A Defense of Platonic Realism in Mathematics

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FLORIDA STATE UNIVERSITY

COLLEGE OF ARTS AND SCIENCES

A DEFENSE OF PLATONIC REALISM IN MATHEMATICS:

PROBLEMS ABOUT THE AXIOM OF CHOICE

By

WATARU ASANUMA

A Dissertation submitted to theDepartment of Philosophy

in partial fulfillment of the

requirements for the degree ofDoctor of Philosophy

Degree Awarded:

Spring Semester, 2009


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The members of the Committee approve the Dissertation of Wataru Asanuma defended on April

7, 2009.

__________________________

Russell M. DancyProfessor Directing Dissertation

__________________________Philip L. BowersOutside Committee Member

__________________________J. Piers RawlingCommittee Member

__________________________Joshua GertCommittee Member

Approved:

_________________________________________________________________

J. Piers Rawling, Chair, Philosophy

_________________________________________________________________Joseph Travis, Dean, College of Arts and Sciences

The Graduate School has verified and approved the above named committee members.
http://www.math.fsu.edu/People/faculty.php?id=2http://www.fsu.edu/~philo/new%20site/staff/gert.htmhttp://www.fsu.edu/~philo/new%20site/staff/gert.htmhttp://www.math.fsu.edu/People/faculty.php?id=2


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ACKNOWLEDGMENTS

First of all, I would like to thank all my dissertation committee members, Dr. Russell Dancy,

Dr. Piers Rawling, Dr. Joshua Gert and Dr. Philip Bowers. Special thanks go to my major

professor, Dr. Dancy, who has guided my dissertation every step of the way. The courses

they offered, especially Dr. Dancys Platos Unwritten Doctrines, Dr. Rawlings

Modern Logic I& II and Dr. Gerts Philosophy of Mathematics, formed the backbone

of my dissertation. I would also like to thank Dr. Bowers (Department of Mathematics)

for serving as an outside committee member. It was exceptionally fortunate that I had an

opportunity to present the parts of my dissertation at the 2008 Logic, Mathematics and

Physics Graduate Philosophy Conference at the University of Western Ontario. The

conference has marked a major turning point for me, and continues to inspire and motivate

further research. Especially Im very grateful to Dr. John Bell and Dr. William Harper for

their kind and insightful words on my presentation. In addition, I had opportunities to

present the parts of my dissertation at the 7thHawaii International Conference on Arts and

Humanities and at the 12thNortheast Florida Student Philosophy Conference. I would like

to express my appreciation to the great audience for their attention.


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TABLE OF CONTENTS

List of Figures .................................................................................................................... viAbstract ............................................................................................................................ viii

INTRODUCTION ........................................................................................................... 1

1. WHAT IS THE AXIOM OF CHOICE? ...................................................................... 4

1.1 The Axioms of ZF Set Theory .......................................................................... 41.2 The Axiom of Choice and its Equivalents ........................................................ 51.3 The Consequences of the Axiom of Choice ...................................................... 81.4 A Weaker Form of the Axiom of Choice .......................................................... 111.5 The Axiom of Choice and the Continuum Hypothesis ..................................... 131.6 Fictionalism or Instrumentalism ....................................................................... 15

2. LEBESGUES THEORY OF MEASURE .................................................................... 20

2.1 Lebesgues Theory of Integration..................................................................... 202.2 Measurable Cardinals ........................................................................................ 31

3. THE BANACH-TARSKI PARADOX ........................................................................ 39

3.1 Preliminaries ..................................................................................................... 403.2 Non-Lebesgue Measurable Sets ........................................................................ 453.3 The Hausdorff Paradox ................................................................................... 473.4 The Banach-Tarski Paradox .............................................................................. 553.5 What is a Paradox? ............................................................................................ 583.6 What cannot Happen? ....................................................................................... 633.7 A Paradox without the Axiom of Choice? ......................................................... 643.8 Some Philosophical Implications ...................................................................... 66

4. GDELS INCOMPLETENESS THEOREMS .......................................................... 69

4.1 Gdels First Incompleteness Theorem............................................................. 694.2 Turing Machines ............................................................................................... 734.3 Recursive Functions and Recursive Sets .......................................................... 784.4 The Halting Problem ......................................................................................... 824.5 The Undecidability of First-order Logic and Arithmetic .................................. 854.6 Undecidability and Incompleteness .................................................................. 874.7 Gdels Second Incompleteness Theorem ........................................................ 904.8 Relative Consistency Proofs ............................................................................. 91


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5. MODEL-THEORETIC ARGUMENTS ...................................................................... 95

5.1 What is a Model? .............................................................................................. 965.2 The Lwenheim-Skolem Theorem and the Skolem Paradox ........................... 97

5.3 Quines Thesis of the Indeterminacy of Translation......................................... 1035.4 Putnams Model-Theoretic Arguments against Metaphysical Realism ............ 1075.5 The Lessons from the Model-Theoretic Arguments ......................................... 111

6. WHAT IS MATHEMATICAL EXISTENCE? .................................................................... 116

6.1 Benacerrafs Challenges to Platonism .............................................................. 1166.2 Some Applications of the Axiom of Choice ..................................................... 1196.3 The Axiom of Determinacy............................................................................... 1206.4 The Axiom of Constructibility .......................................................................... 1216.5 Anselms Argument of the Existence of God.................................................... 1246.6 Locke on Essences ............................................................................................ 1266.7 Essence and Existence in Mathematics ............................................................. 1306.8 What is a Maximally Consistent Theory? ......................................................... 131

CONCLUSION ............................................................................................................... 136

APPENDIX ....................................................................................................................... 138

BIBLIOGRAPHY ............................................................................................................. 140

BIOGRAPHICAL SKETCH ............................................................................................. 148


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LIST OF FIGURES

Figure 1: The Axiom of Choice ......................................................................................... 6

Figure 2: Riemann Integral vs. Lebesgue Integral ............................................................ 28

Figure 3: Example of a partially ordered set ..................................................................... 40

Figure 4: Partial order by inclusion of the power set of a set S{0, 1, 2} ....................... 43

Figure 5: The use of the Axiom of Choice in the proof of Zorns Lemma ........................ 44

Figure 6: Example of a non-Lebesgue measurable set ...................................................... 47

Figure 7: G-paradoxical ..................................................................................................... 48

Figure 8: G-equidecomposable .......................................................................................... 48

Figure 9: The Hausdorff Paradox ...................................................................................... 50

Figure 10: The existence of a set Tcontaining exactly one element from each G-orbit ... 52

Figure 11: Hausdorffs paradoxical decomposition........................................................... 53

Figure 12: The Weak Form of the Banach-Tarski Paradox(Two Spheres from One Version) ...................................................................................... 57

Figure 13: The Weak Form of the Banach-Tarski Paradox(The Pea and the Sun Version) .......................................................................................... 57

Figure 14: Example of a computer program to decide whetherx is an even .................... 76

Figure 15: Example of a computer program to decide whetherx is an odd ...................... 77

Figure 16: Example of a computer program to computexy.......................................... 78

Figure 17: Recursively enumerable (r.e.) .......................................................................... 80

Figure 18: Decidable (recursive) ....................................................................................... 81

Figure 19: Undecidable ..................................................................................................... 81

Figure 20: Example of a recursive set ............................................................................... 82


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Figure 21: Example of a recursively enumerable but not recursive set............................. 82

Figure 22: The analogy between the Halting Problem and Cantors Theorem ................. 84

Figure 23: T(0) is undecidable......................................................................................... 87

Figure 24: Approach to Gdels First Incompleteness Theorem from the theory ofcomputability ..................................................................................................................... 89

Figure 25: The Skolem Paradox ........................................................................................ 101

Figure 26: The comparison of the indeterminacy of translation and the underdeterminationof scientific theory ............................................................................................................. 104

Figure 27: Example of actual practice of mathematics ..................................................... 124

Figure 28: Example of the interrelationships among the models ...................................... 132


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ABSTRACT

The conflict between Platonic realism and Constructivism marks a watershed in philosophy

of mathematics. Among other things, the controversy over the Axiom of Choice is typical

of the conflict. Platonists accept the Axiom of Choice, which allows a set consisting of

the members resulting from infinitely many arbitrary choices, while Constructivists reject

the Axiom of Choice and confine themselves to sets consisting of effectively specifiable

members. Indeed there are seemingly unpleasant consequences of the Axiom of Choice.

The non-constructive nature of the Axiom of Choice leads to the existence of non-Lebesgue

measurable sets, which in turn yields the Banach-Tarski Paradox. But the Banach-Tarski

Paradox is so called in the sense that it is a counter-intuitive theorem. To corroborate my

view that mathematical truths are of non-constructive nature, I shall draw upon Gdels

Incompleteness Theorems. This also shows the limitations inherent in formal methods.

Indeed the Lwenheim-Skolem Theorem and the Skolem Paradox seem to pose a threat to

Platonists. In this light, Quine/Putnams arguments come to take on a clear meaning.

According to the model-theoretic arguments, the Axiom of Choice depends for its

truth-value upon the model in which it is placed. In my view, however, this is another

limitation inherent in formal methods, not a defect for Platonists. To see this, we shall

examine how mathematical models have been developed in the actual practice of

mathematics. I argue that most mathematicians accept the Axiom of Choice because the

existence of non-Lebesgue measurable sets and the Well-Ordering of reals open the

possibility of more fruitful mathematics. Finally, after responding to Benacerrafs

challenge to Platonism, I conclude that in mathematics, as distinct from natural sciences,

there is a close connection between essence and existence. Actual mathematical theories

are the parts of the maximally logically consistent theory that describes mathematical

reality.


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INTRODUCTION

A fundamental problem of philosophy of mathematics boils down to the conflict between

Platonic realism and Constructivism, and the conflict between them marks a watershed in

philosophy of mathematics. By Platonic realism I mean the philosophical view that

posits mathematical entities, such as numbers, sets, functions and so on, as

super-spatio-temporal ones. Indeed, owing to this view, mathematical knowledge was

extended further and further. At the turn of the last century, however, a variety of

paradoxes, such as Russells Paradox, were discovered by mathematicians and logicians in

the wake of the attempts to base the whole of mathematics on set theory, and gave rise to

the so-called crisis in the foundations of mathematics.

Against Platonic realism, there arose an anti-realistic doctrine called

Constructivism. Constructivism avoids positing mathematical entities dogmatically and

restricts them to those that are legitimately constructible in space and time.1 But this view

conceals in itself the danger that we have to pay a high price: the sacrifice of many

productive results of classical mathematics. This is the reason why philosophers of

mathematics take pains to seek some middle ground between the two extreme camps. At

this point the problem of how to deal with the Law of Excluded Middle, impredicative

definition, the Axiom of Choice, actual or potential infinity and so on becomes a

controversial issue.

Among other things, the controversy over the Axiom of Choice is typical of the

conflict between Platonic realism and Constructivism. Not only is the Axiom of Choice

the most interesting axiom in axiomatic set theory, but it also plays an important role in

many other areas of mathematics. So the problem of the Axiom of Choice is one of the

significant topics in philosophy of mathematics.

First of all, we shall see what the Axiom of Choice is and where the problem with the

Axiom lies. Especially, we shall focus on what we can do in the presence of the Axiom of

1 I will use the word Constructivism in a broader sense than Browers Constructivism. In Browers

Constructivism mathematical entities are constructible in our mind. But I will use the word

Constructivism in a narrower sense than Gdels axiom of constructibility. Gdels Axiom of

Constructibility is a much stronger assumption than Constructivism as I call it.


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Choice that we couldntotherwise. Platonists accept the Axiom of Choice, which allows a

set consisting of the members resulting from infinitely many arbitrary choices, while

Constructivists reject the Axiom of Choice and confine themselves to sets consisting of

effectively specifiable members (Chapter 1).

Lebesgues theory of measure will set the stage for discussing the Banach-Tarski

Paradox and the existence of measurable cardinals in later chapters. Also, since Lebesgue

is one of the French Constructivists, it is interesting to see the non-constructive nature of

Lebesgue measure creates an irreconcilable tension with Lebesgues skeptical attitude

toward the Axiom of Choice (Chapter 2).

The Hausdorff Paradox is the prototype of the Banach-Tarski Paradox. Informally,

the Hausdorff Paradox states that a sphere is decomposed into finite number of pieces and

reassembled by rigid motions to form two copies of almost the same size as the original.

Here almost means except on a countable subset. Banach and Tarski made

improvement on the Hausdorff Paradox by eliminating the need to exclude a countable

subset from a sphere. Informally, the Banach-Tarski Paradox states that a sphere is

decomposed into finite number of pieces and reassembled by rigid motions to form two

copies of exactly the same size as the original. The Banach-Tarski Paradox deepened the

skepticism about the Axiom of Choice. But the Banach-Tarski Paradox is so called in the

sense that it is a counter-intuitive theorem, as distinct from a logical contradiction or a

fallacious reasoning. I argue that we should accept the Banach-Tarski Paradox as a

Platonic truth and rejects epistemology based on a mathematical intuition (Chapter 3).

Next, from a slightly different perspective, I corroborate my view that mathematical

truths are of non-constructive nature. Once we got the undecidability of Peano Arithmetic

(PA), GdelsFirst Incompleteness Theorem is immediate. The set of true sentences in PA

is not recursively enumerable. But the set of theorems (provable sentences) in PA is

recursively enumerable. So it is easy to see that there is a sentence that is true but

unprovable. This implies that there are some arithmetical truths we cannot get access to in

an effective way. We also have to note Gdels Incompleteness Theorems show that there

are limitations inherent in formal methods (Chapter 4).

The Lwenheim-Skolem Theorem and the Skolem Paradox seem to pose a threat to


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Platonists. In the light of the Lwenheim-Skolem results, both Quines thesis of the

indeterminacy of translation and Putnams model-theoretic arguments against metaphysical

realism come to take on a clear meaning. According to the model-theoretic arguments, the

Axiom of Choice depends for its truth-value upon the model in which it is placed. In my

view, however, this is another limitation inherent in formal methods, not a defect for

Platonists (Chapter 5).

Finally, I meet Benacerrafs epistemological and ontological challenges to Platonism

by examining how mathematical models have been developed in the actual practice of

mathematics. Most mathematicians prefer the Axiom of Choice to the Axiom of

Determinacy in favor of the existence of non-Lebesgue measurable sets and the

Well-Ordering of reals. Also, most mathematicians reject the Axiom of Constructibility in

favor of the existence of a measurable cardinal. In both cases, working mathematicians

are driven by Platonic realism rather than Constructivism. I conclude that in mathematics,

as distinct from natural sciences, there is a close connection between essence and existence,

actuality and possibility. The actual mathematical theories are the parts of the maximally

logically consistent theory that describes mathematical reality (Chapter 6).2

2 I shall give some credit to the sources from which I got mathematical technicalities. Throughout the

process of writing the dissertation, I referred to Cameron (1998), Hamilton (1988), Jech (1978), Kunen (1980),

Levy (1979). They offer a panoramic view of set theory overall. The former two are concise but useful

introductions, whereas the latter three provide detailed and exhaustive information. For Lebesgues theory

of measure, Hawkins (1975) is a good help to know the historical background. We have seen how the

Lebesgue integral overcomes the difficulties of the Riemann integral. For this, see e.g. Weir (1973), Wilcoxand Myers (1978). For the Banach-Tarski Paradox, one can find technical details in Wagon (1985).

Wapner (2005) gives a more informal presentation of the Banach-Tarski Paradox. When discussing Gdels

First Incompleteness Theorem, I put focus on the approach from the theory of computability. For this

approach, Boolos and Jeffery (1974) is a classic although a wholesale revision has been made in the 4 th

edition of the same title (2002). Also, I consulted Cohen (1987), Cutland (1980), Ebbinghaus, Flum and

Thomas (1994). Franzn (2005) warns against a prevalent misconception of Gdels First Incompleteness

Theorem and a conflation of distinct senses of completeness and undecidability. Manin (1977),

Mendelson (1997) are good guides for the Skolem Paradox.


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CHAPTER 1

WHAT IS THE AXIOM OF CHOICE?

Introduction

In this chapter, first of all, I recapitulate the Axioms of Zermelo-Fraenkel (ZF) set theory

(Section 1). Then, I state the Axiom of Choice and give a couple of its equivalents: the

Well-Ordering Theorem and the Multiplicative Axiom (Section 2). Next, I shall show that

the Axiom of Choice has some useful consequences, e.g., the Aleph Theorem. At the

same time, we shall see that there were many opponents of the Axiom of Choice, and that it

has some unpleasant consequences as well (Section 3). Also, I shall discuss a weaker

form of the Axiom of Choice: the Denumerable Axiom of Choice, and some of its

consequences (Section 4). Moreover, I shall examine the relation of the Axiom of Choice

and the Continuum Hypothesis (Section 5). Finally, I provisionally conclude that the

debate over the Axiom of Choice favors Platonic realism.

1.1 The Axioms of ZF Set Theory

Before I state the Axiom of Choice, I shall see what constitutes the Axioms of ZF set theory.

In 1930 Zermelo proposed ZF set theory in a form closely related to that used today, which

consisted of the following seven Axioms.

(i) Axiom of Extensionality: If the two sets x,yhave the exactly same members, then they

are equal.

(ii) Power Set Axiom: For any setx, the power set ofxis a set.

Here the power set is the set of all subsets ofx.

(iii) Axiom of Union: For any setx, the union ofxis a set.

The union, denoted by , is the set of all members of the members of a setx.

(iv) Axiom of Pairing: For any setsx,y, {x,y} is a set.

(v) Axiom of Separation: If a propositional function P(x) is definite for a setz, there is a set

y containing exactly the members ofzfor whichP(x) holds.

The Axiom allows us to separate the members with some property from a set and form a set

consisting of these members.

(vi) Axiom of Replacement: IfFis a function, then for every setx,F[x] is a set.


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F[x] is called the image ofxunder the mappingF.

(vii) Axiom of Foundation: Ifx0then there existsyxsuch thatyx0.

This means that there is no infinite descending -sequence.

Also, there were two Axioms that were not included in this system but had occurred

in his system of 1908: the Axiom of Infinity and the Axiom of Choice. Since I shall

discuss the Axiom of Choice in detail below, I shall mention just the Axiom of Infinity here.

Axiom of Infinity: There exists a setxsuch that 0x and wheneveryxtheny{y}x.

This means that if we pick up any memberyin a setx, then the immediate successor ofy is

also inx.

Zermelo did not include the Axiom of Infinity in his system of 1930 because he

believed that it did not belong to general theory of set theory. He did not include the

Axiom of Choice on the ground that it differed in nature from the other Axioms. In

contemporary ZFC set theory are included the seven Axioms as postulated above, the

Axiom of Infinity, and the Axiom of Choice.

1.2 The Axiom of Choice and its Equivalents

The Axiom of Choice

First of all, we shall see what the Axiom of Choice says:

For every familyFof disjoint nonempty sets S, there exists a set Ccontaining

exactly one member from each member SofF(i.e., for each SFthe set SC

is a singleton).

Using the notion of a function we can paraphrase this as follows:

For every familyFof disjoint nonempty sets S, there exists a choice functionf

onFsuch thatf(S)Sfor each set Sin the familyF.

For instance, we can classify all natural numbers by the residues that result when they are

divided by 3 (i.e., the set Tof the sets Sof numbers congruent each other, modulo 3).

T{S1{0, 3, 6, },

S2{1, 4, 7, },

S3{2, 5, 8, }}

Then it is easy to see that there exists a set Ccontaining exactly one member from each


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member S1, S2, S3 of T (e.g., C{0, 4, 8}). In fact, the use of the Axiom of Choice is

dispensable in the case of a family of finitely many disjoint non-empty sets, and even in the

case of a family of infinitely many disjoint non-empty sets if we can specify the rule by

which to perform the choices. In our case, we can make sure that there exists such a set

without appealing to the Axiom of Choice, for instance, following the rule of choosing the

least member from the members of Sn(i.e., C{0, 1, 2}). The problem of the Axiom of

Choice is concerned only with infinitely many arbitrary choices.

Figure 1:The Axiom of Choice.

The Well-Ordering Theorem

The most useful form of the Axiom of Choice is the Well-Ordering Theorem: Every

set can be well-ordered. Actually, the Axiom of Choice is equivalent to the Well-Ordering

Theorem. But since this requires proof, we cannot regard the Well-Ordering Theorem

itself as an axiom despite its usefulness. So it is important to show the equivalence of the

Axiom of Choice and the Well-Ordering Theorem. But first we have to define a

well-ordering.


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In order to define a well-ordering exactly, we need to define the notion of an

R-minimal member:

xis anR-minimal member ofAif and only ifxA(y)(yA(yRx)).

Also, we need to define the notion of connected:

Ris connected inAif and only if (x)(y)(x,yAxyxRyyRx).

We shall next define a well-ordering:

Rwell-orders A if and only if every nonempty subset of Ahas an R-minimal

member &Ris connected inA.

Roughly speaking, the notion of anR-minimal memberguarantees us the existence of a

least member of every subset of A under the relation R. The notion of connected

guarantees that there is a linear ordering on Aexcluding the possibility of circularity. In

Appendix (I), I shall show that the Axiom of Choice is equivalent to the Well-Ordering

Theorem.

In 1904 Zermelo explicitly formulated the Axiom of Choice and proved the Axiom of

Choice is equivalent to the Well-Ordering Theorem. As we shall see in Section 3, there

arose much controversy over the non-constructive nature of the Axiom of Choice. In

response to his critics, in 1908 Zermelo reformulated the Axiom of Choice and his proof.

There Zermelo attempted to deprive the Axiom of Choice of all the constructivist

appearances by replacing a system of successive choices by a system of simultaneous ones

and put more emphasis on its super-temporality. We can clearly see the figure of Zermelo

as a Platonic realist here. In the same year Zermelo launched the axiomatization of set

theory. It is often said that the discovery of set-theoretic paradoxes motivated Zermelo to

axiomatize set theory. Under these circumstances, however, we could safely conclude that

Zermelo wanted to secure the status of the Axiom of Choice by creating a rigorous system

of axioms for set theory and lay down firm foundations of set theory and mathematics in

general.

The Multiplicative Axiom

We also have to notice that there are many other equivalents of the Axiom of Choice.

For instance, in abstract algebra one of the equivalents of the Axiom of Choice, Zorns

Lemma, is applied earlier than the Well-Ordering Theorem. This means that the Axiom of


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Choice is not an ad hocprinciple formed in the development of mathematics, but a stable

principle which is widely applicable in many branches of mathematics. But here in

connection with axiomatic set theory I shall confine my attention to Russell s

Multiplicative Axiom. InPrincipia Mathematica Russell introduces the Axiom of Choice

in the following way: If isa class of mutually exclusive classes, no one of which is null,

there is at least one class which takes one and only member from each member of .3

Russell calls it the Multiplicative Axiom, probably because of the Axioms connection

with cardinal multiplication, i.e., the construction of a set for the product of a denumerable

infinity of cardinals.

Russell takes as an example the millionaire who bought 0pairs ofboots and 0pairs

of socks.4 The question is how many boots and how many socks the millionaire had in all.

Although it is natural to suppose that he had 20boots and 20socks, we know that 0is not increased by doubling it, that is, 200. So the answer is that he had 0bootsand 0 socks. In general, the sum of 0 pairs must have 0 members. But we have to

notice that this result presupposes the existence of a set that consists of either of each pair.

In some cases we can have such a set without the Multiplicative Axiom, whereas in other

cases we cannot unless we assume the Axiom. In our case, among a pair of boots we can

distinguish left from right and thus choose all the right boots and then all the left boots.

Since there are no such distinguishing features among a pair of socks, however, we have no

specific rule by which to choose either of each pair of socks. Therefore, in the case of

socks the use of the Multiplicative Axiom is essential to show that there exists a set

consisting of either of each pair of socks.

1.3 The Consequences of the Axiom of Choice

As we have seen above, if we assume the Axiom of Choice, then, by the Well-Ordering

Theorem, every set can be well-ordered. So the set R of all real numbers can be

well-ordered. This is one of the most significant consequences of the Axiom of Choice.

This does not mean that in the absence of the Axiom of Choice we know little about the set

R. Actually, we know that the cardinality of the setRis greater than that of the setN of all

3 Russell, B. and Whitehead, A. N. [1910], vol. I, p. 536.


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natural numbers by the Cantorian diagonal argument, and that the cardinality of the setRor

of the continuum is that of the power set of the set N, i.e., 20. Based on ZF set theory

without the Axiom of Choice, however, we cannot prove whether or not the set Rcan be

well-ordered, therefore we dont even know whether or not the cardinality of the set Ris an

aleph.5 Only in the presence of the Axiom of Choice we do know that the set Rcan be

well-ordered, and that the set R is an aleph. And only then we can ask which aleph is its

cardinal.

The set N of all natural numbers can be well-ordered by the less-than relation.

Using the terminology of ZF set theory, the set Ncan be well-ordered by the membership

relation. One of the strengths of ZF set theory is that the less-than relation can be replaced

by the membership relation. The set N can be well-ordered by the less-than relation

because every nonempty subset of the setNhas a least member. On the other hand, the set

Ncannot be well-ordered by the greater-than relation because there are a bunch of subsets

that do not have a greatest member. The set Q of all rational numbers cannot be

well-ordered by magnitude. But, it is easy to see how the set Qof all rational numbers can

be well-ordered. Because, using the ordering that emerges from the proof that the

cardinality of rational numbers is the same as that of natural numbers, its trivial that there

is some way in which the set Q is put into one-to-one correspondence to the set of all

natural numbers.

But the situation is quite different with the set R of all real numbers. Intuitively

speaking, we dont know how the set R can be well-ordered. Even so by the

Well-Ordering Theorem, which implies that every set can be well-ordered, the setRcan be

well-ordered. As with the set Q, it is obvious that the set R cannot be well-ordered by

magnitude for the same reason as the set Q. But unlike the set Q, there is no obvious

ordering to hand that does the trick. However, the Well-Ordering Theorem tells us that

there is some relation by which the set Rcan be well-ordered, though we dont know what

it is specifically. We can see even from this that the Well-Ordering Theorem indeed makes

4 Russell, B. [1919], p. 126.5 Alephs are the infinite well-ordered cardinals.


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a very strong and powerful claim.

The Aleph Theorem, The Trichotomy of Cardinals

Moreover, since the Well-Ordering Theorem claims that every set can be well-ordered, it is

not just the setR that can be well-ordered. So it follows from the Well-Ordering Theorem

that all the cardinals are ordinals, which leads us to the Aleph Theorem that every infinite

cardinal is an aleph. Thus the Well-Ordering Theorem simplifies addition and

multiplication of infinite cardinal numbers, which would be more complicated otherwise.

Also, all cardinals are taken to be initial ordinals. In particular, any two sets are

comparable in terms of cardinality. Therefore the Trichotomy of Cardinals is true:

For every cardinal m and n, either mn, or mn, or mn.

Furthermore, as a corollary of the Aleph Theorem, the following equalities hold:m2mm2.

In this fashion the fundamental propositions true for alephs are extended to all infinite

cardinals.

If we assume the Axiom of Choice, then by the Well-Ordering Theorem, we dont

have to worry about the existence of sets that cannot be well-ordered. We know much

more about the cardinals of well-orderable sets than about the cardinals of sets that cannot

be well-ordered. As a consequence, once we assume the Axiom of Choice, which implies

the Well-Ordering Theorem, the theory of cardinals is considerably simplified.

But in fact there arose much controversy over the Axiom of Choice and Zermelo s

proof of its equivalence to the Well-Ordering Theorem. Hadamard, Hausdorff, and

Keyser defended the proof in full generality. Roughly speaking, however, German critics

such as Bernstein and Schoenflies disputed the proof on the ground that the Burali-Forti

paradox lies hidden in the proof, while French Constructivists such as Lebesgue, Borel, and

Baire opposed the Axiom of Choice itself on the ground that it does not provide the specific

rule by which to perform the choices.6 Though Zermelo met the first criticism by

rejecting the assertion that the collection Wof all ordinals is a set, the second one was more

6 Poincar, who is often said to be a conventionalist, accepted the Axiom of Choice so he did not reject the

Well-Ordering Theorem but Zermelo's proof of it because it makes use of impredicative definition.


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serious because of the stark philosophical difference underlying that criticism.7

The fundamental opposition in philosophy of mathematics is that between Platonic

realism, which posits mathematical entities outside of space and time, on the one hand, and

anti-realism, which restricts them to those which are legitimately constructible in space and

time, on the other. Under the philosophical background of this sort, the Platonic realists

accept the Axiom of Choice, which allows a set consisting of the members resulting from

infinitely many arbitrary choices, while the anti-realists reject the Axiom of Choice and

confine themselves to sets consisting of the effectively specifiable members. Hence some

mathematicians have claimed that we should avoid the Axiom of Choice wherever possible,

treating it just as a heuristic device for finding a new theorem, which is then to be proved

without appeal to the Axiom.

Though, as we have seen above, the legitimacy of the Axiom of Choice was already

controversial, skepticism about the Axiom of Choice was deepened when in 1914

Hausdorff discovered an unpleasant consequence of it, which is called Hausdorffs

paradox: half of a sphere is congruent to a third of the same sphere. Later Banach and

Tarski established this result as the Banach-Tarski paradox: any sphere S can be

decomposed into a finite number of pieces and reassembled into two spheres with the same

radius as S. In fact, Borel believed Hausdorffs paradox to show that contradictions

follow from the Axiom of Choice and that as a result the Axiom of Choice should be

rejected.

1.4 A Weaker Form of the Axiom of Choice

Given the controversial character of the Axiom of Choice, it is natural to attempt to weaken

it in some way acceptable to its opponents. We can then save some of its consequences,

although we have to sacrifice others. Precisely speaking, I have thus far confined myself

to the so-called full Axiom of Choice in distinction from its weaker form. Since the full

Axiom of Choice is independent of ZF, the weaker form of the Axiom of Choice should be

7 The following two objections against the Axiom of Choice can be expected:

(1) The Axiom of Choice should be constructibly justifiable.

(2) Even if the Axiom of Choice cannot be justified constructibly, we should be able to justify constructibly

the Well-Ordering of reals which is most wanted.

I doubt that both are legitimate criticisms.


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too strong for theorems of ZF, but too weak for the full Axiom of Choice. In other words,

the weaker form of the Axiom of Choice should be a theorem T of ZFC. More

specifically, when we ask firstly whether or not its a theorem of ZF and then whether or

not its equivalent to the full Axiom of Choice, both of the questions should be answered in

the negative. For its supposed to have the intermediate power between the theorems of

ZF and the full Axiom of Choice.

The Denumerable Axiom of Choice, The Principle of Dependent Choices

An example in point is the Denumerable Axiom of Choice, which restricts infinitely many

arbitrary choices to the cases of denumerable many sets. To put it precisely, the

Denumerable Axiom of Choice runs as follows:

Every family of denumerably many nonempty sets has a choice function.

The Denumerable Axiom of Choice is closely related to the Principle of Dependent

Choices:

IfRis a relation on a set Ssuch that for everyxSthere existsySsuch that

xRy, then there is a sequencex0,x1,x2, of members of Ssuch that

x0Rx1,x1Rx2, ,xnRxn1,

This principle enables us to make a countable number of consecutive choices. In

Appendix (II), I shall show that the Principle of Dependent Choices implies the

Denumerable Axiom of Choice.

The Countable Union Axiom

If we assume the Denumerable Axiom of Choice, then we can get the Countable Union

Axiom:8

The union of countably many countable sets is countable.

In Appendix (III), I shall show this.

Every infinite set has a countable subset, Every Dedekind-finite set is finite, The

restricted form of Trichotomy of Cardinals

8 A set is called denumerable if it is equinumerous with . A set is called countable if it is either

equinumerous with or finite.


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Also, if we assume the Denumerable Axiom of Choice, then we can prove that every

infinite set has a denumerable subset. In Appendix (IV), I shall show this.

In sum, I have shown above that the Principle of Dependent Choices implies the

Denumerable Axiom of Choice and this in turn implies that every infinite set has a

countable subset. Incidentally, neither of these implications can be reversed. The last

fact means that every Dedekind-finite set is finite. A set Sis Dedekind-finite if and only if

there is no proper subset of S equipollent to S. It is a matter of significance that if we

dont assume the Denumerable Axiom of Choice we cannot prove the equivalence of the

notions of Dedekind-finite set and finite set. For this means that in the absence of the

Denumerable Axiom of Choice there might exist sets which were infinite in one sense but

were finite in another. Russell and Whitehead were seriously concerned that there might

exist mediate cardinals which were too large to be finite but too small to be

Dedekind-infinite. At the same time, it is worth noting that the Denumerable Axiom of

Choice, instead of the full Axiom of Choice, suffices to reject such a possibility. Thus,

every cardinal number is comparable with 0, and the restricted from of the Trichotomy of

Cardinals does hold, i.e., |x|0, or |x|0, or |x|0 for any x. But the Principle ofDependent Choices has its limitations; it does not, for instance, imply the existence of a

well-ordering of the set Rof all real numbers. Historically speaking, Borel, who rejected

the full Axiom of Choice, accepted only the Denumerable Axiom of Choice, while unlike

Borel, Hobson rejected even denumerably many arbitrary choices, though he was

sympathetic with Borels critique.

1.5 The Axiom of Choice and the Continuum Hypothesis

In Section 3, we have seen that the cardinality of the set Rof all real numbers is greater

than that of the setNof all natural numbers, and that it is that of the power set of the set N,

i.e., 20

. But there we have also seen only in the presence of the Axiom of Choice, which

implies the Well-Ordering Theorem, we know that the set R can be well-ordered and the

cardinality of the set R is thus an aleph, and also we can ask which aleph is its cardinal.

That is, we can ask whether the cardinality of the set Ris the successor cardinal 1 of that of

the setN, or there is the successor cardinal 1between the cardinality of the setNand that

of the set R. The Continuum Hypothesis claims that the cardinality of the set R is the


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successor cardinal 1 of that of the set N, i.e., 120

. This means that the Continuum

Hypothesis presupposes the Axiom of Choice. To generalize this, the Generalized

Continuum Hypothesis claims that the cardinality of the a set Sis the successor cardinal of

that of a set S, i.e., n2n1.

In this connection, it is interesting to see that Brouwer claims that for the intuitionists

the Continuum Hypothesis doesnt make sense.9 0 is the only infinite cardinality of

which the intuitionists can accept the existence. For the intuitionists real numbers are the

rule-governed sequences constructed by a finite number of steps. Therefore for the

intuitionists the set of all real numbers which contains free choice sequences is meaningless.

So Brouwer claims that for the intuitionists it has no meaning to ask whether or not the

cardinality of the set of all real numbers is greater than 1, and whether or not the

cardinality of the set of all real numbers is the second smallest infinite cardinality. Given

that the Continuum Hypothesis presupposes the Axiom of Choice, it comes as no surprise

that Brouwer believes that for the intuitionists the Continuum Hypothesis doesnt make

sense. But it is interesting to see that Brouwer admits that a set Sis infinite if and only if S

is equipollent to one of its subsets. As we have seen in Section 4, this definition is exactly

Dedekind-infinite. This means that even Brouwer uses the Denumerable Axiom of

Choice implicitly.

To see how the Generalized Continuum Hypothesis works, we shall introduce the

function . The letter (beth) is the second letter of the Hebrew alphabet.

00,

12 ,

, where is a limit ordinal.

This definition makes sense only if we assume the Axiom of Choice because only in the

presence of the Axiom of Choice, which implies the Well-Ordering Theorem, every set can

be well-ordered and all cardinals are ordinals. For all , , since 21

9 Brouwer [1999], in Jacquette (ed) (2002), p. 271-4.


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. Especially, if the Generalized Continuum Hypothesis holds, then .Also, we shall see that under the Generalized Continuum Hypothesis an inaccessable

cardinal is the first weakly inaccessable ordinal. An ordinal is called weakly

inaccessable if is a limit cardinal for a limit ordinal . We can get the conceptof an

inaccessable cardinal stronger than that of an weakly inaccessable cardinal by replacing the

moderately increasing sequence from to 1by the exponentially and thus more rapidly

increasing sequence from to 2

. Then we can ask how big an inaccessable cardinal is.

Since an inaccessable cardinal is stronger than a weakly inaccessable cardinal, it is at least

as big as the first weakly inaccessable ordinal. If we assume the Generalized Continuum

Hypothesis, since 12

, an inaccessable cardinal is the first weakly inaccessable

ordinal.

1.6 Fictionalism or Instrumentalism

I believe that a deep-rooted and far-reaching topic in philosophy of mathematics is the

debate between those who claims mathematical objects to exist over and above space and

time (Platonic realism) and those who take them to be constructed within space and time

(Constructivism). Indeed there is a fictionalist or instrumentalist account of mathematical

objects, but I dont believe that fictionalism or instrumentalism is a good account of

mathematical objects. According to fictionalism, mathematical statements are simply

false, whereas according to instrumentalism, mathematical statements are neither true nor

false. The difference between fictionalism and instrumentalism is only in the letter but not

in the spirit. Philosophers of this sort attempt to explain the usefulness of mathematics in

natural science by means of the conservation theorem. This means that the mathematical

theory preserves the truth of the scientific theory, but facilitates the deductions which could

be made at greater length and with greater difficulty otherwise. A mathematical object

plays a role like a catalyst in chemistry that is a substance facilitating a chemical reactionthough itself remaining unchanged.

Fictionalists make their claim by refuting the indispensability argument. Roughly

speaking, indispensabilists accept the existence of mathematical objects, insofar as those

mathematical objects are indispensable to explain natural sciences. So fictionlists attempt


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to show that mathematical objects are dispensable to explain natural sciences. But we

have to note that one can reach the fictionalist conclusion by refuting the indispensability

argument only if the indispensability argument is the most promising argument for

mathematical realism. If there is a better argument for mathematical realism than the

indispensability argument, fictionalists will have much more work to do in order to deny

the existence of mathematical objects. So we shall examine the Quine/Putnam

indispensability argument in detail.

The Quine/Putnam indispensability argument aims to establish a realm of

mathematical objects by showing that if a scientific theory is accepted as true, then any

mathematical theory which is indispensable to formulate that scientific theory must also be

accepted as true.

[Q]uantification over mathematical entities is indispensable for science, both formal and

physical; therefore we should accept such quantification; but this commits us to accepting the

existence of the mathematical entities in question. This type of argument stems, of course,

from Quine, who has for years stressed both the indispensability of quantification over

mathematical entities and the intellectual dishonesty of denying the existence of what one daily

presupposes.10

We must notice that Quine and Putnam are not only claiming that the truth of mathematics

is presupposed by its use in science, but that the mathematics employed in our best

scientific theories enjoys empirical support. The upshot of the Quine/Putnam

indispensability argument is that the mathematics employed in a scientific theory is

confirmed indirectly from the confirming evidence for the scientific theory in which it is

contained.

According to Quine, just as we accept the existence of molecules, atoms, and quarks

if by so doing we have the best scientific theory that organizes and explains our experience,

so we accept the existence of mathematical objects. Putnam stresses that scientific

theories cannot even be formulated without the use of mathematics. Physical laws, such

as Newtons law of gravitation, are formulated using equations. Thus, Putnam claims that

they cannot be stated in a nominalistic language, that is, one in which no reference is made

to numbers, functions, sets, etc. If this is the case, scientific theories refer to mathematical


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objects and so we cannot accept our best scientific theories without accepting the existence

of mathematical objects. Putnam says, mathematics and physics are integrated in such a

way that it is not possible to be a realist with respect to physical theory and a nominalist

with respect to mathematical theory.11

The indispensability argument is supported by the idea that mathematical objects are

on a par with physical objects. Quine in Two Dogmas of Empiricismmaintains that our

statements about the external world face the tribunal of sense experience not individually

but only as a corporate body. This means that logical reflection and sense experience

together shape the total theory. Quine introduces a policy of minimum mutilation in

which we revise less central beliefs rather than more central ones. The logical and

mathematical beliefs are the most central beliefs. This is why we seldom are tempted to

revise them in the light of experience. According to Quine, however, the logical and

mathematical beliefs do not enjoy some special sort of nonempirical justification. The

logical and mathematical beliefs differ just in degree from the empirical beliefs. Thus, our

belief in the validity of modus tollensis just central than our vernacular beliefs.

Quine claims that the logical and mathematical beliefs are central because they apply

to a lot of situations and plays an important role in organizing how we think about these

situations. Even if modus tollensis central, however, I dont think that we could say that

every theorem in pure mathematics is like this. A result in some recondite area of

algebraic topology, for instance, might play little or no general role in organizing how we

think about the world. Likewise, the parts of mathematics, such as advanced set theory,

that go beyond this role are not accepted as true. The drawback of the indispensability

argument is that it conflicts with the actual practice of mathematics. The history of

mathematics after the nineteenth-century shows how mathematics separated and developed

itself independently from natural sciences and took its own course.

I think that, when we say that mathematics is indispensable, its very important that

mathematics is indispensable to either natural sciences or mathematics itself, especially

considering the autonomous developments of actual practice of mathematics after the

10 Putnam,Mathematics, Matter and Method, p. 347.11 Putnam,Mathematics, Matter and Method, p. 74.


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nineteenth century. If we interpret it as indispensable to mathematics itself, such as the

self-organization of mathematics, it amounts to much the same thing as the Platonistic

claim that a consistent mathematical theory describes at least a part of mathematical

universe. But in this case it seems to me that indispensability is not the best way to

represent the characteristic feature of mathematical objects. So if indispensabilists have

something to say different from Platonists, we have to interpret it as indispensable to

natural sciences. For instance, if indispensabilists would accept the existence of

mathematical objects which are indispensable to mathematics itself, since the Axiom of

Choice is indispensable to mathematics itself, especially axiomatization of Cantorian set

theory, they would have to accept the Axiom of Choice. Against Platonists, however,

indispensabilists reject the Axiom of Choice in its own right. If indispensabilists accept

the existence of mathematical objects which are indispensable to natural sciences, since the

Axiom of Choice is dispensable to natural sciences, they can reject the Axiom of Choice as

required. So when we use the indispensability argument to justify the ontological status of

mathematical objects, we have to make it clear that mathematics is indispensable to natural

sciences.

It might be objected that, even if indispensabilists would accept the existence of

mathematical objects which are indispensable to mathematics itself, they could differentiate

themselves from Platonists in the sense that, as we shall see later, the Axiom of Choice is

dispensable to prove the Banach-Tarski Paradox. But I shall claim that, even if the

Banach-Tarski Paradox can be reformulated without the Axiom of Choice, it does not

necessarily deal a blow to the Platonists. I shall ask whether or not the proof without the

Axiom of Choice depends on extremely complex or ad hoc principles, compared with the

proof with the Axiom of Choice. If by invoking the Axiom of Choice the Banach-Tarski

Paradox can be proved in a simpler, more systematic and more unified way, I believe the

proof with the Axiom of Choice reflects the fact of matter rather than the proof without the

Axiom of Choice. In any case, I dont believe that the indispensability argument is the

most promising argument for mathematical realism. If we believe mathematical theories

applicable to natural sciences, in the extension we should believe mathematical theories not

applicable to natural sciences. For instance, if we believe a weaker form of the Axiom of


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Choice, there is no good reason to disbelieve the full Axiom of Choice. In Chapter 6, I

shall submit the argument for mathematical realism I believe is the best. So even if

fictionalists succeed in the nominalization of mathematical objects in natural sciences, since

there is a better argument for mathematical realism, I dont believe that factionalism or

instrumentalism is a correct account of mathematical objects.

Conclusion

The main problem with the Axiom of Choice concerns the issue of whether or not infinitely

many arbitrary choices should be accepted in mathematics. The Platonists admit the

possibility of making a set consisting of indefinable members of a certain kind. On the

other hand, the constructivists allow only the existence of sets consisting of members that

are specifiable by a finite number of steps. But the Axiom of Choice largely contributed

to the systematization of Cantorian set theory. It is interesting to note that even some of

the opponents of the Axiom of Choice used it implicitly. For instance, though Russell was

skeptical of the Well-Ordering Theorem and the Trichotomy of Cardinals, he used the

proposition that every infinite set has a denumerable subset in order to prove that a set is

Dedekind-finite if and only if it is finite. But the proof of this proposition makes essential

use of the Denumerable Axiom of Choice. This is a good example of the deductive power

of the Axiom of Choice. Mathematics, then, is severely curtailed if we reject the Axiom

of Choice. In light of this, I provisionally conclude that the debate over the Axiom of

Choice favors Platonic realism.


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CHAPTER 2

LEBESGUES THEORY OF MEASURE

Introduction

In this chapter, I shall trace back the theory of large cardinals to its origin: Lebesgues

theory of measure, and claim that the non-constructive nature of the Lebesgue measure lies

in the notion of -additivity (or, more generally, -additivity). To that aim, in the first half

of the chapter, we shall see that the Lebesgue integral based upon the Lebesgue measure

was devised in attempts to solve the problems with the Riemann integral based upon

Jordans content. The Lebesgue integral enabled the integration of functions that are not

Riemann integrable, and also made much improvement on Riemanns theory of

convergence properties.

In the second half of this chapter, we shall investigate how Lebesgue s theory of

measure is applied to the theory of large cardinals. The cogent relationship between

Lebesgues theory of measure and the theory of large cardinals can be detected in the

theorem to the effect that if there exist measurable cardinals, they are (strongly)

inaccessible. Finally, I shall point out that the non-constructive nature of the Lebesgue

measure as shown above creates an irreconcilable tension with Lebesgues skeptical attitude

toward the Axiom of Choice.

2.1 Lebesgues Theory of Integration

We can see the nature of the Riemann integral in the method to find the area bounded by a

continuous function f(x) and the x-axis. The Riemann integral involves partitioning the

domain of f(x) and approximating f(x) by means of the upper and lower step functions

bracketingf(x) from without and within respectively. A partitionPof [a, b] is a set {a0, a1,

, an} such that

aa0a1 anb.

Let Si{x| ai1xai}. (i1, 2, n)

Si(x) is the characteristic function of the set Si, defined by


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Si(x)1 ifxSi

0 ifx Si

Also, letMisupSif(x) and miinfSif(x).

Then, the upper step function

(x)M1S1M2S2 MnSni1nMiSi.

Similarly, the lower step function

(x)m1S1m2S2 mnSni1nmiSi.

Now, the upper sum U(f, P) is the area bounded by the upper step function (x) and the

x-axis:

U(f,P)M1(a1a0)M2(a2a1) Mn(anan1)Sni1nMi(aiai1).

In the same way, the lower sumL(f,P) is the area bounded by the lower step function (x)

and thex-axis:

L(f,P)m1(a1a0)m2(a2a1) mn(anan1)Sni1nmi(aiai1).

As n, we have the upper integral

and the lower integral

Finally,fis Riemann integral iff

Although the Riemann integral will do for most practical use, there are some

problems with the Riemann integral when it comes to advanced fields of mathematics.

First of all, there exist a lot of functions that are not Riemann integrable. Secondly, the

Riemann integral contains too strict convergence properties. The Lebesgue integral

extends the range of integrable functions, taking over the nice properties of the Riemann

integral. The turn from the Riemann to Lebesgue integral could be characterized as the

one from constructive to non-constructive mathematics, as it were.

In order to overcome the difficulties with the Riemann integral, Lebesgue substituted

the Lebesgue measure for Jordans content that provided foundation for the Riemann

integral.

Earlier concepts of measure such as Jordans content were onlyfinitely additivein the sense

)},(inf{ PfUfb

a

)},(inf{ PfLfb

a

. b

a

b

aff


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that

m(AB)m(A)m(B)

for any two disjoint measurable sets A, B, and these led to more limited theories of

integration. On the other hand, the Lebesgue measure is -additive (countably additive) in

the sense that

m(X)m(X).

for any pairwise disjoint measurable setsX. The measure defined in this way fits into our

intuition that the measure should be a length in one dimension, an area in two dimensions,

and a volume in three dimensions. The upshot of this definition is that though the

countable union of sets with measure zero is again of measure zero, the uncountable union

of sets with measure zero has positive measure. Due to the property of -additivity of the

Lebesgue measure, the Lebesgue integral based on the Lebesgue measure is more powerful

than the Riemann integral based on Jordans content.

In order to see how significant the notion of -additivity is, it is useful to reconsider

Zenos argument and fifth-century Atomists reaction to that.12

According to Zenos

argument, if finite extension is infinitely divisible, either the resulting least parts have no

size or they have some positive size. If they have no size, however, when put together

they result in something with no size. If they have a positive size, no matter how small it

may be, when an infinite number of them are put together, the result is something of infinite.

Either way, we cannot form the original object by reassembling its parts. So, the Atomists

avoid this argument by claiming that bodies are ultimately composed of indivisibles. To

Zenos argument, Lebesgue would reply that a countable set of points with measure zero

remains of measure zero, and only an uncountable set of points with measure zero can have

positive measure.

We define the Lebesgue measure more precisely. Let Ebe the unit interval [0, 1].

The outer measure can be obtained by approximating the set from without by open sets.

That is, the outer measure ofA is the infimum of open sets containingA. In symbols,

12 For this, see McKirahan,Philosophy before Socrates: an Introduction with Texts and Commentary, p. 310.


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me(A)inf{m(G) AGopen set}13

On the other hand, the inner measure can be obtained by approximating the set from within

by compact (i.e., closed and bounded) sets.14

That is, the inner measure of A is the

supremum of compact sets contained inA. In symbols,

mi(A)sup{m(K) AKcompact set}1me(E A)

E A is the difference ofE andA. Most importantly, a setAEis Lebesgue measurable

if me(A)mi(A).

A set with Lebesgue measure zero is called a null set. Here we must be careful not

to confuse empty setwith null setin this sense. Actually, since we define the empty set

to be of measure zero, the empty set is a null set. But a set containing just a single point,

that is, a singleton is a null set as well. Due to the property of -additivity of the Lebesgue

measure, any countable set of points is also a null set. Only an uncountable set can have

positive measure. But we have to note that some uncountable sets of points can be null

sets. A case in point is the Cantor set. The Cantor set is constructed as follows:

Take the unit interval [0, 1]. Divide it into three equal intervals and remove the middle

open third, leaving the set C1.

C1[0, 1/3][2/3, 1]

Then, divide each of the two intervals into three equal intervals and remove the middle

open third of each interval, leaving the set C2.

C2[0, 1/9][2/9, 1/3][2/3, 7/9][8/9, 1]

..

At the nth step, we get the set Cn.

Cn[0, 1/3n][2/3n, 1/3n1][11/3n1, 12/3n][11/3n, 1]

Repeat this process again and again. After steps, we get the Cantor set C.

13 A set Sis open if, for anyxS, Scontains an open ball of centerx. In symbols, x((x,x)S).14 A set Sis closed if its complement SCis open. A Set S is bounded if it is contained in some ball. Note

that a closed set is not necessarily bounded. According to this definition, there are closed and unbounded

sets. For instance, an infinite half open interval (, 1] is closed because its complement (1, ) is open,but unbounded because it is an infinite interval.


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Now, we prove that the Cantor set is of measure zero. As we can clearly see from

the process of constructing the Cantor set, at the nth step we get 2nmany intervals of the

length 1/3n. Therefore, Cnhas the total length 2

n1/3n(2/3)n. Since (2/3)n0 as n

, the Cantor set is of measure zero. To see the unique nature of the Cantor set, for

instance, we shall divide the unit interval [0, 1] into two equal intervals and remove the one

half, leaving the other half. Indeed, after steps, we get the setAof measure zero since

(1/2)n0 as n. Unlike the Cantor set C, however,Aconverges to a single point, so

it is no wonder thatAis of measure zero.

It remains to show that the Cantor set is uncountable. The upshot of this proof is to

see that the Cantor set is the set of reals in [0, 1] that can be expressed, in the ternary

system, only by 0 and 2 (i.e., without 1). This is the reason why the Cantor set is often

called the Cantor ternaryset.

In general, when an integerNin the decimal system has a ternary expansion:

.. a232a13

1a030 (an0, 1, 2),

Nis written, in the ternary system, as

.. a2a1a0.

Applying this notation to a decimal n[0, 1], when nhas a ternary expansion:

a1/31

a2/32

a3/33

.. (an0, 1, 2),

nis written, in the ternary system, as

0.a0a1a2..

For instance, the number expressed by 34 in the decimal system is expressed by 1021 in the

ternary system because 34 has a ternary expansion: 1.330.322.311.30. Likewise, the

number expressed by 0.5 in the decimal system is expressed by 0.111 in the ternary

system because 0.5 has a ternary expansion: 1/311/321/33 ..

A possible ambiguity is that the end points can be written in two ways. For instance,

1/3 can be written as both 0.1 and 0.022.. and 2/3 can be written as both 0.2 and 0.122

.. But this does not cause much trouble, considering similar cases encountered in the

decimal system, such as 10.999 We shall adopt the rule according to which:

If the last non-zero place is 1, we choose the non-terminating expression.


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Otherwise (i.e., if the last non-zero place is 2), we choose the terminating

expression.

Then, 1/3 is written as 0.022.. and 2/3 is written as 0.2.

Using the terminology of the ternary system, we can put the process of constructing

the Cantor set in a more simple way. Every number in [0, 1], expressed in the ternary

system, is of the form:

0.a0a1a2.. (an0, 1, 2)

The construction of the Cantor set in the ternary system is as follows:

xC1 iff a10, 2

xC2 iff a10, 2 & a20, 2

..

xCn iff a10, 2 & a20, 2 .. an0, 2

Therefore, the Cantor set consists of reals in [0, 1] expressed, in the ternary system, by 0

and 2 (i.e., without 1) as stated above.

Now, from the diagonal argument, we can show that the Cantor set is uncountable.

Suppose that the Cantor set is put in one-to-one correspondence with natural numbers as

follows:

1 0.a11a12a13..2 0.a21a22a23..

..

n 0.an1an2an3..

As we have seen, each aijis 0 or 2. So, let bn2 if ann0 and bn0 if ann2. Then, we

get a number

0.b1b2b3..

This is exactly the number different in the nth place from that corresponding to n, therefore

cannot be found in the list above. This shows that even though it is uncountable, the

Cantor set is so scattered that it is negligible from the Lebesgue measure-theoretical point

of view.

Now, we shall discuss what impact the Lebesgue measure has on the Lebesgue


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integral. The upshot of the Lebesgue integral is that the values of a function f(x) dont

affect the values of the integral f(x)dxat all pointsxthat form a null set. The Lebesgue

integral can give the explicit answer to the question of how many points can be removed

without altering the value of the integral. The answer is as many points as form a null set.

In other words, points that do not form a null set determine the value of the integral. We

already know that due to the property of -additivity of the Lebesgue measure, a countable

set of points forms a null set. Therefore, most importantly, even if the values of a function

f(x) could be altered at a countable set of points, the value of the integral remains the same.

This is another way to show that the set of reals is not countable. This also tells us that the

boundaries make no difference to the area.

If some property holds except on a null set, the property is said to be hold almosteverywhere, (abbreviated a.e), or presque partout (abbreviated p.p.). If there are two

different functions f(x)g(x) almost everywhere, we cannot distinguish f(x) from g(x).

Then, the Lebegue theory tells us that we can regard these two functions as being virtually

equal.

The Lebesgue integral made possible the integration of functions that are not

Riemann integrable. The characteristic function Q(x) of the set Q of rationals is an

example of functions that are not Riemann integrable but Lebesgue integrable. The reason

why Q(x) is not Riemann integrable is as follows. No matter how small the partitions

(x0, x1), (x1, x2), (xn1, xn) of [0, 1] are, rationals and irrationals coexist in the same

partition. Therefore, its possible to choose a rational from any partition and then an

irrational from any partition in such a way that the upper and lower step functions

bracketing Q(x) cannot coincide each other.

The stock-in-trade of the Lebesgue integral is to approximate f(x) not by means of

vertical strips,i.e., the upper and lower step functions but by means of horizontal strips.

The Lebesgue integral involves a partition of the range of f(x) rather than a partition of the

domain as for the Riemann integral. Thus, in the Lebesgue integral we partition the range

off(x) and approximatef(x) by means of the upper and lower step functions bracketing f(x)

from without and within respectively. A partitionPof [a, b] is a set {a0, a1, , an} such


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that

aa0a1 anb.

Let Si{x ai1f(x)ai}. (i1, 2, n)

Then, the upper function

(x)a1S1a2S2 anSni1naiSi.

This function is called asimple function(orgeneralized step function).

Similarly, the lower function

(x)a0S1a1S2 an1Sni1nai1Si.

Now, the upper sum

U(f,P)a1m(S1)a2m(S2) anm(Sn)i1naim(Si).

15

In the same way, the lower sum

L(f,P)a0m(S1)a1m(S2) an1m(Sn)i1nai1m(Si).

As with Riemann integral, f is Lebesgue integrable iff the infimum of the upper sum is the

supremum of the lower sum. Therefore, Q(x) is Lebesgue integrable onR, and

RQ(x)dx1m(Q)0m(R Q)0

because Qis a null set, so m(Q)0 .

15 m(S1) stands for the Lebesgue measure of S1.


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Figure 2: Riemann Integral vs. Lebesgue Integral.


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Also, there are two major convergence theorems involving the Lebesgue integral: the

Monotone Convergence Theorem and the Lebesgue Dominated Convergence Theorem,

neither of which is true with regard to Riemann integrable functions. Both are concerned

with when the limit of the integrals is the integral of the limit, that is, when we can

interchange the order of the limit and the integral.

(Monotone Convergence Theorem)

Let {fn(x)} be a monotonic increasing sequence of measurable functions

0f1(x)f2(x)

such thatfn(x) converges (pointwise) tof(x).

Then, limnfn(x)dx(limnfn(x)dx)f(x)dx.(Lebesgue Dominated Convergence Theorem)

Let {fn(x)} be a sequence of measurable functions such that limnfn(x)f(x).

If the sequence is dominatedby an integrable functiong(x) in the sense that

fn(x) g(x).

Then, limnfn(x)dx(limnfn(x)dx)f(x)dx.

Here I need to clarify the distinction between uniform convergence and pointwise

convergence. The sequence of functions f1(x), f2(x), is said to converge uniformly to

f(x) if

0Nx(if nN, then f(x)fn(x) ).16

On the other hand, the sequence of functionsf1(x),f2(x), is said to converge pointwise to

f(x) if

0xN (if nN, then f(x)fn(x) ).

We note that the order of xand N iscontrary each other. In uniform convergence for

all x the sequence converges to f(x) simultaneously, whereas in pointwise convergence at

16 This would be easier to understand if interpreted in such a way that 0Nx(nNis large enough

so that f(x)fn(x) ).


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eachxthe sequence could converge to f(x) in different ways. This is the reason why N

precedes x in uniform convergence while x precedes N in pointwise convergence.

Obviously, uniform convergence implies pointwise convergence, but this implication

cannot be reversed. Uniform convergence is nice in the sense that if each function of the

sequencef1(x),f2(x), has some property such as continuity or measurability, so does the

limit functionf(x).

We shall take a couple of examples to see in concretothe difference between uniform

convergence and pointwise convergence. The sequence of functionsfn(x)(11/n)x(x

[0, 1]) converges uniformly to the function f(x)x for eachx[0, 1]. To see this, for a

given choose N1/. Then, for all nN for all x[0, 1] f(x)fn(x) as

required. On the other hand, the sequence of functions fn(x)xn (x[0, 1]) converges

pointwise to the function f(x)0 if 0x1 and f(x)0 if x1. This sequence is not

uniform convergence because when 1/2, say, then, no matter how large n is, f(x)

fn(x) 1/2 for x[1/n2, 1). Notice that in the former case each fn(x) and f(x) are

continuous on [0, 1], while in the latter case each fn(x) is indeed continuous on [0, 1] but

f(x) is discontinuous atx1. These examples precisely show that in uniform convergence

f(x) inherits a nice property of eachfn(x), i.e., continuity.

The significance of the Lebesgue integral, not least, Lebesgues Convergence

Theorems, is that it provides foundation for functional analysis. Functional analysis is a

branch of mathematics that discusses Banach space and Hilbert space in a rigorous manner,

and is applied to the theory of integral equation or, beyond mathematics, to quantum

physics. Since functional analysis has to deal with discontinuous functions, the sequence

of functions does not necessarily converge uniformly. Therefore, Lebesgue Convergence

Theorems, which make it possible to interchange the order of the limit and the integral not

only in uniform convergence but also in pointwise convergence, plays a significant role in

functional analysis.


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2.2 Measurable Cardinals

Now we consider how Lebesgues theory of measure is applied to the theory of large

cardinals. Large cardinals are uncountable cardinals that cannot be reached from below.

Actually, mathematicians assume various kinds of large cardinals, e.g., inaccessible, Mahlo,

weakly compact, ineffable, measurable cardinals in the order of magnitude. That is, the

least weakly compact cardinal, if any, is a lot bigger than the least Mahlo cardinal, which is,

in turn, a lot bigger than the least inaccessible cardinal. Measurable cardinals are very

large cardinals, and the least measurable cardinal, if any, is greater than many weakly

compact and even ineffable cardinals. We have the proof that 20

is not a measurable

cardinal. The concept of a measurable cardinal plays a much more major role in the

theory of large cardinals than the weakly compact and the ineffable cardinals.

Now we shall see a measurable cardinal in connection with Lebesgues theory of

measure. We begin with the definition of measure on a set S. A measureon a set Sis a

map mfromP(S) to [0, 1] such that

(i)m()0 and m(S)1

(ii) Monotonicity: IfAB, then m(A)m(B)

(iii) Non-triviality:m({a})0 for aS

(iv) -additivity: If theXs are pairwise disjoint, then m(X)m(X).

As to the Lebesgue measure, it is natural to ask whether or not there is non-trivial

translation-invariant countably additive measure on all subsets of reals. In 1905 Vitali

showed the existence of non-Lebesgue measurable sets of reals by using the Axiom of

Choice. The concept of a measurable cardinal arose in response to Vitalis construction of

a non-Lebesgue measurable set of real numbers.

(1) If the measure does not need to be translation-invariant, is there a non-trivial countably

additive measure on all sets of real numbers?(2) Is there such a measure for all subsets of some set S?

These questions led to the theory of large cardinal numbers, which had a great impact on

both in pure set theory and in descriptive set theory.

We define the notion of -additivity by generalizing the notion of -additivity:


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If 0is regular, , and theXs are pairwise disjoint for any , then

m(X)m(X).

Though m(X)m(X) because measure zero is assigned to singletons, the

upshot of this definition is that the union of fewer than sets with measure zero remains of

measure zero. We are now ready to define measurable cardinals:

is measurable if and only if0has a two-valued, -additive measure.17Intuitively, a cardinal is measurable iff is an uncountable cardinal and the union of

fewer than sets of measure zero is of measure zero. It is worth noting that we put 0in the definition. If it were not for this condition, 0would be measurable. The cardinal

2 would be also measurable. In this case, according to the conditions that a measure must

satisfy, m(0)0, m({0})0, m({1})0, and m({0, 1})1. Obviously, mis two-additive

measure on 2. But the core of the definition of measurable cardinals is that the union of

countably many sets of measure 0 is again of measure zero. Therefore, cardinals 0

should not be considered as measurable. This is why we put 0in the definition.We have to note that there is a slippage between the naming of -additivity and the

naming of -additivity. For -additivity is so called by paying attention to the point up to

which the union of sets with measure zero remains of measure zero (therefore, the

uncountable union of sets with measure zero can have positive measure), whereas

-additivity is so called by paying attention to the point beyond which the union of sets

with measure zero can have positive measure (therefore, the fewer than unio n of sets with

measure zero remains of measure zero). Thus, -additivity is the same as 1-additivity.18

Since 0, 0is a non-measurable cardinal. Then, a question arises: Is 20also a

non-measurable cardinal? We can show that the answer is yes by reductio ad absurdum:

17 Since we put non-triviality into the definition of measure above, we dropped it from the definition of

measurable cardinals here. But we could separate non-triviality from the definition of measure, and put it

into the definition of measurable cardinals: is measurable if and only if0has a two-valued, -additive,non-trivial measure.18 Fortunately, there arises no ambiguity here because 0is the first aleph.


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If we suppose that 20 is a measurable cardinal, we shall reach a contradiction. The proof

runs as follows:

Suppose that 20 is a measurable cardinal, that is, we can assign a two-valued,

-additive measure to all the subsets of 20. Therefore, there are 220many subsets. Now

we take a functionf: 0{0, 1}. But note that this function is still not the one that assigns

a measure to all the subsets of 20. And we have the set of functionfs: 2{f1,f2,f3,,

f2}. Assigning a measure to all the subsets of 2

0boils down to assigning a measure to all

the subsets of2. When we discuss the set of functions, we denote it by

2 in distinction

from cardinal exponentiation 2in order to avoid confusion.

According to the definition (i) of measure above, m(2)1. Here we can divide 2

into the two disjoint sets: the set of functions satisfying f(0)0 and the set of functions

satisfyingf(0)1. That is,

2{f2 f(0)0}{f2 f(0)1}

Then, we have to assign measure 1 to either {f2 f(0)0} or {f2 f(0)1}, no

matter which it may be, because if both of them were of measure zero, they would not add

up to m(

2)1, contrary to the definition (iv) of measure above. Just for the sake ofargument, we shall assume that m({f2 f(0)1})1. Again, we can divide the set

{f2 f(0)1} into the two disjoint sets: the set of functions satisfying f(0)1&f(1)

0 and the set of functions satisfyingf(0)1&f(1)1. That is,

{f2 f(0)1}{f2 f(0)1&f(1)0}{f2 f(0)1&f(1)1}.

For the same reason as before, we have to assign measure 1 to either {f2 f(0)

1&f(1)0} or {f2 f(0)1&f(1)1}. Just for the sake of argument, we shall

assume that m({f2 f(0)1&f(1)1})1. We let this process go on and on.

Here is the upshot of this proof. The Axiom of Choice guarantees that after steps,

we get the set which consists of the unique function:

{fn}{f2 f(0)1&f(1)1& .. &f(0)1}


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Then we ask: Is m({fn})0 or 1? According to -additivity, all the sets to which we have

assigned measure zero so far cannot add up to measure 1, but m(2)1, so m({fn})m(

2)

1. According to the definition (iii) of measure above, however, the singleton is of

measure zero and the set {fn} is a singleton, so m({fn})0. A contradiction. This

completes the proof. This proof also tells us that if2 has a two-valued, -additive

measure, it should be trivial.

It is interesting to capture measurable cardinals in connection with the notion of an

ultrafilter. But first we have to define the notion of a filter:

Afilteron a non-empty set Sis a collectionFof subsets of Ssuch that for anyA,BS,

(i) SFand F.

(ii) IfA,BF, thenABF.

(iii) If AF and AB, then BF (In words, any set B that contains a set A being a

member of a filterFis also in that filter).

A trivial filter F{S}. We define a principal filter. Let X0be a non-empty subset of S.

A principal filterF{XS |X0X}. This means that there is no infinite regress in that

filter. Thus every filter on a finite set is a principal filter. Take as an example the filters

on the set {0, 1, 2, 3}. A trivial filterF{{0, 1, 2, 3}}. A filterF{{0, 1, 2}, {0, 1, 2,

3}}. Another filterF{{0, 1}, {0, 1, 2}, {0, 1, 3}, {0, 1, 2, 3}}.

I shall also define the dual notion of a filter, i.e., an ideal:

An idealon a non-empty set Sis a collectionFof subsets of Ssuch that for anyA,BS,

(i) Iand S I.

(ii) IfA,BI, thenABI.

(iii) IfAI andBA, thenBI(In words, any set Bthat is contained in a set Abeing a

member of an idealIis also in that ideal).

There is a remarkable relationship between a filterFand an idealIon S:I{SX XF}

or equivalently,F{SX XI}. So, for the set {0, 1, 2, 3} mentioned above, an ideal

I{, {3}}. Another idealI{, {2}, {3}, {2, 3}}.


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We now turn to the notion of an ultrafilter that is closely related to measurable

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A Defense of Platonic Realism in Mathematics

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