Heights of Models of ZFC and the Existence of End Elementary Extensions II

Author(s): Andrés Villaveces

Source: The Journal of Symbolic Logic, Vol. 64, No. 3 (Sep., 1999), pp. 1111-1124
Published by: Association for Symbolic Logic
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THE JOURNAL OF SYMBOLICLOGIC
Volume 64. Number 3, Sept. 1999

HEIGHTS OF MODELS OF ZFC AND THE EXISTENCE OF END

ELEMENTARY EXTENSIONS II

ANDRES VILLAVECES

Abstract. The existence of End Elementary Extensions of models M of ZFC is related to the ordinal
height of M, according to classical results due to Keisler, Morley and Silver. In this paper, we further
investigate the connection between the height of M and the existence of End Elementary Extensions of M.
In particular, we prove that the theory 'ZFC + GCH + there exist measurable cardinals + all inaccessible
non weakly compact cardinals are possible heights of models with no End Elementary Extensions' is
consistent relative to the theory 'ZFC + GCH + there exist measurable cardinals + the weakly compact
cardinals are cofinal in ON'. We also provide a simpler coding that destroys GCH but otherwise yields the
same result.

I wish to thank my advisor, Kenneth Kunen, for many helpful conversations and
comments, alwaysfull of interestinginsights. I also wish to thank Ali Enayat for very
helpful discussions about some of the topics treated in this article, Sy Friedman for
interesting questions and comments related to the class forcing construction around
Theorem 3. 1, Mirna Dzamonja and Arnie Miller for various helpful discussions.

?1. Introduction. Let (M, E), (N, F), etc., denote models of 'enough set theory.'
The central notion of extension we use in this article is the well known 'end elemen-
tary extension'. A model (N, F) end extends (M, E) if and only if for every a E M,
the sets aE= { b E M b E a } and aF= { b E N b F a} are the same. In other
words, elements of M are not enlarged by the extension from M to N.
Let (9'M, -<e) denote the structureof all proper elementary end extensions ('eees')
of M (a model of set theory), together with the relation <e (we write 'Il --< A' if
and only if _ is a proper elementary end extension of A). He is an ordering on 9'M.
The kind of ordering relation that -<e is on 9'M depends heavily on certain structural
features of M. Notice that structures like (9'M, -<e) need not be well-founded: as
Kaufmann notes in [7], if X-is weakly compact, X(n), -<e) has infinite descending
chains.

REMARK. A model theoretical result by Rubin (Theorem 2.1 of [12]) implies

that if M is a countable model of ZFC then 9'M is not well-founded. This, and
the previously mentioned result by Kaufmann still leave many cases open for the
question of well-foundedness of the structure ((,n), H<e).

Received October 6, 1996; revised October 26, 1997.

? 1999, Association for Symbolic Logic

0022-4812/99/0010/$2.40

1111

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1112 ANDRES VILLAVECES

In this paper, we concentrate on problems of existence of end elementary exten-

sions of specific models of ZFC, and we study the relationship between the height of
those models and the possibility to obstruct the existence of eees. In a forthcoming
work, we will concentrate on the study of chains in ('M, <e), and their connection
with large cardinal properties of M.
In this paper and in [13], we consider the following general question:
QUESTION1. How does the structure of M affect the structure of (9'M,

In [13], we concentrate on the problem of the existence of arbitrarilylong chains

in e(v ,cs), and investigate the connections with large cardinals properties of ;.
Here, we rather concentrate on the more specific
QUESTION2. Under what conditions on M is 'M + 0?

The earliest results toward a solution to Question 2 were obtained by Keisler,

Silver and Morley in [10], [9] and [8]. All their theorems addressed the specific
question of the existence of eees (when is 'M + 0?). They provided answers for the
two following cases.
THEOREM1.1 (Keisler, Morley [8]). Let M be a model of ZF with cof(M) = co.
Then e'M 4 0. (This takes care in particular of all countablemodels of ZFC.)
THEOREM 1.2 (Keisler [10] and Keisler-Silver [9]). If M isoftheform W(K),where
X is a weakly compact cardinal, thenfor all S c M, :( 4,Cs) 7 0.
In light of those early results, it is natural to ask to what degree does the height of
M determine the structure of 9'M. We begin by the basic question: does the height
of M determine the existence of an eee?
Here are the two main results of this paper.
THEOREM 2.2. The theory 'ZFC + 3] measurable+ Vi- (,- inaccessiblenot weakly
compact -* 3 transitiveM,< l= ZFC such that o(M) = and fM = 0)' is consistent
relative to the theory 'ZFC + 3] measurable'.
THEOREM 3.1. The theory 'ZFC + GCH + ]X (Ameasurable)+ VK [I inaccessible
not weakly compact -* 3 transitiveM, - ZFC such that o(M) = t/ and FM = 0)]'
is consistent relative to the theory 'ZFC + 3] (Ameasurable) + the weakly compact
cardinalsare cofinal in ON'.
The 'conclusion' in both theorems is that the existence of transitive models of the
form 3W(,-)which have eees, yet the inner model M,< c 3W(,-)does not have any
eees (that is, the 'global' negative answer to the Height Problem, later abbreviated
as the 'NED' property) is consistent with fairly reasonable large cardinal axioms.
The proofs in both cases use codings of the obstructions to the existence of eees
by an appropriateinnermodel. Although the results look similar, the codings used in
both cases differ strongly: in the first theorem, the places where GCH holds or fails
provide the main tool for the coding; in the second case, since one needs to obtain
models where GCH holds everywhere, one cannot use anymore such a device for
the coding. In that case, forcing nonreflecting stationary sets at the appropriate
levels to a model previously freed of any of those sets does the trick.
The notation we use is standard. We restrict ourselves to the case of transitive
M. Following two different traditions, we freely switch between the two notations

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 HEIGHTS OF MODELS OF ZFC ... 1113

'V,<'and '3W(-.')' when we denote the set of objects of the universe of rank less than
. Given a model M, o(M) denotes the ordinal height of M.

?2. The height problem.

QUESTION 3 (Height Problem). Do $9g(,) 0 and o(M) = imply together that

FIGURE1. The Height Problem.

When M is definable, the problem is trivial, by the following

2. 1. If M is definablein -(Pi), and 3i(/)
PROPOSITION has an eee, then M has an
eee.

PROOF. Let N elementarily end extend -W Since o(M) = MN, the

interpretation of the definition of M in N, properly (end) extends M. It is easy to
verify that we actually have M-<eMN. A

Of course, this is far from settling the general problem. As Theorem 2.2 shows,
it is quite possible that this question has a negative answer: there are models of
set theory where a certain - (,-) has eees, yet some transitive submodel with the
same height has no eees. This can even be obtained in a quite uniform way: the
'extendability property' only holds at weakly compact cardinals.
Still, if we add new axioms to the theory, the answer to the question may become
positive. This is the case, for example, when V = L holds in M, the question
becomes trivial, since in that case the only transitive submodel of height X-of an
3j1(/j;)is -(p,) itself.
Therefore, the answer to the Height Problem is independent of ZFC.

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1114 ANDRESVILLAVECES

2.1. Consistency of a global negative answer. We will next present some of the
possible situations for answers to a version of the problem. Let G (A) denote any
large cardinal property preserved under Easton-type extensions where the iteration
process is not carried too often (such as measurability, etc.). More precisely, we
mean here properties unaffected by Easton iterations which only act at certain
successor cardinals.
DEFINITION1. Let K- be an inaccessible cardinal. We say that K- is non-end-
determining (or for short NED(X-)) if and only if there is a transitive M,< I=ZFC
such that o (M) = t, and 'M = 0.
Thus, if NED(X-), there may be models of the form t,< which have eees, yet
certain inner models of them do not have well-founded eees. Observe that the NED
property depends strongly on the universe where X-is being considered.
THEOREM 2.2. The theory 'ZFC+3A G (A)+Vi (, inaccessiblenot weaklycompact
NED(X,))' is consistent relative to the theory 'ZFC + 3] G (A)'.
We will devote the remainder of this section to proving Theorem 2.2 and to
draw some corollaries from it. The proof of Theorem 2.2 is based on a forcing
construction which produces codes for various sets which witness the non existence
of eees at the desired cardinals. The '3] G (A)'clause is added to the theory to point
out that the forcing can be done in such a way that large cardinal properties G (A) (at
least, those not destroyed by Easton iterations which only act at certain successor
cardinals) are consistent with the negative answer to Question 3.
In the next lemma, we introduce the forcing construction, and we show that both
weak compactness and non weak compactness are preserved in the extension.
LEMMA2.3. Let M be a model of ZFC + GCH, and let M[G] be forwards Easton
extension obtained by adding a++ Cohen subsets to every successor inaccessible a.
Then M and M[G] have the same weakly compact cardinals.
PROOF. Let first K- be weakly compact. Notice that K- is not an active stage of
the Easton iteration: this one only acts at successor inaccessibles. Let p be El
(say _p 3Xy(X), y first order), and let P = iP, be the iteration defined in the
statement, below K. Let
I Vp (Va < K 3X c a [(a, <, T n a) l ())

for some P-name z in M such that 1 IF~pz c a. Then, since X-is weakly compact in
M, there are N, P*, r* such that
(-W(K), By -, ED) He (N. , * P*).
Now, in N, P* is a forwards Easton iteration which (since P* does not act on
may be regarded as a product
-
P x P'.
A usual argument shows that P' is (< p-+)-closed. Then, P* does not add any new
subsets to K. Now, in N, we have
1 I p*3X c <((,,<z* nF>-,X) )

As forcing with P* is isomorphic to forcing with P' and then with P, and z-*n K=,
1 I[,p, 3 someP-name a [1I[-p ((a, <<, ) =

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 HEIGHTS OF MODELS OF ZFC ... 1115

All we need now is to 'eliminate' P' from the previous sentence. But this is done
by using the (< p+)-closure of P', and the following usual fusion argument: let
(pa, la < a) be a chain in P', such that pa decides whether a E a and pal is stronger
than pa whenever a < a' < i. By the (< K+)-closure of P' there is p,< stronger
than all the pa, for a < a. This condition decides a P-name a in M such that
1 I ,p ((a ,<i)
<, f C)
So, X is also weakly compact in M[G].
Now, if X is not weakly compact in the universe, there is a ;-Aronszajn tree T.
There are two cases:
(i) X is Mahlo: then P is the product of a ;-Knaster and a r+;-closed forcing; thus
1 IF T is r-Aronszajn,
whereby X cannot be weakly compact in M[G]. The Mahloness of X implies that
P<,. is ;-Knaster (i.e., every subset of P<,. of cardinality X contains a (< r)-linked
subset of cardinality a). ;-Knaster forcings cannot add new branches to e-trees,
by a variation of the argument in [2], Theorem 8.5.
(ii) X is not Mahlo. Then, there is a club C of singular cardinals in a, C E M.
None of these singular cardinals becomes regular in M[G]; clearly C is also a club
in X in M[G]. So, X remains non-Mahlo in the extension. -

Having shown this preservation property of P, we are ready to complete the

PROOFOFTHEOREM 2.2. Assume that GCH and 3] (G (A)) both hold in V. Let
V[G] be a forward Easton extension as in the previous Lemma. Then V and V[G]
have the same weakly compact cardinals. If X is not weakly compact (in V or
V[G]), then we need to find a transitive M,< in V[G] with no eee and o(M,) = K.
There are three cases:
(1) K is a successor inaccessible: then let M,< = M(K). This M,< cannot be
end elementary extended: let N >Fe M,<. Then without loss of generality, K is
inaccessible in N: this is clearly true if N \ M happens to have a minimum ordinal.
If this is not the case, then using Theorem 2.1 of [9] we can arrange some well-
founded N, 'sufficiently elementary' (for the purpose of the proof) in N, and end
extending M. We can then continue the argument with N replacing N.
Now, the inaccessibility of K in N implies that the inaccessibles are unbounded
in K, and this is incompatible with the fact that K is a successor inaccessible.
(2) K is a non-Mahlo limit inaccessible: then fix a club C in K with no regular
cardinals, and let M,< code C by the powers of the successor inaccessibles below K.
This M,< is obtained as follows: write K as an increasing sequence (Aa)a<,n where
the a's run over the successor inaccessibles below K. Now, set M,< = the extension
of W(K) where a++ subsets of a are added only when Aa E C. Clearly, we have
(K) v c M,< c _ (K) V[G]and o(M,) = K. Since M,<codes C, it cannot have any
eee (if it had one, say N, then K would be a singular cardinal in N!).
(3) K is Mahlo: then let T be a K-Aronszajn tree in V. Since T remains i-
Aronszajn in V[G], it is enough to let M,< encode T similarly to part 2; this new M,<
cannot be elementarily end extended, for it would then provide a Ki-branch to T in
V[G], contradicting the fact that T is Aronszajn. This completes our proof. A

2.4. The Height Problemmay have a negative answer.

COROLLARY

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1116 ANDRIS VILLAVECES

PROOF. In the model V[G] constructed in Theorem 2.2, let K-be the first inac-
cessible such that e(,) z/ 0. Results in [9] show that X-is not weakly compact.
Then, by Theorem 2.2, there exists a transitive model M,<of ZFC which is of height
X-and has no well-founded eees. This provides a negative answer to the Height
Problem. -

It is interesting to observe the following, in connection to Theorem 2.2.

PROPOSITION2.5. If K- is weakly compact, thenfor all transitive M C W(j-) of
height /., 9'M 0.

PROOF. It suffices to observe that if K-is weakly compact, then (3(,), C,S) has
eees, for all S c _(/). Take any (transitive) M c _(j-) of height a. Then
(g(X), E, M) has a well founded eee (N, E, M'). Clearly,
(i) M Ce M': by endness, all the 'new' elements go 'on top' of M, and
(ii) M -< M': given any sentence v, _ (X) I=(M fv b) if and only if N (M' =
v). This implies M -< M' by the transitivity of N. -

The 'opposite' problem is still open, for cardinals X-which are inaccessible not
weakly compact:

OPEN QUESTION 1. Is the theory 'ZFC + large cardinals + Vi1 inaccessible not
weakly compact ((M F ZFC A o(M) = A () 0) -* em 0)' consistent?
In other words, is it consistent to have large cardinals and simultaneously a
globally positive answer to the Height Problem? Up to now, the only way to get
globally positive answers to the Height Problem is with V = L or similar axioms.
Recently, Leshem has proved a result that provides insight into the difficulty of
Open Question 1. He has obtained the following result.
THEOREM2.6. (Leshem [ 1]) If 00 exists thenfor every cardinalK there is an inner
model M such that

M #= v= ? 0.

This theorem does not solve Open Question 1 in the presence of 00: 9vKis empty
only in the sense of M, not in the sense of the universe. Leshem's proof uses a
decomposition (due to Kunen) of Cohen forcing into adding a Suslin tree and its
own destruction, and a code of the Suslin tree in an inner model via Levy collapses
of certain L-cardinals.
We basically have two extreme situations here: the Height Problem under the very
restrictive V = L (trivial positive answer since there are no strictly inner models),
and the Height Problem in the presence of Large Cardinals (negative answers). It
is natural to ask

QUESTION 4. What happens 'in between' V = L and Large Cardinal Axioms?

More specifically,what is the situation for models of the form L[O] or L[u] (where
,u is a nonprincipal re-completeultrafilter on some uncountable a-)?

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 HEIGHTS OF MODELS OF ZFC ... 1117
?3. A new coding: keeping GCH true throughout the forcing. In the previous
section, the construction of models for NED(X-;)used Mc Aloon's coding method:
coding (in inner models of the generic extension) the Aronszajn trees or the clubs
of singulars needed to destroy eees by using the successor inaccessibles where GCH
holds/fails as the coding device. Naturally, if one wants to obtain a similar result
while keeping GCH true, one must code the construction in a completely different
way.
THEOREM3.1. The theory 'ZFC + GCH + 3] (A measurable)+ Vi- [ri; inaccessible
not weakly compact -* NED(X-;)]' is consistent relative to the theory 'ZFC + 3] (A
measurable)+ the weakly compact cardinalsare cofinal in ON'.
This answers a question that Sy Friedman asked me during the Tenth Latin
American Mathematical Logic Symposium in Bogota'. The proof of this theorem
consists of a two step forcing, followed by a construction of models of height X-with
no eees in the same way as in the end of the proof of Theorem 2.2: coding in M,<
an object that makes it impossible for M,<to have eees. (Depending on the case, a
/-;-Aronszajntree or a club of singulars.)
The two forcing constructions provide enough coding tools to complete the proof
in a way analogous to how coding was used to prove Theorem 2.2. The idea is to use
the existence/non-existence of nonreflecting stationary sets on a++ (those which
will be called Ea?? -sets in the construction), for a successor inaccessible, instead of
the continuum function (cardinals where GCH holds/fails), to do the coding.

c M

V[G1

\ / / (Of /V[G]~~~~~~~~~~~~[H

the
coding

cardinals

FIGURE 2. Coding M by adding only some E-sets (the Ds).

We start with a model of 'ZFC+GCH?+3 (Ameasurable) + the weakly compacts

are cofinal in ON'. We first force the non-existence of 'E-sets', and then force back

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1118 ANDRESVILLAVECES

the existence of them on successor inaccessibles. We will then be able to construe

the inner models we need inside the final generic model by putting in it exactly the
E-sets one needs to carry the coding.
DEFINITION 2 (EO, nonreflecting stationary sets). Let 0 be a regularcardinal, and
X> 0, with cof(/i;)N1. We mean by ES0that

3A c {f < iC cof(b) = 0 }, A stationary in iKsuch that

Vlimit a < i, (A n a is not stationary in a).
Abusing the notation a bit, we sometimes say that the set A is an 'ET-set', or even
an 'F-set'.
PROOF OF THEOREM3.1. Start with M, a model of 'ZFC + GCH + :R (, measur-
able) + Va 3,u > a (u weakly compact)'.
Let IPbe the Easton product of the Levy collapses Li = Coll (ei+, < 0 ) of cardinals
< Oi to <,+,where i E ORDM, and the sequence ( (ei, 0i) i E ORDM ) is defined
as follows: ro is the first successor inaccessible cardinal, and 00 is the first weakly
compact cardinal; if ci, Oi are defined, then 'ci+l is the first successor inaccessible
after Oi and 0i+1 is the first weakly compact cardinal after 'ci+1. For limit i, ci is the
first successor inaccessible after sup{ rjI j < i }, and Oiis the first weakly compact
cardinal after ,j.
More precisely, p E IPif and only if p is a function such that
(i) p(i) E Li,
(ii) for all i E dom(p), for all a inaccessible such that i < a implies &i< a,
dom(p) n a is bounded in a.
IPis ordered in the usual way.
For every en, IPfactors into three forcings: IP IP<,. * P,, * IP>,,, where IP<,<,IP,
IP,, mean respectively the portions of the iteration below a', at iC, and above i'.
By Easton's Lemma and GCH in M, IPcollapses the cardinals in [iQO+, 0) to
K+ (thus, in the generic extension, &ibecomes ,t+j), and preserves all the other
cardinals. In particular, all the Icisremain successor inaccessible in the extension,
as the reader can easily check.
CLAIM. For all i, for all stationary S c { < , <++ cof(/3) = ,i } in VP, there
exists a such that cof(a) = <,+and S n a is stationary. (We say that the stationary
reflection property holds for ci and ,?++ at K,+.)

To show this, it is enough to look at V(?Oi), since this coincides with VP up

to the level 'ci+1. V('?i) can be seen as (VL )P<o, Since the stationary reflection
property holds in V for ci and 0Otat <,+(by Baumgartner [1]), we have that the
stationary reflection property for ei and 0i =K jt+ (at ti+ necessarily) also holds
in VL . On the other hand, IP<o,I < <+ < 0i; hence if S is a stationary subset
of { < <++ I cof(/3) = Kc}, and S E (VLi)P<oi, then there is T c S stationary,
T E VL. By the stationary reflection property in VLj, there exists some y < 0i,
with cof(y) = <+, such that T n y is stationary in VLj. T n y is still stationary in
(VLj)1<oj, since IP<oI < <,+. (Since whenever p IF 'i club in y' (cof y = '<+), we
have that i already is a club in y in VLj .) All this ends the proof of the Claim.

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 HEIGHTSOF MODELSOF ZFC ... 1119

So, the stationary reflection property holds in VP for all ei, 0j. This ends the first
stage of the proof. We have obtained a model V1 = V[G], for G EP-generic,where
Egi-sets do not exist, for any i E ORD.
For the second step of the proof, we will simplify matters by writing 'V' instead
of 'VI' and calling our forcing IPagain. We need to force E,> -sets into e for
each successor inaccessible i'. We will achieve this goal by 'piecewise adding' the
E-sets.
The proof that such E-sets can be added is an Easton iteration. We present
here a variant of the construction by Cummings, Dzamonja and Shelah in [4] of
a generic extension of a model of 'GCH + EU (, measurable)', where a regular
cardinal 0 'strongly non-reflects' at Rj,where i < 0, cof(Ri) = ,j < A+ < 0. Their
construction yields in particular Ei -sets in the generic extension (see Lemma 2 and
Theorem 1 of [4]).
The building blocks. Given any successor inaccessible cardinal iK.let IP,<consist
of conditions p = (ap, Ep, Cp), where ap < e++, cof(ap)=Kc+, Ep c ap,
C: SCP 'U 9(A

where SP= {f < cof(/3) = e'+i}, and for all f E SV,C+p (/) is a club on
ap I
f. We also require that for all fi E S'P, C13(f) n Ep = 0, and Ep c So"'. So, Ep
behaves like an 'approximation' to a non-reflecting stationary subset of S? . We
define the ordering of IP, by
(a) aq > ap,
q< p <> (b) Eq n ap = Ep,
t (c) CqFS; = C
We need only worry at ordinals of cofinality en+:
LEMMA3.2.If cof (a) E [RI, K+), then there is a club C c a such that for all
y EC, cof (y) <EK.
PROOF. Fix a with RI < cof (a) < E, and let f: cof (a) -* a be increasing
and continuous. Then all the ys in the domain of f are < eK.As f is continuous
increasing, ran(f) is a club in a, and for every fi E ran(f), we have cof(/3)
cof(f-1 ()) <e. A

Let now G be IP,-generic, and let EG =

UpCG EP.

CLAIM. EG is a nonreflecting stationary subset of S?++.

To show that EG is nonreflecting, let a < C++be of cofinality EC+.Then a < ap

for some p E G, and thus CP (a) n EP = 0. But by (b) in the definition of <,
EGn ap = Eoap Cap (a) c a < ap. So, CaP(a), which is a club on a, does
not intersect EG: CGP(a) n EG = (Cap(a) n ap) n EG = Cap(a) n (ap n EG)
Cap(a) nEaP 0.
To see that EG is stationary, let F be a canonical name for EG, and suppose that
p HF(z club of K++A r n F = 0). Now construct a decreasing K-chainof conditions
p >no > p< > * > pq > s p< E P,, such thatforU < a, pT q is E a
and agl < A,, < ahs+ .~Le sp<< g U~<KEpe, Up<,< Cp, ). This q is not a

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1120 ANDRES VILLAVECES

condition, since the cofinality of its support is < i'+. But we just have to take any
condition r that extends q in a natural way.
[How? Just let

r =(sup ?+ +, U Ep U {a*}, U Cp U ),

where
(i) a* = sup,<,oap,, and
(ii) C(4 is any club on ,Bwith min(C(,B > or*,for B (E S. +\o*
It is not difficult to check that this r E IP,,.]
Then, the limit of the 5is (= a*) is forced by r to belong to the club z. On
the other hand, a* E E, and r BFa* E F. But this contradicts the fact that
pHV(zclubAznFO0).
CLAIM. IP,.is (< r,+)-closed.

This Claim is established by the same argument as the previous one, although
taking now sequences of arbitrarylengths below e'+.
REMARK. Clearly, IP,,is not (< r,++)-closed, since IP,<adds r,++-sequences.
The IPs correspond thus to the 'Building Blocks' of our construction. The re-
maining part consists of the iteration through all the successor inaccessible cardinals,
and of the proof that measurables are preserved.
CLAIM. IP,j < +i++.

(Just observe that in V1, ,++. 2', 2' e -

3.1. The iteration. The iteration is in the style of Backward Easton. The sup-
ports are bounded below regular cardinals. We add new subsets only at successor
inaccessible stages. Formally, this corresponds to defining IP< as the forcing up
to stage a and {2 E Vet as the forcing at stage a. Set 0, = {0}, if a is not a
successor inaccessible, and Q-(IP) v<, otherwise, where V< stands for V1<-.
By the rc-closure of the forcing IP,.,this iteration yields a model of ZFC. In order
to finish the proof, we need only show that the measurability of A is preserved by
this iteration.
Let A be a measurable cardinal in V, and let j: V -? M be the ultrapower
map arising from a normal measure U on e,. We first observe that it is enough
to prove that A remains measurable in the extension by P<K,the rest of the forcing
being (< i+)-closed,by the previous claim. (The power set of A is not changed by
IP>A IP>.)
As is well-known, j has the following properties:
1. crit(j) = A.
2. AM C M.
3. A+ < j(A) < jLLA+)< i+
4. M={j(F)( F
FEVAdom(F)=i}.
We want to prove that j 'lifts' to an embedding j: V[G] N C V[G] (which -

we shall also denote by 'I', abusing notation), where G is IP-genericover V, thereby

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 HEIGHTSOF MODELSOF ZFC ... 1121

automatically ensuring that A is also measurable in V[G]. The idea is to prolong

the generic G to a generic H for the forcing j (IPx,).
We start by comparing IR< to j (1K2). j (IP,) is an iteration defined in M, forcing
the existence of non-reflecting stationary sets on each SKj , for each e' successor
inaccessible < j (A). If we compute the iteration j (IP<A)up to stage A,we get exactly
P<K. We can now compute a generic extension M[G] of M using the V-generic
filter, since IP<A-genericsover V are also generic over M. Since JIP<K< A, every
canonical Ix<-name for a A-sequenceof ordinals is in M, so is also in V[G]. Thus,
we have
CLAIM. V[G] i A(M[G]) c M[G].
In M[G], we will prolong G to a j(I2<A)-generic. Call the 'remainder forcing'
JR= 1Rj(A),that is, j(IP<2) = IP< * R. JRis (< A+ )-closed in M[G]; hence, by the
previous claim, JRis also (< A+ )-closedin V[G]. In M[G], the forcing JRis j(A)-cc
and has size j (A)).There are then at most j (A)maximal antichains in IRtin the model
M[G], as [j (A) -(A) = j ())]M, by elementarity and since IP<AI < A. In V[G], we can
enumerate those antichains as (A, : a < A+), since A+ < j(A) < j(A+) < i++.
Using the closure to meet all these antichains, it is clear that in V[G], we can build
H which is JR-genericover M[G]. Letting G+ = G * H, G+ is j (PA)-genericover
M. We define j: V[G] -, M[G+] by j(QG) =j()G?. This is a well-defined
elementary embedding, as follows from the following fact, whose proof uses the
Truth Lemma and the elementarity of j.
FACT. Let k: M -, N be an elementary embedding between two transitive
models of ZFC. Let P E M be a forcing notion, let k(P) = Q, and suppose that
G is lP-genericover M and H is Q-generic over N. Let also k"G c H. Then
the definition k (iG) k(i)H for every i E M' gives a well-defined elementary
-

embedding k: M[G] - N[H], which extends k: M -* N and is such that k(G) =

H.
This ends the proof of measurabilityof A in V[G], and thus the proof of Theorem
3.1: At this point, to witness the non-end-determining property of the appropriate
As, exactly the same argument used in the proof of Theorem 2.2 works. Instead of
using the failure/non-failure of GCH at the successor inaccessibles below e' to code
the corresponding object (club of singulars or ';-Aronszajntree), here, the coding
is achieved by the inclusion/non-inclusion of the corresponding E-set (see figure
3). -1

?4. Eees of inner models. In this final section, we push the study of the con-
nections between the existence of eees and the structure of the basic model one
step further: We look at inner models of some specific models related to 00 or to
measures, and we look at some instances of the 'wider model problem'.
In the last section, we remarked that for L[,u], the situation is similar to the one
obtained via Theorem 2.2: if e, is not weakly compact, then there is a model M of
height e, with no eees. Contrasting this, we have the following fact for the inner
model LM, in presence of 00.
PROPOSITION4.1. If M t 'O0exists', and M is a set model of ZFC, thenXLM =& 0.

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1122 ANDRES VILLAVECES

PROOF.The existence of 00 in M allows to obtain eees of LM: as was done by

Enayat in Remark 2.21 of [6], one stretches the Silver indiscernibles to get eees of
LM. -]

The converse to this is clearly false: just take M = L,< such that (for example), em
is weakly compact. Then, XLAM + 0, yet M t 'O0does not exist'.
We have the following situation, asking the same question for wellfounded eees
(compare the following result and Proposition 4. 1, and to Kunen's result in Kauf-
mann [7]).
PROPOSITION4.2. Suppose that M satisfies
(i) M is countable,and
(ii) M t '00exists',
and that M has minimalordinalheight amongall the models whichhavesimultaneously
theproperties (i) and (ii). Then, M has no wellfounded eee.

PROOF.We can use a S1 formula to say that the height of a certain model M is
a and that M thinks that 00 exists: let u(a) '3 transitive M (o(M) = a A M t
00 exists)'. This is clearly E'. Now, we also have by Ldwenheim-Skolem that
ZFC F- [3a q/(a) -* Ca < w1 co, (a)]. The formula y/ (a) relativises down to models
N of ZFC when aN is countable. So, working in V, we can fix M, a transitive
model such that M t '0oexists', o(M) =- , and 6 < w1 is the least possible among
the heights of such models. By Keisler-Morley, we know that 9'LM + 0, and thus
LM = Lb t ZFC + 'inaccessibles are cofinal in ORD'.
Suppose then that 9'LAf has well-founded elements; let y < 0w be such that
LY, s Lo. We then have that Ly t am = a. A collapse of 6 to o does the trick:
working in Ly, let IP= Coll(w), 6), and let G be IP-generic over LI,. So, on one hand,
in LJJG], 6 is countable, and on the other hand, y(b) holds in V. Then, y(b)
also holds in Ly[G], by El-absoluteness. But then, since Ly and LJ[G] have the
same ordinals, y/(b) holds in Ly. Finally, by elementarity, La t 3O (y(0)). This
contradicts the minimality of 6. -A

So far, the Height Problem has only been looked at for inner models of a given
W(r,) which is known to have non trivial eees. Yet a natural question arises con-
cerning 'wider' models. More specifically, we have the following

QUESTION 5 (eees for wider models). Let 2Abe an inner model, X(,S ) 0, M D
M (r,) , M t- ZFC, o(M) = r, Ir, > co_. When is FM:+ 0?

In such generality, there is no simple answer to this question. Yet, we have that
in particular, when M is a generic extension of S(,4), obtained via a set-forcing
P, M must also have nontrivial eees. This is easily established by observing that
if N E 2 and M = ( then N[G] F-e M. This was implicitly used
in the proof of Theorem 2.2. There, we did not really have a set-forcing. Yet, the
decomposition properties of Easton forcing acting only on successor inaccessible
cardinals imply that in that case the forcing extension necessarily has eees.
An example of a class forcing that destroys the property of M(r,) having eees
can be obtained from Boos's Easton style forcing construction from [3]. This was

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 HEIGHTSOF MODELSOF ZFC ... 1123

FIGURE
3. The 'wider models problem'.

pointed out to me by Ali Enayat. Boos's forcing, a variation on a thin-set forcing

notion by Jensen, allows one to efface the Mahlo property:
THEOREM 4.3 (Boos [3]). If M is a transitive model of ZFC + GCH, and f, is
r,-Mahlo in M, thenfor each a < X there is a forcing notion B, such that MIBQt
is strongly a-Mahlo but not (a + 1)-Mahlo'.
Using the B1,from this theorem, any weakly compact cardinal in M becomes the
first Mahlo in a generic extension. So, we have
COROLLARY4.4. Thereare Easton-typeforcings whichdo not preserve theproperty
of havingeees: let s, be a weaklycompact cardinal. Then,although we knowby Keisler
and Silver [8] that q(,K)M + 0, we have 9q(,)M[G]- 0,for G 113I-generic
over (K)M.
REMARKS.
(1) Boos's construction-a variant of a construction by Jensen-is an Easton-
type iteration. By [8], if r, is the first Mahlo cardinal, 3g(r,) cannot have any eees.
The weakly compact is of the corollary (in M) becomes the first Mahlo in the
extension by B I.
(2) Of course, if B,, is used instead of B1 here, then is becomes the first n-
Mahlo cardinal. The corresponding M (,s)M[G] cannot either have elementary end
extensions: if N >We RH(l)M[G] then N tb VA 3C c A (C club and C does not
contain any (n - 1)-Mahlos. But then, this is also true about ,s, and ,s would not
be n-Mahlo.

REFERENCES

[1] J. BAUMGARTNER,
A new class of ordertypes, Annals of Mathematical Logic, vol. 9 (1976), pp. 187-
222.

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1124 ANDRES VILLAVECES

[2] , Iteratedforcing, Surveys in set theory (A. R. D. Mathias, editor), London Mathematical
Society Lecture Note Series, no. 87, Cambridge University Press, 1983, pp. 1-59.
[3] W Boos, Boolean extensions whichefface the Mahlo property, this JOURNAL, vol. 39 (1974), no. 2,
pp. 254-268.
[4] J. CUMMINGS, M. D'AMONJA, and S. SHELAH, A consistencyresult on weak reflection, accepted by
FundamentaMathematicae.
[5] A. ENAYAT, Countingcountablemodels of set theory, preprint.
[6] , On certain elementary extensions of models of set theory, Transactionsof the American
Mathematical Society (1984).
[7] M. KAUFMANN, Blunt and topless extensions of models of set theory, this JOURNAL, vol. 48 (1983),
pp. 1053-1073.
[8] H. J. KEISLER and M. MORLEY, Elementary extensions of models of set theory, Israel Journal of
Mathematics, vol. 6 (1968).
[9] H. J. KEISLER and J. SILVER, End extensions of models of set theory, Proceedings of Symposia in
Pure Mathematics, vol. 13 (1970), pp. 177-187.
[10] J. KEISLER, Some applications of the theory of models to set theory, Logic, methodology and
philosophy of science, Proceedings of the 1960 InternationalCongress, Stanford University Press, 1962,
pp. 80-86.
[11] A. LESHEM,00 and elementaryend extensions of V,<,preprint.
[12] S. SBELAH, End extensions and the number of countable models, this JOURNAL, vol. 43 (1978),
pp. 550-562.
[13] A. VILLAVECES, Chainsof end elementaryextensions of models of set theory, this JOURNAL, vol. 63
(I998),pp. 1116-1136.

DEPARTMENT OF MATHEMATICS
THE HEBREW UNIVERSITY OF JERUSALEM
JERUSALEM, ISRAEL
and
DPTO. DE MATEMATICAS
UNIV. NACIONAL DE COLOMBIA
SANTA FE DE BOGOTA, COLOMBIA
E-mail: villavec~math.huji.ac.il

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