 ∪ {j} and
K = K0 ∪ {k}. Then ΦJ0 ∪{k},K0 ∪{j} (y) is consistent and the formula (with
parameters) ^
ΦJ0 ,K0 (y) ∧ ¬ϕ(x, y)
has the strict order property.
∗
Exercise 8.2.7. a) ⇒ b): A type p ∈ S(A) is determined by the family of
all pϕ , the ϕ-parts of p. Hence
|T |
Y Y
|S(A)| ≤ |Sϕ (A)| ≤ |A| = |A| .
ϕ ϕ
APPENDIX D. SOLUTIONS TO EXERCISES 220

So if |A| = λ and λ|T | = λ, then |S(A)| ≤ λ.

b) ⇒ c): Clear.
c) ⇒ a): Follows directly from Theorem 8.2.3, a) ⇒b).
The last assertion follows from Lemma 5.2.2.
Exercise 8.2.8. See the proof of 8.2.3, a) ⇒ d). Let A = 0, 12 , 1 . We


embed I into µ A by extending sequences from I0 to a sequence of length µ with

constant value 21 . We order I by the ordering induced from the lexicographic
ordering of µ A.
Exercise 8.2.9. If ψ(x, y) is a Boolean combination of ϕ0 (x, y) and ϕ1 (x, y),
show that the ψ-type of a tuple over B is determined by its ϕ0 (x, y)-type and
its ϕ1 (x, y)-type. So we have | Sψ (B)| ≤ | Sϕ0 (B)| · | Sϕ1 (B)|.
Exercise 8.2.10. 1. Since Rϕ (ψ) = ∞, as in Theorem 6.2.7, there is a
binary tree of consistent formulas of the form ψ ∧ δ, δ ∈ Φ. Now we follow
the proof of Theorem 5.2.6 and conclude first that over some countable A there
are uncountable many ϕ-types which contain ψ. This implies then that ϕ(x, y)
has the binary tree property. Again by the proof of 6.2.7 this implies that
Rϕ (ψ) = ∞.
2. If β < Rϕ (ψ), there is a formula ϕ(x, a) such that ψ(x) ∧ ϕ(x, a) and
ψ(x) ∧ ϕ(x, a) have rank at least β. So if ω ≤ Rϕ (ψ), then ϕ would have binary
trees of arbitrary finite height and so would have the binary tree property.
∗
Exercise 8.2.11. Assume that the tree property of ϕ is witnessed by the
parameters A = {as | s ∈ <ω ω}. If ϕ has the tree property with respect to
k = 2, it is easy to see that ϕ has the binary tree property.
For the general case we make use of Exercise 8.2.10: if ϕ is stable, all ϕ-ranks
are less than ω. VIt follows that there is a sequence σ ∈ ω ω such that the ϕ-ranks
of the formulas 1≤i<n ϕ(x, aσi ) are strictly decreasing, which is impossible.
Exercise 8.3.1. Let T be the theory of an equivalence relations with three
infinite classes. There is only one 1-type over the empty set, and this does not
have a good definition.
Exercise 8.3.2. Six of the eight possible cases are realised by 1-types. For
the cases (¬D, C, I, ¬H), (¬D, ¬C, I, H) use 2-types and for the case (¬D, ¬C,
I, ¬H) a 3-type.
Exercise 8.3.3. Let πn : Sn (B) → Sn+1 (B) be a continuous section. For
any n-tuple c if πn maps tp(c/B) to p(x, y), then pc = p(x, c) is a type over cB.
Continuity implies that for every ϕ(x, y) there is a B–formula ψ(y) such that for
S ϕ(x, c) ∈ pc ⇔ |= ψ(c). The following coherence condition ensures
all c we have
that p = c∈C pc is consistent. For any map s : {1, . . . , m} → {1, . . . , n} let
s# : Sn (B) → Sm (B) and s∗ : Sn+1 (B) → Sm+1 (B) be the associated natural
restriction maps. Then coherence means πm ◦ s# = s∗ ◦ πn .
Exercise 8.3.4. Consider the Boolean algebra of all M -definable subsets
of ϕ(M )n and the subalgebra of ϕ(M )-definable subsets. The two algebras
coincide if and only if they have the same Stone spaces (see p. 49). For the
APPENDIX D. SOLUTIONS TO EXERCISES 221

second part note that – if ϕ(C) has a least two elements – then for every ψ(x, y)
there is a formula χ(x, z) such that every class ψ(x, b) which is a subclass of
ϕ(C)n has the form χ(C, c) for some c ∈ ϕ(C).
Exercise 8.3.5. 1. Prove that formulas with Morley rank are stable. The
proof that totally transcendental theories are stable on page 132 is similar.
2. It is enough to find dp x ϕ(x, y) for stable ϕ(x, y). Use the proof of
Theorem 8.3.1.
3. Use the proof of Corollary 8.3.3 and Remark 8.2.2.
Exercise 8.3.6. The proof follows the pattern of the proof of the Erdős–
Makkai Theorem C.2.1. Assume that B ∗ = {b ∈ B | ϕ(x, b) ∈ p} is not a
positive Boolean combination of sets of the form {b ∈ B ||= ϕ(c, b)}, c ∈ C.
Construct three sequences (b0i | i < n) in B ∗ , (b00i | i < n) in B \ B ∗ and
(ci | i < n) in C such that for all i < n

|= ϕ(ci , b0n ) ⇒ |= ϕ(ci , b00n )

and for all i ≤ n

|= ϕ(cn , b0i ) and |= ¬ϕ(cn , b00i ).

Exercise 8.3.7. Show first that there is a finite sequence ∆1 , . . . , ∆n such

that every ϕ-type is definable by an instance of one of the ∆i . (Otherwise the
L ∪ {P, c}-theory stating that in a model of T the ϕ-part of the type of c over
P is not definable would be consistent.)
Exercise 8.3.8. Hint: If q is a weak heir of p, then Dϕ (q) = Dϕ (p) where
Dϕ (p) is defined as the minimum of Dϕ (θ) for θ ∈ p. The argument in Theo-
rem 8.3.1 now shows that q is definable over M .
Exercise 8.3.9. Let p be the global extension of tp(a/M ) which is de-
finable over M . By Lemma 8.1.5 tp(a/M b) is an heir of tp(a/M ) if and only
if tp(a/M b) ⊆ q, i.e., if and only if ϕ(x, b) ∈ tp(a/M b) ⇔ |= dp xϕ(x, b). So
if q is the global M -definable extension of tp(b/M ), we have that tp(a/M b) is
an heir of tp(a/M ) if and only if ϕ(x, b) ∈ tp(a/M b) ⇔ |= dp xϕ(x, y) ∈ q(y).
Lemma 8.3.4 implies now that tp(a/M b) is an heir of tp(a/M ) if and only if
tp(b/M a) is an heir of tp(b/M ).
Exercise 8.3.10. One direction follows from Exercise 8.1.3. For the other
direction assume that ϕ does not fork over A. Then ϕ is contained in a global
type p which does not fork over A. Apply Corollary 8.3.7.
Exercise 8.4.1. D is definable from some some tuple d ∈ D. Any such d
is a canonical parameter for D.
Exercise 8.4.2. Let e be an imaginary and A the smallest algebraically
closed set in the home sort over which e is definable. Then e is definable from a
finite tuple a ∈ A. Since every automorphism which fixes e leaves A invariant,
all elements of A are algebraic over e.
APPENDIX D. SOLUTIONS TO EXERCISES 222

For the converse let a ∈ acl(e) be a real tuple over which e is definable. Then
A = acl(a) is the smallest algebraically closed set over which e is definable.
Exercise 8.4.3. Infset and DLO have the following property: if A, B are
finite sets and if the tuples a and b have the same type over A ∩ B, then there
is a sequence a = a0 , b0 , . . . , an , bn = b such that ai and bi have the same type
over A and bi and ai+1 have the same type over B. This implies that for every
definable class D there is a smallest set over which D is definable.
Infset does not eliminate imaginaries since no finite set with at least two
elements has a canonical parameter.
∗
Exercise 8.4.4. Let q1 and q2 be extensions of p to B. Choose realisations
a1 , a2 of q1 , q2 , respectively. There is an α ∈ Aut(C/A) taking a1 to a2 . Since
α(B) = B, we have α(q1 ) = q2 .
Exercise 8.4.5. Let p ∈ S(A) be algebraic. If p is realised by b ∈ dcl(B),
then dp xϕ(x, y) = ϕ(b, y) is a good definition of p over B. Conversely, if q is an
.
extension of p to M , then q is realised by some b in M and dq x(x = y) defines
the set {b}. So if q is definable over B, then b ∈ dcl(B).
Exercise 8.4.6. Let d be a canonical parameter of D = ϕ(C, d). If d0 has
the same type as d, we have

ϕ(C, d0 ) = D ⇒ d0 = d.

By compactness this is true for all d0 which satisfy some ψ(x) ∈ tp(d). Consider
the L ∪ {P }-formula θ(x, P ) = ψ(x) ∧ ∀y (ϕ(y, x) ↔ P (y)).
∗
Exercise 8.4.7. 1. Let ϕ(x, a) ∈ p have the same Morley rank as p and
be of degree 1. Then pdx ϕ(x, y)q is a canonical base of p.
2. Use Part 1 and Exercise 8.4.1.
Exercise 8.4.9. If stp(a/A) 6= stp(b/A) there is an acl(A)-definable class
D = ϕ(x, a) with |= ϕ(a, a) and 6|= ϕ(b, a). By Lemma 8.4.4, D is the union of
equivalence classes of an A-definable finite equivalence relation Ea , proving the
claim.
Exercise 8.4.10. The correspondence is given by H = Stab(A) and A =
Fix(H). That A = Fix(Stab(A)) for definably closed A follows from Corol-
lary 6.1.12(1). To see that H = Stab(Fix(H)) for closed subgroups H, we have
to show that every g ∈ Stab(Fix(H)) agrees on every finite tuple b with some
element h of H. To this end let a be a canonical parameter of the finite set Hb.
Then a is fixed by H and therefore also fixed under g. So g(Hb) = Hb, which
implies that gb = hb for some h ∈ H.
Exercise 8.5.1. Show that the two conditions are equivalent to each of the
following
1. tp(a/K) has a unique extension to K sep ;
2. K(a) ∩ K sep = K.
APPENDIX D. SOLUTIONS TO EXERCISES 223

Exercise 8.5.2. Let P be the set of all strong types over A which are
consistent with p, and Q be the set of all strong types which are consistent with
q. Both P and Q are closed subsets of S(acleq (A)) and are disjoint since strong
types are stationary. So they can be separated by a formula ϕ(x) over acleq (A).
By Lemma 8.4.4, ϕ(C) is a union of classes of a finite A-definable equivalence
relation E(x, y).
Exercise 8.5.4. Use the second part of Exercise 8.3.5. The first claim can
now be proved like Theorem 8.5.1.
The second claim is proved like Corollary 8.5.3, but we must be more careful
and use the ϕ-rank introduced in Exercise 8.2.10. We call the minimal ϕ-rank of
a formula in a type p the ϕ-rank of p. Let A = acleq (A), p ∈ S(A) a stable type
and p0 and p00 two non-forking global extensions. Then p0 and p00 are definable
over A. We want to show that ϕ(x, b) ∈ p0 ⇔ ϕ(x, b) ∈ p00 . We may assume that
ϕ(x, y) is stable (containing parameters from A). Let q(y) be a global extension
of tp(b/A) which has the same ϕ∼ -rank, where ϕ(x, y)∼ = ϕ(y, x). Since there
are only finitely many possibilities for the ϕ-part of q, the ϕ-part is definable
over A. Now the claim follows from an adapted version of Lemma 8.3.4.
We still have to show that p has a non-forking global extension, i.e., an
extension which is definable over A. Choose for every stable ϕ(x, y) a global ex-
tension of p with the same ϕ-rank. By the above the ϕ-part pϕ of this extension
is definable over A and as such is uniquely determined. It remains to show that
the union of all pϕ is consistent. Consider a finite sequence ϕ1 (x, y), . . . , ϕn (x, y)
of stable formulas. Choose a stable formula ϕ(x, y, z) such that every instance
ϕi (x, b) has the form ϕ(x, b, c) for some choice of c. Then pϕ contains all pϕi for
i = 1, . . . , n.
∗
Exercise 8.5.5. 1. Argue as in the second part of the proof of Theorem
8.5.10. Replace p < q by MR(p) = MR(q).
2. By the first part of Exercise 8.3.5 p is stable. Let q be a global extension of
p. If MR(p) = MR(q), then q has only finitely many conjugates over A. Since q
is definable, this implies that q is definable over acleq (A). So by Exercise 8.5.4,
q does not fork over A. Now assume that q does not fork over A. Using
Exercise 6.2.8 we see that we can assume that A = acleq (A). Let q 0 be an
extension of p with the same Morley rank. Then q 0 does not fork over A. So by
Exercise 8.5.4 q = q 0 .
∗
Exercise 8.5.6. Choose A1 ⊆ A of cardinality at most |T | over which p
does not fork. Let (pi ) be the non-forking extensions of p  A1 to A. For each
L-formula ϕ(y, y) there are only finitely many different piϕ . Hence there is a
finite subset Aϕ of A such that for all i

(pi  Aϕ )ϕ = (p  Aϕ )ϕ =⇒ piϕ = pϕ .
S
Now put A0 = A1 ∪ ϕ Aϕ .
If p has Morley rank, choose ϕ ∈ p having the same Morley rank and degree
as p. Any set A0 containing the parameters of ϕ does the job.
APPENDIX D. SOLUTIONS TO EXERCISES 224

Exercise 8.5.7. 1: Easy.

2: Exercise 8.5.6 shows that it is enough to consider types p over a countable
set A. The multiplicity is the number of extensions of p to acl(A). These
extensions form a separable compact space. By Exercise 8.4.4, either all or
none of them are isolated. In the first case the space is finite; in the second, it
has cardinality 2ℵ0 .
3: Use Exercise 8.5.5.
Exercise 8.5.8. 1) Both a and a · b are interalgebraic over b. This implies
MR(a) = MR(a/b) = MR(a · b/b) ≤ MR(a · b). If MR(a) = MR(a · b), we have
MR(a · b/b) = MR(a · b) and a · b and b are independent.
2) Let b be independent from a. If a is generic, MR(a · b) cannot be bigger
than MR(a), so a · b and b are independent by part 1). For the converse we
choose b generic. Part 1) (with sides reversed) implies that a · b is also generic.
If a · b and b are independent, it follows that MR(a) = MR(a · b) and a is generic.
Exercise 8.5.9. For 1) notice that each line Ai consists of two elements
and their product.
For 2) note that by Exercise 8.5.8, if a, b, c are independent generics, then
a, b, b · c are again independent generics. If one applies this rule repeatedly
starting with a1 , a2 , a3 , one obtains every non-collinear triple of our diagram.
Exercise 8.6.1. It follows from Remark 7.1.3 and Symmetry that a type
is algebraic if and only if it has no forking extensions. A type has SU-rank 1
if and only if the algebraic and the forking extensions coincide. So a type is
minimal if and only if it has SU-rank 1 and has only one non-forking extension
to every set of parameters.
Exercise 8.6.2. Use Exercise 7.1.5.
Exercise 8.6.4. This follows from Exercise 7.1.2.
Exercise 8.6.5. This follows easily from Exercise 7.2.5. Prove by induction
on α and γ

SU(a/C) ≥ α ⇒ SU(ab/C) ≥ SU(b/aC) + α

SU(ab/C) ≥ γ ⇒ SU(b/aC) ⊕ SU(a/C) ≥ γ.

Exercise 8.6.6. The first claim follows from Lemma 7.2.4(2) and the re-
mark thereafter. The second claim is easily proved using the Diamond Lemma
(Exercise 7.2.2) and Exercise 7.1.7.
Exercise 8.6.7. Totally transcendental theories are superstable by Corol-
lary 8.5.11. It follows also that the multiplicity of a type over arbitrary sets is
finite, namely equal to its Morley degree. If T is superstable, one can compute
an upper bound for the number of types over a set A of cardinality κ as in the
proof of Theorem 8.6.5(2). If T is small, there are only countably many types
over a finite set E. If we know also that all p ∈ S(E) have finite multiplicity, we
have | S(A)| ≤ κ · ℵ0 · ℵ0 = κ.
APPENDIX D. SOLUTIONS TO EXERCISES 225

Exercise 8.6.9. We can assume that all Ei are 0-definable. Choose a

sequence a0 , a1 , . . . such that |= Ei (ai , ai+1 ) and pai /Ei q is not algebraic over
a0 . . . ai−1 . Let b be an element in the intersection of all ai /Ei , A = {a0 , a1 , . . .}
and p = tp(b/A). Then for all i we have b ^ 6 | a ,...,a pai /Ei q by Remark 7.1.3.
0 i−1
This shows that p forks over each finite subset of A.
Exercise 8.6.10. Define Ei (x, y) as xy −1 ∈ Gi . For any imperfect field K
i i+1
of finite characteristic p set Gi = K p . x − y ∈ K p .
Exercise 8.6.11. Half of the claim follows from Remark 6.2.8 and Exer-
cise 8.6.10. Assume that M has the dcc on pp-definable subgroups. Then for
every element a and every set B of parameters the positive type tp+ (a/B)
contains a smallest element ϕ0 (x, b). So there are at most max(|T |, |A|) many
types over A. This shows that M  R0 is ω-stable for every countable subring,
so M is totally transcendental (see Exercise 5.2.5). Now assume that there is
no infinite sequence of pp-subgroups with infinite index in each other. Then
tp+ (a/B) contains a formula ϕ0 (x, b0 ) such that tp+ (a/B) is axiomatised by
formulas ϕ(x, b) where ϕ(M, 0) is a subgroup of finite index in ϕ0 (M, 0). There
are max(|T |, |A|) many possibilities for ϕ0 (x, b0 ) and for each ϕ(x, y) finitely
many possibilities. So the number of types over A is bounded by max(2|T | , |A|)
and M must be superstable. Indeed, the proof of Theorem 8.6.5(3) shows that
otherwise for every κ there would be a set A of cardinality κ with | S(A)| ≥ κℵ0 .

Chapter 9. Prime extensions

∗
Exercise 9.1.1. Let p ∈ S(A) and q a non-forking extension to B. Let I be
a Morley sequence of q, so I is independent over B. Since every element of I is
independent from B over A, it follows that I is independent over A as well (see
Exercise 7.2.6), hence a Morley sequence of p.
∗
Exercise 9.1.2. a): Since I \ I0 is independent from B over A, it is
independent over B. The elements of I realise the non-forking extension of p
to B.
b): Let B ⊇ A and q the non-forking extension of p. We extend I to a very
long sequence I 0 indiscernible over A. Then I 0 is still a Morley sequence of p.
If we choose I0 ⊆ I 0 with |I0 | ≤ |T | + |B| and B ^ | AI I 0 , then I 0 \ I0 is an
0
infinite Morley sequence of q having the same average type as I, so q ⊆ Av(I).
Exercise 9.1.3. If I0 and I1 are parallel, hence I0 J and I1 J indiscernible,
then
Av(I0 ) = Av(I0 J ) = Av(I1 J ) = Av(I1 ).
If conversely p = Av(I0 ) = Av(I1 ), note that by the proof of Theorem 9.1.2
there are sets B0 and B1 over which p does not fork and such that I0 and I1
are Morley sequences of the stationary types p  B0 and p  B1 . Let J be a
Morley sequence of p  B0 I0 B1 I1 . Then I0 J and I1 J are Morley sequences of
p  B0 and p  B1 , respectively.
APPENDIX D. SOLUTIONS TO EXERCISES 226

∗
Exercise 9.1.4. Let p and q be stationary types with infinite Morley
sequences I and J . Then the average types Av(I) and Av(J ) are the global
non-forking extensions of p and q, respectively. Now p and q are parallel if and
only if Av I = Av J , i.e., if and only if I and J are parallel.
∗
Exercise 9.2.1. Let p1 , . . . , pn be the extensions of p to acleq (A) and let
Ii be the elements of I which realise pi .

Chapter 10. The fine structure of ℵ1 -categorical

theories
Exercise 10.1.2. To prove part 1 either use Remark 6.2.8 to obtain a finite
subset of E which has trivial stabiliser in Aut(E/F) or apply Corollary 8.3.3.
Exercise 10.1.1. Let N be an elementary submodel which contains F(M ).
By Lemma 10.1.4 there is a definable surjection f : Fn → C. Write f (x) =
g(x, a) for a 0-definable function g and a parameter tuple a. Since N is an
elementary substructure, we may assume that a ∈ N . Then M = g(Fn (M ), a) =
g(Fn (N ), a) = N .
Exercise 10.1.3. If d is in dcleq (F), the set {d } is definable over F. The
proof of Theorem 8.4.3 shows that {d} has a canonical parameter in Feq .
Exercise 10.1.5. a) ⇒ b) was implicitly proved in Lemma 10.1.5.
b) ⇒ a): This is the proof of Corollary 8.3.3.
b) ⇔ c): This is Exercise 10.1.4.
b) ⇒ d): Same as the proof of Lemma 10.1.5.
d) ⇒ a): Assume that ϕ(a, F) is not definable with parameters from F. Let
ai be an enumeration of C. Construct a sequence ϕi of partial automorphisms of
F so that the domain of ϕi contains some f with |= ϕ(a, f ) ⇔|= ¬ϕ(ai , ϕi (f )).
∗
Exercise 10.2.1. Use induction on MR(A).
Exercise 10.2.3. Show first that if p, r are two types over A and A0 is
an extension of A, then p and q are almost orthogonal if any two non-forking
extensions of p and q to A0 are almost orthogonal. This implies that q is or-
thogonal to p if and only if q is orthogonal to all non-forking extensions of p to
A0 .
Exercise 10.2.4. 1. Let p ∈ S(A) be stationary and p0 ∈ S(A0 ) be a regular
non-forking extension. We want to show that p is orthogonal to every forking
extension q ∈ S(B). For this we may assume that B = acleq (B) so that q
is stationary. By the Diamond Lemma q has a non-forking extension q 0 which
extends an A-conjugate of p0 . So p0 and q 0 are orthogonal and by Exercise 10.2.3
so are p and q.
2. Let us check the four properties in Definition 5.6.5: A ⊆ cl(A) is true
since p is non-algebraic. Finite Character follows from the finite charac-
ter of forking, Exchange from forking symmetry. Regularity is used only for
APPENDIX D. SOLUTIONS TO EXERCISES 227

Transitivity. Show the following: assume that C ^ 6 | A d and that for all c ∈ C
all extensions of tp(c/AB) are orthogonal to tp(d/A). Then B ^ 6 | A d.
3. We assume A = ∅ to simplify notation. Assume b ^ 6 | c, c ^6 | d and b ^
| d.
Choose an d-independent sequence (bα cα )α<|T |+ of realisations of tp(bc/d).
Since bα ^ | d, it follows that each cα is independent from B = {bβ | β < α}.
Since all cβ are dependent from B, we conclude by regularity that cα is inde-
pendent from {cβ | β < α} over A. So (cα )α<|T |+ is independent. But we have
cα ^6 | d for all α, contradicting Exercise 7.2.1.
∗
Exercise 10.4.1. If A ≤ M and C is a finite subset of N which contains
A ∩ N we have δ(AC/A) ≤ δ(C/A ∩ N ). This proves A ≤ N .
Assume A ≤ B ≤ M . Consider a finite extension C of A. Then B ∩ C ≤ C.
This implies δ(A) ≤ δ(B ∩ C) ≤ δ(C) and therefore A ≤ C.
The last implication follows directly from the first two.
For the last part of the exercise choose a finite extension B of A with minimal
δ(B). Then B is strong in M . So we may take for cl(A) the intersection of all
finite extensions of A which are strong in M .
∗
Exercise 10.4.2. Let E(A) denote the set of edges of A. If

X = E(A1 ∪ · · · ∪ Ak ) \ (E(A1 ) ∪ · · · ∪ E(Ak )),

we have

δ(A1 ∪ · · · ∪ Ak ) = 2|A1 ∪ · · · ∪ Ak | − |E(A1 ) ∪ · · · ∪ E(Ak )| − |X|.

Now apply Lemma 3.3.10.

Exercise 10.4.3. This follows because any path (x1 , a, x2 ) is strong in
Mµ .

Appendices
Exercise B.3.1. Let L be an elementary extension of K. Show first that if
S is an integral domain which contains K, then L ⊗K S is again an integral
domain. So L ⊗K K alg is an integral domain.
Exercise C.1.1. 1. Symmetry is clear from the definition. For the other
properties show first that a finite set A is independent from B over C if and only
if dim(A/BC) = dim(A/C). This implies Monotonicity and Transitivity.
Finite Character and Local Character follow from the fact that for finite
A and any D there is a finite D0 ⊆ D such that dim(A/D) = dim(A/D0 ).
2. Symmetry, Finite Character and Weak Monotonicity are clear.
If A is finite and D is any set, let D0 be a basis of cl(A) ∩ D. So D0 is finite
and A ^ |0 D D. This shows Local Character. Always A ^ |cl C B implies
0
0 0
A^ | C B. If ^| satisfies Monotonicity, the converse is true: we may assume
that B = {b1 , . . . , bn } is finite. Then A ^ |0 Cb ...b bi+1 for all i.
|0 C B implies A ^
1 i
cl
But this is the same as A ^ | Cb ...b bi+1 , from which follows that A ^ |cl C B.
1 i
APPENDIX D. SOLUTIONS TO EXERCISES 228

Exercise C.1.3. Choose a, b, x, c ∈ K p-independent. Set F0 = K p (c),

F1 = K p (c, a, b) and F2 = K p (c, x, ax + b). Then F0 has p-dimension 1, F1 and
F2 have p-dimension 3 and F1 F2 has p-dimension 4. To show that F0 = F1 ∩ F2 ,
prove that dimF0 F1 = dimF0 F2 = p2 and dimF0 (F1 + F2 ) = 2p2 − 1.
Exercise C.1.5. First show that in any pregeometry for any closed A ⊆ B:
if dim(A/B) is finite, then it is the longest length n of a proper chain B = C0 ⊆
C1 ⊆ · · · ⊆ Cn = B of closed sets Ci .
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Index

0, 183 A  K, 4
0-definable set, 9 AB , 4
0-dimensional space, 49 A ⊆ B, 2
1, 183 Aut(A), 2
2, 183 Aut(A/B), 4
<ω
2, 185 Av(I), 155
|A|, 2 ~b a , 8
x
(A, a1 , . . . , an ), 4 B-definable set, 10
AbG, 14 iα , 186
Abs(L), 124 Cb(a/B), 169
ACF, 16, 41, 210 Cb(p), 145
ACF0 , 41 cf, 186
ACFp , 41 clS , 203
acl(A), 79 clS , 203
acleq (A), 139 pDq, 137
∀∃-formula, 29 DCF0 , 45
A ≡ B, 15 dcl(A), 91
A ≺ B, 17 dcleq (A), 139
A ≺1 B, 34 deg(p), 79
A ⇒∆ B, 28 Diag(A), 11
A ⇒∃ B, 28 dimϕ (M), 84
A^ | C B, 112 dimϕ (N/M ), 99
a^ | C B, 144 dim(N/M ), 100
A∼ = B, 2 dim(X), 204
ℵ0 , 186 dim(X/S), 204
ℵα , 184 DLO, 16, 209
∀, 8 dp xϕ(x, y), 128
∼α , 95 D(p, ϕ, k), 153
α-minimal formula, 97 ∃∀-formula, 30
α + 1, 183 ∃>n , 77
α + n, 183 EM(I/A), 64
α-strongly minimal formula, 95 h∅iB , 3
A |= ϕ[~b], 8 .
=, 7
n
[A]
V , 64 ↔, 8
i<m , 10 ∃, 7
∧, 7 ∃∞ , 77
(A, R), 4

234
INDEX 235

⊥, 10 LSkolem , 66
f : A →∆ B, 28 Lstp(a/A), 121
Feq , 163, 164 MDα ϕ, 95
[ϕ], 48 MD(a/A), 96
pϕq, 137 MD ϕ, 95
ϕ(A), 9 MD(p), 96
ϕ-dimension, 84 M eq , 138
over parameters, 84 |=, 8, 90
Field, 14 Mod(R), 38
ϕ(π), 12 MR(a/A), 96
ϕ-rank, 133, 223 MR ϕ, 94
ϕ(t1 , . . . , tn ), 10 MR(p), 96
ϕ-type, 130 MR(T ), 94
ϕ(x1 , . . . , xn ), 9 MB , see AB
f (p), 50 mult(p), 148
G(K), 199 N(B/A), 145
G(k), 199 ncA , 116, 121
≺ ¬, 7
h : A −→ B, 17
∼ n-type, 23
h : A → B, 2 W
h : A → B, 2 i<m , 10
→, 8 ω, 183
Infset, 38 ω-homogeneous, 53
<ω
κ-categorical, 25, 26 x, 185
κ-homogeneous, 88 ω-stable, 68
κ+ , 184 On,
W 183
κ-saturated, 69 i<m , 10
κ-stable, 67, 131, 132 ∨, 8
κ-universal, 88 ω-saturated, 51
k-inconsistent family, 107 otp(x, <), 183
K-saturated, 55 P(a), 181
LAbG , 1 p-basis, 195, 196
Lalg , 80 PC∆ -class, 91
λν , 44 p-closure, 195
L(B), 4 pϕ , 219
L(C), 4 π ` σ, 90
L∅ , 1 p  A, 50
Leq , 138 `, 14, 15, 90
LGroup , 1 ` ϕ, 14
LSet , 1 P(V ), 203
LMod (R), 38 RG, 46
LNumbers , 1 RCF, 42
LOrder , 1 Rϕ , 133
LORing , 1 Ring, 14
LRing , 1 hSiB , 3
Lsep , 44 SA (B), 23
SAn (B), 23
INDEX 236

S(B), 23 absolute part of a field, 124

Sn (B), 23 absolutely prime, 195
S(C), 90, 98 additivity of Morley rank, 104, 166
SCFp,e , 44 affine subspace, 206
SCFp (c1 , · · · , ce ), 44 age, 55
SCFp,∞ , 44 Aleph function, 184
Sϕ (B), 130 algebraic
Skolem(L), 66 closure, 79
S(T ), 48 element, 79
Sn (T ), 48 formula, 79
stp(a/A), 142 type, 79
SU(p), 149 algebraically closed field, 26, 41, 79, 81,
supi∈I αi , 183 129
supi∈I κi , 184 almost orthogonal type, 165
T eq , 138 almost strongly minimal, 105, 166
|T |, 15 Amalgamation Property, 56
tA [a1 , . . . , an ], 6 analysable type, 164
tA [~b], 6 antichain, 116
T∀∃ , 31 assignment, 6
T∀ , 29 atom, 50
T -ec, 35 atomic
T eq , 138 diagram, 11
T ≡ S, 15 extension, 73
Th(A), 15 formula, 8
T KH , 36 structure, 59
tp(a), 23 tuple, 59
tpA (a/B), 23 automorphism, 2
tp(a/B), 23 group, 2
tpqf (a), 58 average type, 155
T ` ϕ, 14 axiom, 13
T ` S, 15
tr. deg(L/F ), 44 back-and-forth property, 16
Tree, 61 1-based formula, 169
T  K, 70 basic formula, 10
>, 10 basis, 203, 203
t(t1 , . . . , tn ), 6 Bernays–Gödel set theory, 97, 181
t(x Beth function, 186
S 1 , . . . , xn ), 6 Beth’s Interpolation Theorem, 91
S a, 181 BG, 181
i∈I Ai , 4
U(p), 152 binary tree, 61, 68
v0 , v1 , . . ., 5 property, 131
y
x, 185 binding group, 164
b 198
Z, binding strength, 8
Boolean algebra, 49
abelian group, 14 bounded relation, 123
absolute Galois group, 199
INDEX 237

canonical theory, 14
base, 137, 145, 169 constant, 1
parameter, 137 constant term, 7
Cantor’s Theorem on DLO, 26 constructible set, 71
Cantor–Bendixson rank, 98 construction, 157
cardinal, 184 continuous chain, 24
arithmetic, 184 Continuum Hypothesis, 186
limit, 184 countable, 184
Ramsey, 208 theory, 47
regular, 186 curves, 170
successor, 184 cyclical order, 110
weakly compact, 208
weakly inaccessible, 186, 208 definable
cardinality, 184 bijection, 81
of a structure, 2 class, 90
categoricity, 25, 26 closure, 91
CH, 186 Morley rank, 103
chain multiplicity property, 106
continuous, 24 set, 9, 10
elementary, 19 type, 128
Lemma, 19 degree
of structures, 3 of a type, 79
class of imperfection, 44, 195, 204
PC∆ -class, 91 density of isolated types, 60
elementary, 16, 19, 36 derivation, 191
closed sets of types, 50 descending chain condition (dcc), 96
club, 156 diagram
cofinal, 186 atomic, 11
cofinality, 186 elementary, 17
coheir, 127, 135 Diamond lemma, 115
commutative ring, 14 differential field, 191
Compactness Theorem, 19 differentially closed field, 45, 103, 147
complete dimension
formula, 50 ϕ-dimension, 84
theory, 15, 16 over parameters, 84
complexity of a pregeometry, 204
of a formula, 8 of extensions of differential fields,
of a term, 6 103
components of a formula, 95 directed
conjugacy, 144 family, 3
conjugate over, 90 elementary, 19
conjunction, 7 partial order, 3
conjunctive normal form, 13 disjoint formulas, 61
conservative extension, 51 disjunction, 8
consistent disjunctive normal form, 13
set of formulas, 14, 90 dividing
INDEX 238

formula, 107 atomic, 73

sequence, 111 conservative, 51
set of formulas, 107 elementary, 18
domain, 2 forking, 112
of a type, 23 minimal, 73
dual numbers, 191 of a structure, 2
of types, 50
Ehrenfeucht–Mostowski type, 64 prime, 70
elementary
chain, 19 field, 14
class, 16, 19, 36 algebraically closed, 26, 41, 79, 81,
diagram, 17 129
directed family, 19 differential, 191
embedding, 17 differentially closed, 45, 103, 147
equivalence, 15 formally real, 187
extension, 18 1-free, 199
map, 17, 49 ordered, 187
property, 82 PAC, 197
substructure, 17 procyclic, 199
elimination pseudo algebraically closed, 197
of ∃∞ , 77 pseudo-finite, 199
of finite imaginaries, 141 real closed, 188, 189
of imaginaries, 137, 139 separably closed, 43, 153
weak, 141 field extension
of quantifiers, 32 Galois, 194
embedding, 2 normal, 194
elementary, 17 regular, 194
equality symbol, 7 separable, 194, 195
equivalence, 8 filter, 13
equivalent finite character of forking, 109, 144
formulas, 10, 29, 90 finite equivalence relation, 138
modulo T , 29 Finite Equivalence Relation Theorem,
theories, 15 147
Exchange, 80, 202 Finite Character, 80, 119, 202
Existence, 119, 146 finitely axiomatisable, 24, 24
existence of non-forking extensions, 112, finitely complete theory, 20
144 finitely generated substructure, 3
existential formula, 11 finitely satisfiable
primitive, 32 set of formulas, 22
simple, 32 theory, 19
existential quantifier, 7 forking
existentially closed extension, 112
structure, 35 independence, 112
substructure, 34 multiplicity, 148
expansion, 4 set of formulas, 109
extension, see field extension symmetry, 114, 144
INDEX 239

formally real field, 187 geometry, 203

formula global type, 90
1-based, 169 good definition of a type, 143
α-minimal over A, 97 graph, 24
α-strongly minimal, 95 connected, 24
∀∃, 29 random, 46, 53, 59, 120, 132
∃∀, 30 group configuration, 149
algebraic, 79 groupoid, 162
atomic, 8 connected, 162
basic, 10 definable, 162
complete, 50
complexity, 8 heir, 127, 135
dividing, 107 Henkin constants, 20
existential, 11 Henkin theory, 20
primitive, 32 Hereditary Property, 56
simple, 32 Hilbert Basis Theorem, 68
isolating, 47 Hilbert’s 17th Problem, 43
locally modular, 168 Hilbert’s Nullstellensatz, 42
positive primitive, 39 home sort, 138
quantifier-free, 10 homomorphism, 2
stable, 130, 136
stably embedded, 134, 164 imaginary elements, 138
strongly minimal, 81 implication, 8
symmetric, 117 independence
thick, 116, 121 forking, 112
universal, 11 geometric, 205
with Morley rank, 94 independence property, 132
with the binary tree property, 131 Independence Theorem, 118, 122
with the order property, 131 Independence over Models, 119
with the tree property, 110 independent
formulas family, 115
disjoint, 61 sequence, 113
equivalent, 10, 29, 90 set, 115, 203
Fraı̈ssé limit, 56 indiscernibles, 63
1-free field, 199 parallel, 156
free occurrence, 9 total, 154
function symbol, 1 induced theory, 82
Fundamental Theorem of Algebra, 189 induction, 182
inductive theory, 30
Galois field extension, 194 infinitesimal, 24
GCH, 186 interdefinable, 91
generated internal type, 161
substructure, 3 Interpolation Theorem, 91
type, 211 interpretation, 2
generating set, 203 Invariance, 146
generic element, 105, 148, 192, 198 invariant
INDEX 240

class, 90, 92, 124 model companion, 35

relation, 55 model complete theory, 34
type, 110, 113 modular
isolated lattice, 206
set of formulas, 47 pregeometry, 204
type, 50 theory, 168
isolating formula, 47 module, 38
isomorphic over a set, 73 monotonicity, 115, 144
isomorphic structures, 2 monotonicity, 119
isomorphism, 2 weak, 146
partial, 16 monster model, 89–93
Morley degree
Joint Consistency, 91 of a formula, 95
Joint Embedding Property, 56 of a type, 96
Morley rank
Kaiser hull, 36 of T , 94
of a formula, 94
Löwenheim–Skolem
of a type, 96
Theorem of, 24
Morley sequence, 113, 155
Lachlan’s Theorem, 73
in p, 113
λ-functions, 44
of q over A, 113
language, 1
Morley’s Theorem, 86
Lascar rank, 152
Morleyisation, 32
Lascar strong type, 121
multiplicity, 148
limit cardinal, 184
limit ordinal, 183 negation normal form, 11
Lindström’s Theorem, 37 negation symbol, 7
linearly disjoint, 193, 194 non-forking extension
local character of forking, 111, 144 existence, 112, 119, 144
Local Character, 119, 146 non-principal ultrafilter, 67
locally finite, 54 normal field extension, 194
locally modular normal form
formula, 168 conjunctive, 13
pregeometry, 206 disjunctive, 13
logical symbols, 7 negation, 11
prenex, 13
many-sorted structure, 5
normal subset, 142, 158
matroid, 80
nowhere dense set, 50
meagre set, 50
Nullstellensatz, 42
minimal
prime extension, 73 omitting a set of formulas, 47
set, 81 Omitting Types, 47
type, 83, 152 order property, 131
model, 10, 90 order type, 183
consisting of constants, 20 ordered field, 187
prime, 59 ordinal, 182
INDEX 241

addition, 105 quantifier elimination, 32

limit, 183 quantifier-free formula, 10
orthogonal type, 165
Ramsey cardinal, 208
PAC field, 197 Ramsey’s Theorem, 64
parallel random graph, 46, 53, 59, 120, 132
indiscernibles, 156 rank
types, 145 SU, 149
parameter set, 90 U, 152
parentheses, 7, 8 Cantor–Bendixson, 98
partial type, 47, 90 ϕ-rank, 133, 223
partially isomorphic, 16 Lascar, 152
perfect hull, 92, 195 Morley, 94
plane curves, 170 real closed field, 188, 189
Poizat, Bruno, 241 real closure, 188
positive primitive formula, 39 real elements, 138
pp-definable subgroup, 39 realisation of a set of formulas, 22
pp-formula, 39 realisation set, 9
predicate, 1 recursion theorem, 182
pregeometry, 80, 202 reduct
modular, 204 of a structure, 4
prenex normal form, 13 of a theory, 70
preservation theorems, 27 Reflexivity, 80, 202
prime regular action, 163
absolutely, 195 regular cardinal, 186
extension, 70 regular field extension, 194
minimal, 73 regular type, 168
model, 59 relation symbol, 1
structure, 32 relativisation of a pregeometry, 203
primitive existential formula, 32 restriction
procyclic of a pregeometry, 203
field, 199 of a type (parameters), 50
group, 198 of a type (variables), 51
profinite group, 198 ring, 14
projective Robinson’s
plane, 171 Joint Consistency Lemma, 91
space, 203 Test, 34
property Ryll–Nardzewski Theorem, 51
independence, 132
order, 131 satisfaction, 8
pseudo algebraically closed field, 197 saturated structure, 51, 69, 87–89
pseudo-finite field, 199 sentence, 10
separable field extension, 194, 195
quantifier separably closed field, 43, 152, 153
existential, 7 separating sentence, 27
universal, 8 Separation Lemma, 27
INDEX 242

Shelah’s Lemma about indiscernibles, sublanguage, 4

113 submodular law, 172, 204
simple existential formula, 32 substitution, 6
simple theory, 111 lemma, 7, 10
skeleton, 55 substructure, 2
Skolem function, 66 elementary, 17
Skolem theory, 66 existentially closed, 34
small theory, 53 finitely generated, 3
SOP, 132 generated, 3
sort, 138 substructure complete theory, 34
home, 138 successor cardinal, 184
special structure, 87 supersimple theory, 150
stability spectrum, 150 superstable theory, 150
stable SU-rank, 149
ω-stable, 68 Sylvester, J. J., 189
formula, 130, 136 symmetric formula, 117
κ-stable, 67, 131, 132 symmetry, 119
theory, 130 symmetry of forking, 114, 144
type, 136
stably embedded formula, 134, 136, 164 Tarski’s
Standard Lemma on indiscernibles, 64, Chain Lemma, 19
107 Test, 18
stationary type, 121, 143 term, 5
Stone duality, 49 complexity, 6
Stone space, 49 constant, 7
strict order property, 132 Theorem
strong type, 142 Beth’s Interpolation, 91
Lascar strong type, 121 Finite Equivalence Relation, 147
strongly κ-homogeneous structure, 88 Lindström’s, 37
strongly minimal of Cantor on DLO, 26
almost, 105, 166 of Löwenheim–Skolem, 24
formula, 81 of Lachlan, 73
theory, 81 of Morley, 86
type, 81 downwards, 74
structure, 2 of Ryll–Nardzewski, 51
κ-homogeneous, 88 of Vaught, 54
κ-saturated, 69 Ramsey’s, 64
κ-universal, 88 Ressayre’s, 159
ω-saturated, 51 Vaught’s Two-cardinal Theorem,
atomic, 59 75
existentially closed, 35 theory, 14
K-saturated, 55 ω-stable, 68
many-sorted, 5, 138 complete, 15, 16
saturated, 69 consistent, 14
special, 87 countable, 47
strongly κ-homogeneous, 88 equivalent, 15
INDEX 243

finitely satisfiable, 19 partial, 47, 90

induced, 82 quantifier-free, 58
inductive, 30 regular, 168
κ-categorical, 25, 26 stable, 136
κ-stable, 67, 131, 132 stationary, 121, 143
model complete, 34 strong, 142
modular, 168 strongly minimal, 81
of a class of structures, 19 weakly orthogonal, 168
of abelian groups, 14 type-definable class, 92
of commutative rings, 14
of fields, 14 ultra-homogeneous, 56
simple, 111 ultrafilter, 13
small, 53 non-principal, 67
stable, 130 unidimensional theory, 165
substructure complete, 34 unique decomposition
supersimple, 150 of formulas, 9
superstable, 150 of terms, 6
totally transcendental, 68 uniqueness of non-forking extensions,
unidimensional, 165 143
universal, 29 universal
with prime extensions, 73 formula, 11
thick formula, 116, 121 quantifier, 8
totally transcendental theory, 68 theory, 29
trace, 190 universe, 2
transcendence degree, 204 U-rank, 152
transitivity, 115, 144
valid sentence, 14
Transitivity, 80, 119, 146, 202
vanishing ideal, 195
tree, 61
variable, 5
tree property, 110
Vaught’s Conjecture, 54
trivial pregeometry, 204
Vaught’s Test, 25
type, 23, 48
Vaught’s Theorem, 54
algebraic, 79
Vaught’s Two-cardinal Theorem, 75
almost orthogonal, 165
Vaughtian pair, 75
analysable, 164
vector space, 39
based on, 145
definable over C, 128 weak elimination of imaginaries, 141
generated, 211 Weak Boundedness, 146
global, 90 weakly compact cardinal, 208
internal, 161 weakly inaccessible cardinal, 186, 208
invariant, 110, 113 weakly orthogonal type, 168
Lascar strong type, 121 Weak Monotonicity, 146
minimal, 83, 152 well-ordering, 182
of a set, 23 theorem, 183
of an element, 23
orthogonal, 165 Zilber’s conjecture, 171
parallel, 145 Zorn’s Lemma, 184