Real Analysis in Reverse

Author(s): James Propp

Source: The American Mathematical Monthly, Vol. 120, No. 5 (May 2013), pp. 392-408
Published by: Mathematical Association of America
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 Real Analysis in Reverse
James Propp

Abstract. Many of the theorems of real analysis, against the background of the ordered
field axioms, are equivalent to Dedekind completeness, and hence can serve as completeness
axioms for the reals. In the course of demonstrating this, the article offers a tour of some
less-familiar ordered fields, provides some of the relevant history, and considers pedagogical
implications.

1. INTRODUCTION. Every mathematician learns that the axioms of an ordered

field (the properties of the numbers 0 and 1, the binary relation <, and the binary
operations +, −, ×, ÷) don’t suffice as a basis for real analysis; some sort of heavier-
duty axiom is required. Unlike the axioms of an ordered field, which involve only
quantification over elements of the field, the heavy-duty axioms require quantification
over more complicated objects such as nonempty bounded subsets of the field, Cauchy
sequences of elements of the field, continuous functions from the field to itself, ways
of cutting the field into “left” and “right” components, and so on.
Many authors of treatises on real analysis remark upon (and prove) the equivalence
of various different axiomatic developments of the theory; for instance, Korner [15]
shows that the Dedekind Completeness Property (every nonempty set that is bounded
above has a least upper bound) is equivalent to the Bolzano–Weierstrass Theorem in
the presence of the ordered field axioms. There are also a number of essays, such as
Hall’s [11] and Hall and Todorov’s [12], that focus on establishing the equivalence of
several axioms, each of which asserts in its own way that the real number line has no
holes in it. Inasmuch as one of these axioms is the Dedekind Completeness Property,
we call such axioms completeness properties for the reals. (In this article, “complete”
will always mean “Dedekind complete”, except in subsection 5.1. Readers should be
warned that other authors use “complete” to mean “Cauchy complete”.) More recently,
Teismann [26] has written an article very similar to this one, with overlapping aims but
differing emphases, building on an unpublished manuscript by the author [21].
One purpose of the current article is to stress that, to a much greater extent than
is commonly recognized, many theorems of real analysis are completeness proper-
ties. The process of developing this observation is in some ways akin to the enterprise
of “reverse mathematics” pioneered by Harvey Friedman and Stephen Simpson (see,
e.g., [25]), wherein we deduce axioms from theorems instead of the other way around.
However, the methods and aims are rather different. Reverse mathematics avoids the
unrestricted use of ordinary set theory and replaces it by something tamer, namely,
second-order arithmetic, or rather various sub-systems of second-order arithmetic (and
part of the richness of reverse mathematics arises from the fact that it can matter very
much which subsystem of second-order arithmetic we use). In this article, following
the tradition of Halmos’ classic text “Naive Set Theory” [13], we engage in what might
be called naive reverse mathematics, where we blithely quantify over all kinds of infi-
nite sets without worrying about what our universe of sets looks like.
Why might a non-logician care about reverse mathematics at all (naive or other-
wise)? One reason is that it sheds light on the landscape of mathematical theories and
http://dx.doi.org/10.4169/amer.math.monthly.120.05.392
MSC: Primary 12J15, Secondary 97F50; 97I99

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structures. Arguably the oldest form of mathematics in reverse is the centuries-old
attempt to determine which theorems of Euclidean geometry are equivalent to the par-
allel postulate (see [19, pp. 276–280] for a list of such theorems). The philosophical
import of this work might be summarized informally as “Anything that isn’t Euclidean
geometry is very different from Euclidean geometry.” In a similar way, the main theme
of this essay is that anything that isn’t the real number system must be different from
the real number system in many ways. Speaking metaphorically, we might say that, in
the landscape of mathematical theories, real analysis is an isolated point; or, switching
metaphors, we might say that the real number system is rigid in the sense that it cannot
be subjected to slight deformations.
An entertaining feature of real analysis in reverse is that it doesn’t merely show us
how some theorems of the calculus that look different are in a sense equivalent; it also
shows us how some theorems that look fairly similar are not equivalent. Consider, for
instance, the following three propositions taught in calculus classes.
(A) P
The Alternating Series Test: If a1 ≥ a2 ≥ a3 ≥ · · · and an → 0, then
n=1 (−1) an converges.
∞ n

(B) The Absolute Convergence Theorem: If ∞

P P∞
n=1 |an | converges, then n=1 an
converges.
(C) The Ratio Test: If |an+1 /an | → L as n → ∞, with L < 1, then ∞
P
n=1 an con-
verges.
To our students (and perhaps to ourselves) the three results can seem much of a
muchness, yet there is a sense in which one of the three theorems is stronger than the
other two. Specifically, one and only one of them is equivalent to completeness (and
therefore implies the other two). How quickly can you figure out which is the odd
one out?
At this point, some readers may be wondering what I mean by equivalence. (“If
two propositions are theorems, don’t they automatically imply each other, since by the
rules of logic every true proposition materially implies every other true proposition?”)
Every proposition P in real analysis, being an assertion about R, can be viewed more
broadly as a family of assertions P(R) about ordered fields R; we simply take each
explicit or implicit reference to R in the proposition P and replace it with a reference to
some unspecified ordered field R. Thus, every theorem P is associated with a property
P(·) satisfied by R, and possibly other ordered fields as well. What we mean when we
say that one proposition of real analysis P implies another proposition of real analysis
P 0 , is that P 0 (R) holds whenever P(R) holds, where R varies over all ordered fields;
and what we mean by the equivalence of P and P 0 , is that this relation holds in both
directions. In particular, when we say that P is a completeness property, or that it can
serve as an axiom of completeness, what we mean is that for any ordered field R, P(R)
holds if and only if R satisfies Dedekind completeness. (In fact, Dedekind proved [2,
p. 33] that any two ordered fields that are Dedekind complete are isomorphic; that is,
the axioms of a Dedekind complete ordered field are categorical.)
To prove that a property P satisfied by the real numbers is not equivalent to com-
pleteness, we need to show that there exists an ordered field that satisfies property P
but not the completeness property. So it’s very useful to have on hand a number of dif-
ferent ordered fields that are almost the real numbers, but not quite. The second major
purpose of the current article is to introduce the reader to some ordered fields of this
kind. We will often call their elements “numbers”, since they behave like numbers in
many ways. (This extension of the word “number” is standard when speaking of a dif-
ferent variant of real numbers, namely p-adic numbers; however, this article is about
ordered fields, so we will have nothing to say about p-adics.)

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 A third purpose of this article is to bring attention to Dedekind’s Cut Property (prop-
erty (3) of section 2). Dedekind singled out this property of the real numbers as en-
capsulating what makes R a continuum, and if the history of mathematics had gone
slightly differently, this principle would be part of the standard approach to the subject.
However, Dedekind never used this property as the basis of an axiomatic approach to
the real numbers; instead, he constructed the real numbers from the rational numbers
via Dedekind cuts and then verified that the Cut Property holds. Subsequently, most
writers of treatises and textbooks on real analysis adopted the Least Upper Bound
Property (aka the Dedekind Completeness Property) as the heavy-duty second-order
axiom that distinguishes the real number system from its near kin. And indeed, the
Least Upper Bound Property is more efficient than the Cut Property for purposes of
getting the theory of the calculus off the ground. But the Cut Property has a high mea-
sure of symmetry and simplicity that is missing from its rival. You can explain it to
average calculus students, and even lead them to conjecture it on their own; the only
thing that’s hard is convincing them that it’s nontrivial! The Cut Property hasn’t been
entirely forgotten ([1], [16, p. 53], and [20]) and it’s well-known among people who
study the axiomatization of Euclidean geometry [10] or the theory of partially ordered
sets and lattices [28]. But it deserves to be better known among the mathematical com-
munity at large.
This brings me to my fourth purpose, which is pedagogical. There is an argument
to be made that, in the name of intellectual honesty, we who teach more rigorous
calculus courses (often billed as “honors” courses) should try to make it clear on what
assumptions the theorems of the calculus depend, even if we skip some (or most) of the
proofs in the chain of reasoning that leads from the assumptions to the central theorems
of the subject, and even if the importance of the assumptions will not be fully clear
until the students have taken more advanced courses. It is most common to use the
Dedekind Completeness Property or Monotone Sequence Convergence Property for
this purpose, and to introduce it explicitly only late in the course, after differentiation
and integration have been studied, when the subject shifts to infinite sequences and
series. I will suggest some underused alternatives.
Note that this article is not about ways of constructing the real numbers. (The
Wikipedia page [27] gives both well-known constructions and more obscure ones;
undoubtedly many others have been proposed.) This article is about the axiomatic
approach to real analysis, and the ways in which the real number system can be char-
acterized by its internal properties.
The non-introductory sections of this article are structured as follows. In the sec-
ond section, I’ll state some properties of ordered fields R that hold when R = R. In
the third section, I’ll give some examples of ordered fields that resemble (but aren’t
isomorphic to) the field of real numbers. In the fourth section, I’ll show which of
the properties from the second section are equivalent to (Dedekind) completeness and
which aren’t. In the fifth section, I’ll discuss some tentative pedagogical implications.
It’s been fun for me to write this article, and I have a concrete suggestion for how the
reader may have the same kind of fun. Start by reading the second section and trying
to decide on your own which of the properties are equivalent to completeness and
which aren’t. If you’re stumped in some cases, and suspect but cannot prove that some
particular properties aren’t equivalent to completeness, read the third section to see if
any of the ordered fields discussed there provide a way to see the inequivalence. And if
you’re still stumped, read the fourth section. You can treat the article as a collection of
math puzzles, some (but not all) of which can be profitably contemplated in your head.
A reminder about the ground rules for these puzzles is in order. Remember that we
are to interpret every theorem of real analysis as the particular case R = R of a family

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of propositions P(R) about ordered fields R. An ordered field is a collection of ele-
ments (two of which are named 0 R and 1 R , or just 0 and 1 for short), equipped with the
relations < and operations +, −, ×, ÷ satisfying all the usual “high school math” prop-
erties. (Note that, by including subtraction and division as primitives, we have removed
the need for existential quantifiers in our axioms; e.g., instead of asserting that for all
x 6= 0 there exists a y such that x × y = y × x = 1, we simply assert that for all x 6 = 0,
x × (1 ÷ x) = (1 ÷ x) × x = 1. It can be argued that, instead of minimizing the num-
ber of primitive notions or the number of axioms, axiomatic presentations of theories
should minimize the number of existential quantifiers, and indeed this is standard prac-
tice in universal algebra.) In any ordered field R we can define notions like |x| R (the
unique y ≥ 0 in R with y = ±x) and (a, b) R (the set of x in R with a < x < b). More
complicated notions from real analysis can be defined as well: For instance, given a
function f from R to R, we can define f 0 (a) R , the “derivative of f at a relative to
R”, as the L ∈ R (unique if it exists) such that for every  > 0 R there exists δ > 0 R
such that |( f (x) − f (a))/(x − a) − L| R <  for all x in (a − δ, a) R ∪ (a, a + δ) R .
The subscripts are distracting, so we will omit them, but it should be borne in mind
that they are conceptually present. What goes for derivatives of functions applies to
other notions of real analysis as well, such as the notion of a convergent sequence: The
qualifier “relative to R”, even if unstated, should always be kept in mind.
The version of naive set theory we will use includes the axiom of countable choice.
A reasonably large subset of real analysis can be set up without countable choice, but
many important theorems, such as Bolzano–Weierstrass, rely on countable choice in
an essential way. Whether or not we believe that the axiom of countable choice is true,
distrusting countable choice requires a fair amount of foundational sophistication, and
therefore cannot (in my opinion) be considered a truly “naive” stance.
Every ordered field R contains an abelian semigroup N R = {1 R , 1 R + 1 R ,
1 R + 1 R + 1 R , . . .} isomorphic to N; N R may be described as the intersection of
all subsets of R that contain 1 R and are closed under the operation that sends x to
x + 1 R . Likewise, every ordered field R contains an abelian group Z R isomorphic to
Z, and a subfield Q R isomorphic to Q. We shall endow an ordered field R with the
order topology, that is, the topology generated by basic open sets of the form (a, b) R .

2. SOME THEOREMS OF ANALYSIS. The following propositions about an or-

dered field R (and about associated structures such as [a, b] = [a, b] R = {x ∈ R : a ≤
x ≤ b}) are true when the ordered field R is taken to be R, the field of real numbers.
(1) The Dedekind Completeness Property: Suppose that S is a nonempty subset of
R that is bounded above. Then there exists a number c that is an upper bound
of S such that every upper bound of S is greater than or equal to c.
(2) The Archimedean Property: For every x ∈ R there exists n ∈ N R with n > x.
Equivalently, for every x ∈ R with x > 0 there exists n ∈ N R with 1/n < x.
(3) The Cut Property: Suppose that A and B are nonempty disjoint subsets of
R whose union is all of R, such that every element of A is less than every
element of B. Then there exists a cutpoint c ∈ R such that every x < c is
in A and every x > c is in B. (Or, if you prefer: Every x ∈ A is ≤ c, and
every x ∈ B is ≥ c. It’s easy to check that the two versions are equivalent.)
Since this property may be unfamiliar, we remark that the Cut Property follows
immediately from Dedekind completeness (take c to be the least upper bound
of A).
(4) Topological Connectedness: Say S ⊆ R is open if for every x in S there exists
 > 0 so that every y with |y − x| <  is also in S. Then there is no way to

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 express R as a union of two disjoint nonempty open sets. That is, if R = A ∪ B
with A, B nonempty and open, then A ∩ B is nonempty.
(5) The Intermediate Value Property: If f is a continuous function from [a, b] to
R, with f (a) < 0 and f (b) > 0, then there exists c in (a, b) with f (c) = 0.
(6) The Bounded Value Property: If f is a continuous function from [a, b] to R,
there exists B in R with f (x) ≤ B for all x in [a, b].
(7) The Extreme Value Property: If f is a continuous function from [a, b] to R,
there exists c in [a, b] with f (x) ≤ f (c) for all x in [a, b].
(8) The Mean Value Property: Suppose that f : [a, b] → R is continuous on
[a, b] and differentiable on (a, b). Then there exists c in (a, b) such that
f 0 (c) = ( f (b) − f (a))/(b − a).
(9) The Constant Value Property: Suppose that f : [a, b] → R is continuous on
[a, b] and differentiable on (a, b), with f 0 (x) = 0 for all x in (a, b). Then
f (x) is constant on [a, b].
(10) The Convergence of Bounded Monotone Sequences: Every monotone increas-
ing (or decreasing) sequence in R that is bounded converges to some limit.
(11) The Convergence of Cauchy Sequences: Every Cauchy sequence in R is con-
vergent.
(12) The Fixed Point Property for Closed Bounded Intervals: Suppose that f is a
continuous map from [a, b] ⊂ R to itself. Then there exists x in [a, b] such
that f (x) = x.
(13) The Contraction Map Property: Suppose that f is a map from R to itself such
that for some constant c < 1, | f (x) − f (y)| ≤ c|x − y| for all x, y. Then
there exists x in R such that f (x) = x.
(14) The Alternating Series Test: If a1 ≥ a2 ≥ a3 ≥ · · · and an → 0, then
n=1 (−1) an converges.
P ∞ n
P∞
(15) The Absolute Convergence Property: If n=1 |an | converges in R, then
P ∞
n=1 a n converges in R.
The Ratio Test: If |an+1 /an | → L in R as n → ∞, with L < 1, then ∞
P
(16) n=1 an
converges in R.
(17) The Shrinking Interval Property: Suppose that I1 ⊇ I2 ⊇ · · · are bounded
closed intervals in R with lengths decreasing to 0. Then the intersection of
the In ’s is nonempty.
(18) The Nested Interval Property: Suppose that I1 ⊇ I2 ⊇ · · · are bounded closed
intervals in R. Then the intersection of the In ’s is nonempty.

3. SOME ORDERED FIELDS. The categoricity of the axioms for R tells us that
any ordered field that is Dedekind-complete must be isomorphic to R. So one (slightly
roundabout) way to see that the ordered field of rational numbers Q fails to satisfy
completeness is to note that it contains too few numbers to be isomorphic to R. The
same goes for the field of real algebraic numbers. There are even bigger proper sub-
fields of the R; for instance, Zorn’s Lemma implies that, among the ordered subfields
of R that don’t contain π, there exists one that is maximal with respect to this property.
But most of the ordered fields we’ll wish to consider have the opposite problem: They
contain too many “numbers”.
Such fields may be unfamiliar, but logic tells us that number systems of this kind
must exist. (Readers averse to “theological” arguments might prefer to skip this para-
graph and the next and proceed directly to a concrete construction of such a number

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system two paragraphs below; but I think there is value in an approach that convinces
us ahead of time that the goal we seek is not an illusory one, and shows that such
number systems exist without commiting to one such system in particular.) Take the
real numbers and adjoin a new number n, satisfying the infinitely many axioms n > 1,
n > 2, n > 3, etc. Every finite subset of this infinite set of first-order axioms (together
with the set of ordered-field axioms) has a model, so by the compactness principle
of first-order logic (see, e.g., [18]), these infinitely many axioms must have a model.
(Indeed, I propose that the compactness principle is the core of validity inside the
widespread student misconception that 0.999. . . is different from 1; on some level, stu-
dents may be reasoning that if the intervals [0.9, 1.0), [0.99, 1.00), [0.999, 1.000),
etc., are all nonempty, then their intersection is nonempty as well. The compactness
principle tells us that there must exist ordered fields in which the intersection of these
intervals is nonempty. Perhaps we should give these students credit for intuiting, in a
murky way, the existence of non-Archimedean ordered fields!)
But what does an ordered field with infinite elements (and their infinitesimal recip-
rocals) look like?
One such model is given by rational functions in one variable, ordered by their
behavior at infinity; we call this variable ω rather than the customary x, since it will
turn out to be bigger than every real number, under the natural imbedding of R in R.
Given two rational functions q(ω) and q 0 (ω), decree that q(ω) > q 0 (ω) if and only if
q(r ) > q 0 (r ) for all sufficiently large real numbers r . We can show (see, e.g., [15])
that this turns R(ω) into an ordered field. We may think of our construction as the
process of adjoining a formal infinity to R. Alternatively, we can construct an ordered
field isomorphic to R(ω) by adjoining a formal infinitesimal  (which the isomorphism
identifies with 1/ω): Given two rational functions q(ε) and q 0 (ε), decree that q(ε) >
q 0 (ε) if and only if q(r ) > q 0 (r ) for all positive real numbers r sufficiently close to
0. Note that this ordered field is non-Archimedean: Just as ω is bigger than every real
number, the positive element ε is less than every positive real number.
We turn next P to the field of formal Laurent series. A formal Laurent series is a formal
expression n≥N an εn , where N is someP non-positive integer and the an ’s are arbitrary
real numbers; the associated finite sum N ≤n<0 an εn is called its principal part. We
can define field operations on such expressions by mimicking the ordinary rules of
adding, subtracting, multiplying, and dividing Laurent series, without regard to issues
of convergence. The leading term of such an expression is the nonvanishing term an εn
for which n is as small as possible, and we call the expression positive or negative
according to the sign of the leading term. In this way we obtain an ordered field. This
field is denoted by R((ε)), and the field R(ε) discussed above may be identified with
a subfield of it. In this larger field, a sequence of Laurent series converges if and only
if the sequence of principal parts stabilizes (i.e., is eventually constant) and for every
integer n ≥ 0 the sequence of coefficients of εn stabilizes. In particular, 1, ε, ε 2 , ε3 , . . .
converges to 0 but 1, 12 , 41 , 18 , . . . does not. The same holds for R = R(ε); the sequence
1, 21 , 14 , 18 , . . . does not converge to 0 relative to R because every term differs from 0
by more than ε.
Then come the really large non-Archimedean ordered fields. There are non-
Archimedean ordered fields so large (that is, equipped with so many infinitesimal
elements) that the ordinary notion of convergence of sequences becomes trivial: All
convergent sequences are eventually constant. In such an ordered field, the only way to
“sneak up” on an element from above or below is with a generalized sequence whose
terms are indexed by some uncountable ordinal, rather than the countable ordinal ω.
Define the cofinality of an ordered field as the smallest possible cardinality of an un-
bounded subset of the field (so that, for instance, the real numbers, although uncount-

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able, have countable cofinality). The cardinality and cofinality of a non-Archimedean
ordered field can be as large as you like (or dislike!); along with Cantor’s hierarchies
of infinities comes an even more complicated hierarchy of non-Archimedean ordered
fields. The cofinality of an ordered field is easily shown to be a regular cardinal, where
a cardinal κ is called regular if and only if a set of cardinality κ cannot be written as
the union of fewer than κ sets, each of cardinality less than κ. In an ordered field R
of cofinality κ, the right notion of a sequence is a “κ-sequence”, defined as a function
from the ordinal κ (that is, from the set of all ordinals α < κ) to R. A sequence whose
length is less than the cofinality of R can converge only if it is eventually constant.
Curiously, if we use this generalized notion of a sequence, some of the large ordered
fields can be seen to have properties of generalized compactness reminiscent of the
real numbers. More specifically, for κ regular, say that an ordered field R of cofinality
κ satisfies the κ-Bolzano–Weierstrass Property if every bounded κ-sequence (xα )α<κ
in R has a convergent κ-subsequence. Then a theorem of Sikorski [24] states that,
for every uncountable regular cardinal κ, there is an ordered field of cardinality and
cofinality κ that satisfies the κ-Bolzano–Weierstrass Property. For more background
on non-Archimedean ordered fields (and generalizations of the Bolzano–Weierstrass
Property in particular), readers can consult [14] and/or [22].
Lastly, there is the Field of surreal numbers No, which contains all ordered fields
as subfields. Following Conway [5] we call it a Field rather than a field because its
elements form a proper class rather than a set. One distinguishing property of the
surreal numbers is the fact that, for any two sets of surreal numbers A, B such that
every element of A is less than every element of B, there exists a surreal number that
is greater than every element of A and less than every element of B. (This does not
apply if A and B are proper classes, as we can see by letting A consist of 0 and the
negative surreal numbers and B consist of the positive surreal numbers.)
See the Wikipedia page [29] for information on other ordered fields, such as the
Levi–Civita field and the field of hyper-real numbers.

4. SOME PROOFS. Here we give (sometimes abbreviated) versions of the proofs of

equivalence and inequivalence.

The Archimedean Property (2) does not imply the Dedekind Completeness Property
(1): The ordered field of rational numbers satisfies the former but not the latter. (Note,
however, that (1) does imply (2): N R is nonempty, so if N R were bounded above,
it would have a least upper bound c by (1). Then for every n ∈ N R we would have
n + 1 ≤ c (since n + 1 is in N R and c is an upper bound for N R ), implying n ≤ c − 1.
But then c − 1 would be an upper bound for N R , contradicting our choice of c as a
least upper bound for N R . This shows that N R is not bounded above, which is (2).)

The Cut Property (3) implies completeness (1): Given a nonempty set S ⊆ R that
is bounded above, let B be the set of upper bounds of S and A be its complement. A
and B satisfy the hypotheses of (3), so there exists a number c such that everything
less than c is in A and everything greater than c is in B. It is easy to check that c is a
least upper bound of S. (To show that c is an upper bound of S, suppose some s in S
exceeds c. Since (s + c)/2 exceeds c, it belongs to B, so by the definition of B it must
be an upper bound of S, which is impossible since s > (s + c)/2. To show that c is a
least upper bound of S, suppose that some a < c is an upper bound of S. But a (being
less than c) is in A, so it can’t be an upper bound of S.)

In view of the preceding result, the Cut Property is a completeness property, and to
prove that some other property is a completeness property, it suffices to show that it

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implies the Cut Property. Hereafter we will write “Property (n) implies completeness
by way of the Cut Property (3)” to mean “(n) ⇒ (3) ⇒ (1).” When a detour through
the Archimedean Property is required as a lemma to the proof of the Cut Property, we
will write “Property (n) implies completeness by way of the Archimedean Property
(2) and the Cut Property (3)” to give a road-map of the argument that follows.
Topological Connectedness (4) implies completeness by way of the Cut Property
(3): We prove the contrapositive. Let A and B be sets satisfying the hypotheses of the
Cut Property but violating its conclusion: There exists no c such that everything less
than c is in A and everything greater than c is in B. (That is, suppose A, B is a “bad
cut”, which we also call a gap.) Then, for every a in A, there exists a 0 in A with a 0 > a,
and for every b in B there exists b0 in B with b0 < b. From this it readily follows that
the sets A and B are open, so that Topological Connectedness must fail.
The Intermediate Value Property (5) implies completeness by way of the Cut Prop-
erty (3): We again prove the contrapositive. (Indeed, we will use this mode of proof
so often that henceforth we will omit the preceding prefatory sentence.) Let A, B be a
gap. The function that is −1 on A and 1 on B is continuous and violates the conclusion
of the Intermediate Value Property.
The Bounded Value Property (6) does not imply completeness: Counterexamples
are provided by the theorem of Sikorski referred to earlier (near the end of Section 3),
once we prove that the κ-Bolzano–Weierstrass Property implies the Bounded Value
Property. (I am indebted to Ali Enayat for suggesting this approach and for supplying
all the details that appear below.)
Suppose that κ is a regular cardinal and that R is an ordered field with cofinal-
ity κ satisfying the κ-Bolzano–Weierstrass Property. Then I claim that R satisfies the
Bounded Value Property. Choose an increasing unbounded sequence (xα : α ∈ κ) of
elements of R. Suppose that f is continuous on [a, b] but that there exists no B with
f (x) ≤ B for all x ∈ [a, b]. For each α ∈ κ, choose some tα ∈ [a, b] with f (tα ) > xα .
The tα ’s are bounded, so by the κ-Bolzano–Weierstrass Property there exists some
subset U of κ such that (tα : α ∈ U ) is a κ-subsequence that converges to some
c ∈ [a, b]. (Note that U must be unbounded; otherwise (tα : α ∈ U ) would be a β-
sequence for some β < κ rather than a κ-sequence.) By the continuity of f , the se-
quence ( f (tα ) : α ∈ U ) converges to f (c).
We now digress to prove a small lemma, namely, that every convergent κ-sequence
(rα ∈ R : α ∈ κ) is bounded. Since (rα : α ∈ κ) converges to some limit r , there exists
a β < κ such that, for all α ≥ β, rα lies in (r − 1, r + 1); then the tail-set {rα : α ≥ β}
is bounded. On the other hand, since κ is a regular cardinal, and since R has cofinality
κ, the complementary set {rα : α < β} is too small to be unbounded. Taking the union
of these two sets, we see that the set {rα ∈ R : α ∈ κ} is bounded.
Applying this lemma to the convergent sequence ( f (tα ) : α ∈ U ), we see that
( f (tα ) : α ∈ U ) is bounded. But this is impossible, since the set U is unbounded and
since our original increasing sequence (xα : α ∈ κ) was unbounded. This contradiction
shows that f ([a, b]) is bounded above. Hence, R has the Bounded Value Property, as
claimed.
(Note the resemblance between the preceding proof and the usual real-analysis
proof that every continuous real-valued function on an interval [a, b] is bounded.)
One detail omitted from the above argument is a proof that uncountable regular
cardinals exist (without which Sikorski’s theorem is vacuous). The axiom of countable
choice implies that a countable union of countable sets is countable, so ℵ1 , the first
uncountable cardinal, is a regular cardinal.

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 The Extreme Value Property (7) implies completeness by way of the Cut Property
(3): Suppose A, B is a gap, and for convenience assume that 1 ∈ A and 2 ∈ B (the
general case may be obtained from this special case by straightforward algebraic mod-
ifications). Define
(
x if x ∈ A,
f (x) =
0 if x ∈ B,

for x in [0, 3]. Then f is continuous on [0, 3] but there does not exist c ∈ [0, 3] with
f (x) ≤ f (c) for all x in [0, 3]. Such a c would have to be in A (since f takes positive
values on [0, 3] ∩ A, e.g., at x = 1, while f vanishes on [0, 3] ∩ B), but for any c ∈
[0, 3] ∩ A there exists c0 ∈ [0, 3] ∩ A with c0 > c, so that f (c0 ) > f (c).

The Mean Value Property (8) implies the Constant Value Property (9): Trivial.

The Constant Value Property (9) implies completeness by way of the Cut Property
(3): Suppose A, B is a gap. Again consider the function f that equals −1 on A and 1
on B. It has derivative 0 everywhere, yet it isn’t constant on [a, b] if we take a ∈ A
and b ∈ B.

The Convergence of Bounded Monotone Sequences (10) implies completeness by

way of the Archimedean Property (2) and the Cut Property (3): If R does not sat-
isfy the Archimedean Property, then there must exist an element of R that is greater
than every term of the sequence 1, 2, 3, . . . . By the Convergence of Bounded Mono-
tone Sequences, this sequence must converge, say to r . This implies that 0, 1, 2, . . .
also converges to r . Now subtract the two sequences; by the algebraic limit laws
that are easily derived from the ordered field axioms and the definition of limits,
we find that 1, 1, 1 . . . converges to 0, which is impossible. Therefore R must sat-
isfy the Archimedean Property. Now suppose that we are given a cut A, B. For n ≥ 0
in N, let an be the largest element of 2−n Z R in A, and bn be the smallest element of
2−n Z R in B. (an ) and (bn ) are bounded monotone sequence, so by the Convergence of
Bounded Monotone Sequences they converge, and since (by the Archimedean Prop-
erty) an − bn converges to 0, an and bn must converge to the same limit; call it c.
We have an ≤ c ≤ bn for all n, so |an − c| and |bn − c| are both at most 2−n . From the
Archimedean Property it follows that, for every  > 0, there exists n with 2−n < , and
for this n we have an ∈ A and bn ∈ B satisfying an > c −  and bn < c + . Hence,
for every  > 0, every number less than or equal to c −  is in A and every number
greater than or equal to c +  is in B. Therefore every number less than c is in A and
every number greater than c is in B, which verifies the Cut Property.

The Convergence of Cauchy Sequences (11) does not imply completeness: Every
Cauchy sequence in the field of formal Laurent series converges, but the field does
P satisfyn the Cut Property. Call a formal Laurent series finite if it is of the form
not
n≥0 an ε ; otherwise, call it positively infinite or negatively infinite according to the
sign of its leading term an (n < 0). If we let A be the set of all finite or negatively
infinite formal Laurent series, and we let B be the set of all positively infinite formal
Laurent series, then A, B is a gap. On the other hand, it is easy to show that this ordered
field satisfies property (11). Note also that if we define the norm of a formal Laurent
series n≥N an εn (with a N 6 = 0) as 2−N and define the distance between two series as
P
the norm of their difference, then we obtain a complete metric space, whose metric
topology coincides with the order topology introduced above.

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 The Fixed Point Property for Closed Bounded Intervals (12) implies completeness
by way of the Cut Property (3): Let A, B be a gap of R. Pick a in A and b in B, and
define f : [a, b] → R by putting f (x) = b for x ∈ A and f (x) = a for x ∈ B. Then
f is continuous but has no fixed point.
The Contraction Map Property (13) implies completeness by way of the Archi-
medean Property (2) and the Cut Property (3): Here is an adaptation of a solution
found by George Lowther [17]. First we will show that R is Archimedean. Suppose
not. Call x in R finite if −n < x < n for some n in N R , and infinite otherwise. Let
(
1
x if x is infinite,
f (x) = 2 1
x + 2 g(x) if x is finite,

with
x
g(x) = 1 − ,
1 + |x|
a decreasing function of x taking values in (0, 2). For all finite x, y with x > y, we
have (g(y) − g(x))/(x − y) ≥ 1/(1 + |x|)(1 + |y|) (indeed, the left-hand side mi-
nus the right-hand side equals 0 in the cases x > y ≥ 0 or 0 ≥ x > y and equals
−2x y/(1 + x)(1 − y)(x − y) > 0 in the case x > 0 > y), so for all x > y in [−a, a]
we have (g(y) − g(x))/(x − y) ≥ 1/(1 + a)2 , implying that |( f (x) − f (y))/(x −
y)| ≤ 1 − 12 /(1 + a)2 for all x, y in [−a, a]. Taking c = 1 − 12 /ω2 with ω > n for all
n in N R , we obtain | f (x) − f (y)| < c|x − y| for all finite x, y. This inequality can
also be shown to hold when one or both of x, y is infinite. Hence f is a contraction
map, yet it has no finite or infinite fixed points, a contradiction.
Now we want to prove that R satisfies the Cut Property. Suppose not. Let A, B be
a gap of R, and let an = max A ∩ 2−n Z R and bn = min B ∩ 2−n Z R . Since A, B is a
gap, neither of the sequences (an )∞ n=1 and (bn )n=1 can be eventually constant, so there
∞

exist n 1 < n 2 < n 3 < · · · such that the sequences (xk )∞k=1 and (yk )k=1 with x k = an k
∞

and yk = bnk are strictly monotone, with 0 < yk − xk ≤ 2 . By the Archimedean

−k

Property, every element of A lies in (−∞, x1 ] or in one of the intervals [xk , xk+1 ], and
every element of B lies in [y1 , ∞) or in one of the intervals [yk+1 , xk ]. Now consider
the continuous map h that has slope 12 on (−∞, x1 ] and on [y1 , ∞), sends xk to xk+1
and yk to yk+1 for all k, and is piecewise linear away from the points xk , yk ; it is well-
defined because these intervals cover R, and by looking at its behavior on each of those
intervals we can see that it has no fixed points. On the other hand, h is a contraction
map with contraction constant 12 . Contradiction.
The Alternating Series Test (14) does not imply completeness: In the field of formal
Laurent series, every series whose terms tend to zero (whether or not they alternate in
sign) is summable, so the Alternating Series Test holds even though the Cut Property
doesn’t.
The Absolute Convergence Property (15) does not imply completeness: The field
of formal Laurent series has the property that every absolutely convergent series is
convergent (and indeed the reverse is true as well!), but it does not satisfy the Cut
Property.
The Ratio Test (16) implies completeness by way of the Archimedean Property (2)
and the Cut Property (3): Note that the Ratio Test implies that 21 + 14 + 18 + · · · con-
verges, implying that R is Archimedean (the sequence of partial sums 21 , 43 , 78 , . . . isn’t

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even a Cauchy sequence if there exists an  > 0 that is less than 1/n for all n). Now we
make use of the important fact (which we have avoided making use of up till now, for
esthetic reasons, but which could be used to expedite some of the preceding proofs)
that every Archimedean ordered field is isomorphic to a subfield of the reals. (See
the next paragraph for a proof.) To show that a subfield of the reals that satisfies the
Ratio Test must contain every real number, it suffices to note that every real number
can be written as a sum n ± 12 ± 14 ± 18 ± · · · that satisfies the hypotheses of the Ratio
Test.

Every Archimedean ordered field is isomorphic to a subfield of the reals: For every
x in R, let Sx be the set of elements of Q whose counterparts in Q R are less than x,
and let φ(x) be the least upper bound of Sx . The Archimedean Property can be used
to show that φ is an injection, and with some work, we can verify that it is also a field
homomorphism. For more on completion of ordered fields, see [23].

The Shrinking Interval Property (17) does not imply completeness: The field of
formal Laurent series satisfies the former but not the latter. For details, see [7, pp.
212–215].

The Nested Interval Property (18) does not imply completeness: The surreal num-
bers are a counterexample. (Note, however, that the field of formal Laurent series is
not a counterexample; although it satisfies the Shrinking Interval Property, it does not
satisfy the Nested Interval Property, since, for instance, the nested closed intervals
[n, ω/n] have empty intersection. This shows that, as an ordered field property, (18)
is strictly stronger than (17).) To verify that the surreal numbers satisfy (18), con-
sider a sequence of nested intervals [a1 , b1 ] ⊇ [a2 , b2 ] ⊇ · · · If ai = bi for some i,
say ai = bi = c, then a j = b j = c for all j > i, and c lies in all the intervals. If
ai < bi for every i, then ai ≤ amax{i, j} < bmax{i, j} ≤ b j for all i, j, so every element of
A = {a1 , a2 , . . .} is less than every element of B = {b1 , b2 , . . .}. Hence there exists a
surreal number that is greater than every element of A and less than every element of
B, and this surreal number lies in all the intervals [ai , bi ]. Thus No satisfies the Nested
Interval Property but, being non-Archimedean, does not satisfy completeness.
If we dislike this counterexample because the surreal numbers are a Field rather than
a field, we can instead use the field of surreal numbers that “are created before Day ω1 ”,
where ω1 is the first uncountable ordinal. See [5] for a discussion of the “birthdays” of
surreal numbers. For a self-contained explanation of a related counterexample that pre-
dates Conway’s theory of surreal numbers, see [6]. For a counterexample arising from
non-standard analysis, see the discussion of the Cantor completeness of Robinson’s
valuation field ρ R in [11] and [12].
Because these counterexamples are abstruse, we can find in the literature and on the
web assertions like “The Nested Interval Property implies the Bolzano–Weierstrass
Theorem and vice versa”. It’s easy for students to appeal to the Archimedean Prop-
erty without realizing they are doing so, especially because concrete examples of non-
Archimedean ordered fields are unfamiliar to them.

To summarize: Properties (1), (3), (4), (5), (7), (8), (9), (10), (12), (13), and (16)
imply completeness, while properties (2), (6), (11), (14), (15), (17), and (18) don’t.
The ordered field of formal Laurent series witnesses the fact that (11), (14), (15), and
(17) don’t imply completeness; some much bigger ordered fields witness the fact that
(6) and (18) don’t imply completeness; and every non-Archimedean ordered field wit-
nesses the fact that (2) doesn’t imply completeness.

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 One of the referees asked which of the properties (6), (11), (14), (15), (17), and (18)
imply completeness in the presence of the Archimedean Property (2). The answer is,
all of them. It is easy to show this in the case of properties (11), (14), (15), (17), and
(18), using the fact that every Archimedean ordered field is isomorphic to a subfield of
the reals (see the discussion of property (16) in section 4). The case of (6) is slightly
more challenging.

Claim. Every Archimedean ordered field with the Bounded Value Property is Dede-
kind complete.

Proof. Suppose not; let R be a counterexample, and let A, B be a bad cut of R. Let
an = max A ∩ 2−n Z R and bn = min B ∩ 2−n Z R , so that |an − bn | = 2−n . Since A, B
is a bad cut, neither of the sequences (an )∞n=1 and (bn )n=1 can be eventually constant.
∞

Let f n (x) be the continuous function that is 0 on (−∞, an −2−n ], 1 on [an , bn ], 0 on

[bn +2−n , ∞), and piecewise linear on [an −2−n , an ] and [bn , bn +2−n ]. The interval
[an − 2−n , bn + 2−n ] has length ≤ 3 · 2−n , which goes to 0 in R as n → ∞ since
R is Archimedean. Any c belonging to all the intervals [an − 2−n , bn + 2−n ] would
be a cutpoint for A, B, and since the cut A, B has been assumed to have no cutpoint,
∩∞ n=1 [an −2 , bn +2 ] is empty. It follows that, for every
−n −n
P x in R, only finitely many of
the intervals [an −2 , bn +2−n ] contain x, so f (x) = ∞
−n
n=1 f n (x) is well-defined for
all x (since all but finitely many of the summands vanish). Furthermore, the function
f (x) is continuous, since a finite sum of continuous functions is continuous, and since
for every x we can find an m and an  > 0 such that f n (y) = 0 for all P n > m and
all y with |x − y| <  (so that f agrees with the continuous function n≤m f n on a
neighborhood of x). Finally, note that f is unbounded, since, e.g., f (x) ≥ n for all x
in [an , bn ].
Alternatively, we can argue as follows: The Archimedean Property implies count-
able cofinality (specifically, N R is a countable unbounded set), and an argument of
Teismann [26] shows that every ordered field with countable cofinality that satisfies
property (6) is complete.
It is worth noting that for all 18 of the propositions listed in section 2, the answer to
the question “Does it imply the Dedekind Completeness Property?” (1) is the same as
the answer to the question “Does it imply the Archimedean Property?” (2). A priori, we
might have imagined that one or more of properties (3) through (18) would be strong
enough to imply the Archimedean Property yet, not so strong as to be a completeness
property for the reals.

This is not the end of the story of real analysis in reverse; there are other theorems in
analysis with which we could play the same game. Indeed, some readers may already
be wondering “What about the Fundamental Theorem of Calculus?” Actually, the FTC
is really two theorems, not one (sometimes called FTC I and FTC II in textbooks).
They are not treated here because this essay is already on the long side for a M ONTHLY
article, and a digression into the theory of the Riemann integral would require a whole
section in itself. Indeed, there are different ways of defining the Riemann integral
(Darboux’s and Riemann’s come immediately to mind), and while they are equivalent
in the case of the real numbers, it is possible that different definitions of the Riemann
integral that are equivalent over the reals might turn out to be different over ordered
fields in general; thus we might obtain different varieties of FTC I and FTC II, some
of which would be completeness properties and others of which would not. It seemed
best to leave this topic for others to explore.

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 An additional completeness axiom for the reals is the “principle of real induc-
tion” [4].

5. SOME ODDS AND ENDS.

History and terminology. It’s unfortunate that the word completeness is used to sig-
nify both Cauchy completeness and Dedekind completeness; no doubt this duality of
meaning has contributed to the misimpression that the two are equivalent in the pres-
ence of the ordered field axioms. It’s therefore tempting to sidestep the ambiguity of
the word “complete” by resurrecting Dedekind’s own terminology (“Stetigkeit”) and
referring to the completeness property of the reals as the continuity property of the
reals—where here we are to understand the adjective “continuous” not in its usual
sense, as a description of a certain kind of function, but rather as a description of a
certain kind of set, namely, the kind of set that deserves to be called a continuum.
However, it seems a bit late in the day to try to get people to change their terminology.
It’s worth pausing here to explain what Hilbert had in mind when he referred to the
real numbers as the “complete Archimedean ordered field”. What he meant by this is
that the real numbers can be characterized by a property referred to earlier in this article
(after the discussion of property (16) in section 4): Every Archimedean ordered field
is isomorphic to a subfield of the real numbers. That is, every Archimedean ordered
field can be embedded in an Archimedean ordered field that is isomorphic to the reals,
and no further extension to a larger ordered field is possible without sacrificing the
Archimedean Property. Hilbert was saying that the real number system is the (up to
isomorphism) unique Archimedean ordered field that is not a proper subfield of a larger
Archimedean ordered field; vis-a-vis the ordered field axioms and the Archimedean
Property, R is complete in the sense that nothing can be added to it. In particular,
Hilbert was not asserting any properties of R as a metric space.
Readers interested in the original essays of Dedekind and Hilbert may wish to read
Dedekind’s “Continuity and irrational numbers” and Hilbert’s “On the concept of num-
ber”, both of which can be found in [9].

Advantages and disadvantages of the Cut Property. The symmetry and simplicity
of the Cut Property have already been mentioned. Another advantage is shallowness.
Although the word has a pejorative sound, shallowness in the logical sense can be a
good thing; a proposition with too many levels of quantifiers in it is hard for the mind to
grasp. The proposition “c is an upper bound of S” (i.e., “for all s ∈ S, s ≤ c”) involves
a universal quantifier, so the proposition “c is the least upper bound of S” involves two
levels of quantifiers, and the proposition that, for every nonempty bounded set S of
reals, there exists a real number c such that c is a least upper bound of S, therefore
involves four levels of quantifiers.
In contrast, the assertion that A, B is a cut of R involves one level of quantifiers,
and the assertion that c is a cutpoint of A, B involves two levels of quantifiers, so
the assertion that every cut of R determines a cutpoint involves only three levels of
quantifiers.
Note also that the objects with which the Dedekind Completeness Property is
concerned—arbitrary nonempty bounded subsets of the reals—are hard to picture,
whereas the objects with which the Cut Property is concerned—ways of dividing the
number line into a left-set and a right-set—are easy to picture. Indeed, in the context
of foundations of geometry, it is widely acknowledged that some version of the Cut
Property is the right way to capture what Dedekind called the continuity property of
the line.

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 It should also be mentioned here that the Cut Property can be viewed as a special
case of the Least Upper Bound Property, where the set S has a very special structure.
This makes the former more suitable for doing naive reverse mathematics (since a
weaker property is easier to verify) but also makes it slightly less convenient for doing
forward mathematics (since a weaker property is harder to use). If we start to rewrite
a real analysis textbook, replacing every appeal to the Least Upper Bound Property by
an appeal to the Cut Property, we quickly see that we end up mimicking the textbook
proofs but with extra, routinized steps (“. . . and let B be the complement of that set”)
that take up extra space on the page and add no extra insight. So even if we want to
assign primacy to the Cut Property, we would not want to throw away the Least Upper
Bound Property; we would introduce it as a valuable consequence of the Cut Property.
Lastly, we mention a variant of the Cut Property, Tarski’s Axiom 3 [30], that drops
the hypothesis that the union of the two sets is the whole ordered field R. This stronger
version of the axiom is equivalent to the one presented above. Like the Least Up-
per Bound Property, Tarski’s version of the Cut Property is superior for the purpose
of constructing the theory of the reals but less handy for the purpose of “decon-
structing” it.

Implications for pedagogy. As every thoughtful teacher knows, logical equivalence

is not the same as pedagogical equivalence. Which completeness property of the reals
should we teach to our various student audiences, assuming we teach one at all?
Here the author drops the authorial “we” (appropriate for statements of a mathemat-
ical and historical nature that are, as far as the author has been able to assess, accurate)
and adopts an authorial “I” more appropriate to statements of opinion.
I think the reader already knows that I am quite taken with the Cut Property as an
axiom for the reals, and will not be surprised to hear that I would like to see more
teachers of calculus, and all teachers of real analysis, adopt it as part of the explicit
foundation of the subject. What may come as a bigger surprise is that I see advantages
to a different completeness axiom that has not been mentioned earlier in the article,
largely because I have not seen it stated in any textbook (although Burns [3] does
something similar, as I describe below).
When we write .3333 . . . , what we mean (or at least one thing we mean) is “The
number that lies between .3 and .4, and lies between .33 and .34, and lies between .333
and .334, etc.” That is, a decimal expansion is an “address” of a point of the number
line. Implicit in the notation is the assumption that for every decimal expansion, such
a number exists and is unique. These assumptions of existence and uniqueness are part
of the mathematical undermind (the mathematical subconscious, if you prefer) of the
typical high schooler. After all, it never occurs to a typical high school student whether
there might be more than one number 0.5, or whether there might be no such number
at all (though balking at fractions is common for thoughtful students at an earlier age);
so it’s tempting to carry over the assumption of existence and uniqueness when the
teacher makes the transition from finite decimal to infinite decimals.
Part of what an honors calculus teacher should do is undermine the mathematical
undermind, and convince the students that they’ve been uncritically accepting pre-
cepts that have not yet been fully justified. The flip side of von Neumann’s adage, “In
mathematics you don’t understand things; you just get used to them”, is that once you
get used to something, you may mistakenly come to believe you understand it! Infi-
nite decimals can come to seem intuitive, on the strength of their analogy with finite
decimals, and the usefulness of infinite decimals makes us reluctant to question the
assumptions on which they are based. But mathematics is a liberal art, and that means
we should bring difficulties into the light and either solve them honestly or duck them

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honestly. And the way a mathematician ducks a problem honestly is to formulate the
problematic assumption as precisely and narrowly as possible and call it an axiom.
Specifically, I would argue that one very pedagogically appropriate axiom for the
completeness of the reals is one that our students have been implicitly relying on for
years: The Strong Nested Decimal Interval Property, which asserts that for all infinite
strings d0 , d1 , d2 , . . . of digits between 0 and 9, there exists a unique real number in
the intervals [.d0 , .d0 + .1], [.d0 d1 , .d0 d1 + .01], [.d0 d1 d2 , .d0 d1 d2 + .001], etc. (I call
it “Strong” because, unlike the ordinary Nested Interval Property, it asserts unique-
ness as well as existence.) The reader who has made it through the article thus far
should have no trouble verifying that this is indeed a completeness property of the
reals, and we can use it to give expeditious proofs of some of the important theo-
rems of the calculus, at least in special cases. (Example: To show that the Intermediate
Value Theorem holds for weakly increasing functions, home in on the place where
the function vanishes by considering decimal approximations from both sides.) This
choice of axiom does not affect the fact that the main theorems of the calculus have
proofs that are hard to understand for someone who is taking calculus for the first
or even the second time and who does not have much practice in reading proofs; in-
deed, I would say that the art of reading proofs goes hand-in-hand with the art of
writing them, and very few calculus students understand the forces at work and the
constraints that a mathematician labors under when devising a proof. But if we ac-
knowledge early in the course that the Strong Nested Decimal Interval Property (or
something like it) is an assumption that our theorems rely upon, and stress that it can-
not be proved by mere algebra, we will be giving our students a truer picture of the
subject.
Furthermore, the students will encounter infinite decimals near the end of the two-
semester course when infinite series are considered; now an expression like .3333 . . .
means 3 × 10−1 + 3 × 10−2 + 3 × 10−3 + 3 × 10−4 + · · · . (Burns [3] adopts as his
completeness axioms the Archimedean Property plus the assertion that every decimal
converges.) The double meaning of infinite decimals hides a nontrivial theorem: Every
infinite decimal, construed as an infinite series, converges to a limit, specifically, the
unique number that lies in all the associated nested decimal intervals. We do not need to
prove this assertion to give our students the knowledge that this assertion has nontrivial
content; we can lead them to see that the calculus gives them, for the first time, an
honest way of seeing why .9999 . . . is the same number as 1.0000 . . . (a fact that they
may have learned to parrot but probably don’t feel entirely comfortable with).
Students should also be led to see that the question “But how do we know that the
square root of two really exists, if we can’t write down all its digits or give a pattern for
them?” is a fairly intelligent question. In what sense do we know that such a number
exists? We can construct the square root of two as the length of the diagonal of a square
of side-length one, but that trick won’t work if we change the question to “How do we
know that the cube root of two really exists?”
To those who would be inclined to show students a construction of the real num-
bers (via Dedekind cuts and Cauchy sequences), I would argue that a student’s first
exposure to rigorous calculus should focus on other things. It takes a good deal of
mathematical sophistication to even appreciate why someone would want to prove
that the theory of the real numbers is consistent, and even more sophistication to ap-
preciate why we can do so by making a “model” of the theory. Most students enter
our classrooms with two workable models of the real numbers, one geometrical (the
number line) and one algebraic (the set of infinite decimals). Instead of giving them a
third picture of the reals, it seems better to clarify the pictures that they already have,
and to assert the link between them.

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 In fact, I think that for pedagogical purposes, it’s best to present both the Cut Prop-
erty and the Strong Nested Decimal Interval Property, reflecting the two main ways
students think about real numbers. And it’s also a good idea to mention that, despite
their very different appearances, the two axioms are deducible from one another, even
though neither is derivable from the principles of high school mathematics. This will
give the students a foretaste of a refreshing phenomenon that they will encounter over
and over if they continue their mathematical education: Two mathematical journeys
that take off in quite different directions can unexpectedly lead to the same place.

ACKNOWLEDGMENTS. Thanks to Matt Baker, Mark Bennet, Robin Chapman, Pete Clark, Ricky De-
mer, Ali Enayat, James Hall, Lionel Levine, George Lowther, David Speyer, and other contributors to
MathOverflow (http://www.mathoverflow.net) for helpful comments; thanks to Wilfried Sieg for
his historical insights; thanks to the referees for their numerous suggestions of ways to make this article better;
and special thanks to John Conway for helpful conversations. Thanks also to my honors calculus students,
2006–2012, whose diligence and intellectual curiosity led me to become interested in the foundations of the
subject.

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JAMES G. PROPP received his A.B. from Harvard College in 1982, where he took both honors freshman
calculus and real analysis with Andrew Gleason. He received his Ph.D. from the University of California–
Berkeley in 1987. His research is mostly in the areas of combinatorics and probability, and he currently teaches
at the University of Massachusetts–Lowell.
Department of Mathematics, University of Massachusetts–Lowell, Lowell, MA 01854
Website: http://jamespropp.org

Famous American Mathematical Monthly Authors

N R M E A D R Y S I E G E L H M N T C H O R
O K N U T H S H A P L E Y E N T I H W G A R
M Y T E M E H T L R E G R E B N E T S R U F
I M H L R F T E G G E D L O N R A L R I E E
S C D A L R O H U N F L I I N O S O O F T R
M N O Y O E E R O I U O R J E R W D F F N D
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Erdos, Fefferman, Floyd, Frisch, Furstenberg, Griffiths, Hamming, Hartmanis, Hoare,
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—Submitted by Vadim Ponomarenko, San Diego State University

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