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    Axiomatic Set Theory
    Axiomatic Set Theory
    Axiomatic Set Theory
    Ebook479 pages4 hours

    Axiomatic Set Theory

    By Patrick Suppes

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    One of the most pressingproblems of mathematics over the last hundred years has been the question: What is a number? One of the most impressive answers has been the axiomatic development of set theory. The question raised is: "Exactly what assumptions, beyond those of elementary logic, are required as a basis for modern mathematics?" Answering this question by means of the Zermelo-Fraenkel system, Professor Suppes' coverage is the best treatment of axiomatic set theory for the mathematics student on the upper undergraduate or graduate level.
    The opening chapter covers the basic paradoxes and the history of set theory and provides a motivation for the study. The second and third chapters cover the basic definitions and axioms and the theory of relations and functions. Beginning with the fourth chapter, equipollence, finite sets and cardinal numbers are dealt with. Chapter five continues the development with finite ordinals and denumerable sets. Chapter six, on rational numbers and real numbers, has been arranged so that it can be omitted without loss of continuity. In chapter seven, transfinite induction and ordinal arithmetic are introduced and the system of axioms is revised. The final chapter deals with the axiom of choice. Throughout, emphasis is on axioms and theorems; proofs are informal. Exercises supplement the text. Much coverage is given to intuitive ideas as well as to comparative development of other systems of set theory. Although a degree of mathematical sophistication is necessary, especially for the final two chapters, no previous work in mathematical logic or set theory is required.
    For the student of mathematics, set theory is necessary for the proper understanding of the foundations of mathematics. Professor Suppes in Axiomatic Set Theory provides a very clear and well-developed approach. For those with more than a classroom interest in set theory, the historical references and the coverage of the rationale behind the axioms will provide a strong background to the major developments in the field. 1960 edition.

    LanguageEnglish
    PublisherDover Publications
    Release dateMay 4, 2012
    ISBN9780486136875
    Axiomatic Set Theory

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      Axiomatic Set Theory - Patrick Suppes

      THEORY

      AXIOMATIC SET THEORY

      by

      PATRICK SUPPES

      Professor of Philosophy and Statistics Stanford University

      DOVER PUBLICATIONS, INC.

      NEW YORK

      Copyright © 1972 by Dover Publications, Inc.

      Copyright © 1960 by Patrick Suppes.

      All rights reserved.

      This Dover edition, first published in 1972, is an unabridged and corrected republication of the work originally published by D. Van Nostrand Company in 1960, to which the author has added a new preface and a new section (8.4).

      International Standard Book Number: 0-486-61630-4

      Library of Congress Catalog Card Number: 72-86226

      Manufactured in the United States by Courier Corporation

      61630413

      www.doverpublications.com

      To

      MY PARENTS

      PREFACE TO THE DOVER EDITION

      The present edition differs from the first edition in the correction of all known errors, and in the addition of a final section (8.4) on the independence of the axiom of choice and the generalized continuum hypothesis. For bringing to my attention numerous misprints and minor mistakes I am indebted to a great many people, in fact, too many to list here; but I do want to mention Nuel Belnap, Theodore Hailperin, Craig Harrison, Milton Levy, Elliott Mendelson, N. C. K. Phillips, D. H. Potts, Hilary Staniland, and John Wallace.

      PATRICK SUPPES

      Stanford, California

      June, 1972

      PREFACE TO THE FIRST EDITION

      This book is intended primarily to serve as a textbook for courses in axiomatic set theory. The Zermelo-Fraenkel system is developed in detail. The mathematical prerequisites are minimal; in particular, no previous knowledge of set theory or mathematical logic is assumed. On the other hand, students will need a certain degree of general mathematical sophistication, especially to master the last two chapters. Although some logical notation is used throughout the book, proofs are written in an informal style and an attempt has been made to avoid excessive symbolism. A glossary of the more frequently used symbols is provided.

      The eight chapters are organized as follows. Chapter 1 provides a brief introduction. Chapter 2 is concerned with general developments, and Chapter 3 with relations and functions. There is not a single difficult theorem in these first three chapters, and they can be covered very rapidly in an advanced class. The main pedagogical emphasis is on the exact role of the various axioms introduced in Chapter 2, which are summarized at the end of the chapter.

      Chapter 4 considers the classical topics of equipollence of sets, finite sets and cardinal numbers. The Schröder-Bernstein Theorem is proved early in the chapter. The development of the theory of finite sets follows closely Alfred Tarski’s well known article of 1924. The theory of cardinal numbers is simplified by introducing a special axiom to the effect that the cardinal numbers of two equipollent sets are identical. This axiom is not part of the standard Zermelo-Fraenkel system, and so every definition or theorem which depends on it has been marked by the symbol ‘†’, but it leads to such a simple and natural development that I feel its introduction is fully justified. Chapter 5 covers some of the same ground from another viewpoint. The natural numbers are defined as von Neumann ordinals and the theory of recursive definitions is developed. The axiom of infinity is introduced, and the final section covers the basic facts about denumerable sets.

      In Chapter 6 the standard construction of the rational and real numbers is given in detail. Cauchy sequences of rational numbers rather than Dedekind cuts are used to define the real numbers. Most of the elementary facts about sets of the power of the continuum are proved in the final section. Because for many courses in set theory it will not be feasible in the time allotted to include the construction of the real numbers or because the topic may be assigned to other courses, the book has been written so that this chapter may be omitted without loss of continuity.

      Chapter 7 is concerned with transfinite induction and ordinal arithmetic. The treatment of transfinite induction and definition by transfinite recursion is one of the most detailed in print. Numerous variant formulations have been given in the hope that successive consideration of them will clarify for the student the essential character of transfinite processes. The axiom schema of replacement is introduced in connection with establishing an appropriate recursion schema to define ordinal addition. The more familiar facts about alephs and well-ordered sets are proved in the latter part of the chapter.

      Chapter 8 deals mainly with the axiom of choice and its equivalents, like Hausdorff’s Maximal Principle and Zorn’s Lemma. Important facts whose known proofs require the axiom of choice are also deferred consideration until this chapter. A typical example is the identity of ordinary and Dedekind infinity.

      Although the list of references at the end of the book is small compared to that given in Fraenkel’s Abstract Set Theory, I have attempted to refer to many of the more important papers for the topics considered. Because set theory, even perhaps axiomatic set theory, is finally coming to be a staple of every young mathematician’s intellectual fare, it is of some historical interest to note that the majority of the papers referred to in the text were published before 1930.

      I hope that this book will prove useful in connection with several kinds of courses. A semester mathematics course in set theory for seniors or first-year graduate students should be able to cover the whole book with the exception perhaps of Chapter 6. Philosophy courses in the foundations of mathematics may profitably cover only the first four chapters, which end with the construction of the natural numbers as finite cardinals. The material in the first six chapters, ending with the construction of the real numbers, is suitable for an undergraduate mathematics course in the foundations of analysis, or as auxiliary reading for the course in the theory of functions of a real variable.

      This book was begun in 1954 as a set of lecture notes. I have revised the original version at least four times and in the process have benefited greatly from the criticisms and remarks of many people. I want particularly to thank Michael Dummett, John W. Gray, Robert McNaughton, John Myhill, Raphael M. Robinson, Richard Robinson, Herman and Jean Rubin, Dana Scott, J. F. Thomson and Karol Valpreda Walsh. During the academic year 1957-58 Joseph Ullian used a mimeographed version at Stanford, and most of his many useful criticisms and corrections have been incorporated. As in the case of my Introduction to Logic, I owe a debt of gratitude too large to measure to Robert Vaught for his detailed and penetrating comments on the next-to-final draft. David Lipsich read the galley proofs and suggested several desirable changes most of which could be accommodated. Mrs. Louise Thursby has typed with accuracy and patience more versions of the text than I care to remember. Mrs. Blair McKnight has been of much assistance in reading proofs and preparing the indexes. Any errors which remain are my sole responsibility.

      PATRICK SUPPES

      Stanford, California

      January, 1960

      TABLE OF CONTENTS

      CHAPTER

      1.INTRODUCTION

      1.1Set Theory and the Foundations of Mathematics

      1.2Logic and Notation

      1.3Axiom Schema of Abstraction and Russell’s Paradox

      1.4More Paradoxes

      1.5Preview of Axioms

      2.GENERAL DEVELOPMENTS

      2.1Preliminaries: Formulas and Definitions

      2.2Axioms of Extensionality and Separation

      2.3Intersection, Union, and Difference of Sets

      2.4Pairing Axiom and Ordered Pairs

      2.5Definition by Abstraction

      2.6Sum Axiom and Families of Sets

      2.7Power Set Axiom

      2.8Cartesian Product of Sets

      2.9Axiom of Regularity

      2.10Summary of Axioms

      3.RELATIONS AND FUNCTIONS

      3.1Operations on Binary Relations

      3.2Ordering Relations

      3.3Equivalence Relations and Partitions

      3.4Functions

      4.EQUIPOLLENCE, FINITE SETS, AND CARDINAL NUMBERS

      4.1Equipollence

      4.2Finite Sets

      4.3Cardinal Numbers

      4.4Finite Cardinals

      5.FINITE ORDINALS AND DENUMERABLE SETS

      5.1Definition and General Properties of Ordinals

      5.2Finite Ordinals and Recursive Definitions

      5.3Denumerable Sets

      6.RATIONAL NUMBERS AND REAL NUMBERS

      6.1Introduction

      6.2Fractions

      6.3Non-negative Rational Numbers

      6.4Rational Numbers

      6.5Cauchy Sequences of Rational Numbers

      6.6Real Numbers

      6.7Sets of the Power of the Continuum

      7.TRANSFINITE INDUCTION AND ORDINAL ARITHMETIC

      7.1Transfinite Induction and Definition by Transfinite Recursion

      7.2Elements of Ordinal Arithmetic

      7.3Cardinal Numbers Again and Alephs

      7.4Well-Ordered Sets

      7.5Revised Summary of Axioms

      8.THE AXIOM OF CHOICE

      8.1Some Applications of the Axiom of Choice

      8.2Equivalents of the Axiom of Choice

      8.3Axioms Which Imply the Axiom of Choice

      8.4Independence of the Axiom of Choice and the Generalized Continuum Hypothesis

      REFERENCES

      GLOSSARY OF SYMBOLS

      AUTHOR INDEX

      SUBJECT INDEX

      CHAPTER 1

      INTRODUCTION

      § 1.1 Set Theory and the Foundations of Mathematics. Among the many branches of modern mathematics set theory occupies a unique place: with a few rare exceptions the entities which are studied and analyzed in mathematics may be regarded as certain particular sets or classes of objects.* This means that the various branches of mathematics may be formally defined within set theory. As a consequence, many fundamental questions about the nature of mathematics may be reduced to questions about set theory.

      The working mathematician, as well as the man in the street, is seldom concerned with the unusual question: What is a number? But the attempt to answer this question precisely has motivated much of the work by mathematicians and philosophers in the foundations of mathematics during the past hundred years. Characterization of the integers, rational numbers and real numbers has been a central problem for the classical researches of Weierstrass, Dedekind, Kronecker, Frege, Peano, Russell, Whitehead, Brouwer, and others. Perplexities about the nature of number did not originate in the nineteenth century. One of the most magnificent contributions of ancient Greek mathematics was Eudoxus’ theory of proportion, expounded in Book V of Euclid’s Elements; the main aim of Eudoxus was to give a rigorous treatment of irrational quantities like the geometric mean of 1 and 2. It may indeed be said that the detailed development from the general axioms of set theory of number theory and analysis is very much in the spirit of Eudoxus.

      Yet the real development of set theory was not generated directly by an attempt to answer this central problem of the nature of number, but by the researches of Georg Cantor around 1870 in the theory of infinite series and related topics of analysis.* Cantor, who is usually considered the founder of set theory as a mathematical discipline, was led by his work into a consideration of infinite sets or classes of arbitrary character. In 1874 he published his famous proof that the set of real numbers cannot be put into one-one correspondence with the set of natural numbers (the non-negative integers). In 1878 he introduced the fundamental notion of two sets being equipollent or having the same power (Mächtigkeit) if they can be put into one-one correspondence with each other. Clearly two finite sets have the same power just when they have the same number of members. Thus the notion of power leads in the case of infinite sets to a generalization of the notion of a natural number to that of an infinite cardinal number. Development of the general theory of transfinite numbers was one of the great accomplishments of Cantor’s mathematical researches.

      Technical consideration of the many basic concepts of set theory introduced by Cantor will be given in due course. From the standpoint of the foundations of mathematics the philosophically revolutionary aspect of Cantor’s work was his bold insistence on the actual infinite, that is, on the existence of infinite sets as mathematical objects on a par with numbers and finite sets. Historically the concept of infinity has played a role in the literature of the foundations of mathematics as important as that of the concept of number. There is scarcely a serious philosopher of mathematics since Aristotle who has not been much exercised about this difficult concept.

      Any book on set theory is naturally expected to provide an exact analysis of the concepts of number and infinity. But other topics, some controversial and important in foundations research, are also a traditional part of the subject and are consequently treated in the chapters that follow. Typical are algebra of sets, general theory of relations, ordering relations in particular, functions, finite sets, cardinal numbers, infinite sets, ordinal arithmetic, transfinite induction, definition by transfinite recursion, axiom of choice, Zorn’s Lemma. At this point the reader is not expected to know what these phrases mean, but such a list may still give a clue to the more detailed contents of this book.

      In this book set theory is developed axiomatically rather than intuitively. Several considerations have guided the choice of an axiomatic approach. One is the author’s opinion that the axiomatic development of set theory is among the most impressive accomplishments of modern mathematics. Concepts which were vague and unpleasantly inexact for decades and sometimes even centuries can be given a precise meaning. Adequate axioms for set theory provide one clear, constructive answer to the question: Exactly what assumptions, beyond those of elementary logic, are required as a basis for modern mathematics? The most pressing consideration, however, is the discovery, made around 1900, of various paradoxes in naive, intuitive set theory, which admits the existence of sets of objects having any definite property whatsoever. Some particular restricted axiomatic approach is needed to avoid these paradoxes, which are discussed in §§ 1.3 and 1.4 below.

      § 1.2 Logic and Notation. We shall use symbols of logic extensively for purposes of precision and brevity, particularly in the early chapters. But proofs are mainly written in an informal style. The theory developed is treated as an axiomatic theory of the sort familiar from geometry and other parts of mathematics, and not as a formal logistic system for which exact rules of syntax and semantics are given. The explicitness of proofs is sufficient to make it a routine matter for any reader familiar with mathematical logic to provide formalized proofs in some standard system of logic. However, familiarity with mathematical logic is not required for understanding any part of the book.

      At this point we introduce the few logical symbols which will be used. We first consider five symbols for the five most common sentential connectives. The negation of a formula P is written as – P. The conjunction of two formulas P and Q is written as P & Q. The disjunction of P and Q as P V Q. The implication with P as antecedent and Q as consequent as P ↔ Q. The equivalence P if and only if Q as P → Q. The universal quantifier For every v as (∀v), and the existential quantifier For some v as (∃v). We also use the symbol (E!v) for There is exactly one v such that. This notation may be summarized in the following table.

      LOGICAL NOTATION

      Thus the sentence:

      For every x there is a y such that x < y

      is symbolized:

      The sentence:

      there is a δ such that for every y

      if |x − y| < δ then |f(x) − f(y

      is symbolized:

      The sentence:

      For every x there is exactly one y such that x + y = 0

      is symbolized:

      (∀x)(E!y)(x + y = 0).

      A given logical symbol may correspond to several English idioms. Thus (∀v)P may be read For all v, P as well as For every v, P. Sentences (1) and (2) illustrate the use of parentheses for purposes of punctuation. No formal explanation seems necessary. However, one convention concerning the relative dominance of the sentential connectives &, V, → and will ↔ reduce considerably the number of parentheses. The convention is that ↔ and → dominate & and V. Thus, the formula:

      (x < y & y < z) → x < z

      may be written without parentheses:

      Similarly,

      x + y ≠ 0 ↔ (x ≠ 0 v y ≠ 0)

      may be written:

      x + y ≠ 0 ↔ x ≠ 0 v y ≠ 0

      Principles of logic which are needed in the sequel and which may not be familiar to some readers will be intuitively explained when used. One principle used, concerning which there is some disagreement in practice among mathematicians, is that the double bar ‘=’ is taken as the sign of identity. The formula ‘x = y’ may be read ‘x is the same as y’, ‘x is identical with y’ or ‘x is equal to y’. The last reading is permissible here only if it is understood that equality means sameness of identity (which is what it does mean in almost all ordinary mathematical contexts). The exact status of the relation of identity within set theory is discussed in §2.2.

      A few remarks concerning quantifiers may also be helpful. The scope of a quantifier is the quantifier itself together with the smallest formula immediately following the quantifier. What the smallest formula is, is always indicated by parentheses. Thus in the formula

      the scope of the quantifier ‘(∃x)’ is the formula ‘(∃x)(x < y)’. Following an almost universal practice in mathematics, we shall omit, in the formulation of axioms and theorems, any universal quantifier whose scope is the whole formula. For instance, instead of (1) above, we would write: (∃y) (x < y).

      In a few places we shall need the notions of bound and free variables. An occurrence of a variable in a formula is bound if and only if this occurrence is within the scope of a quantifier using this variable. An occurrence of a variable in a formula is free if not bound. Finally, a variable is a bound variable in a formula if and only if at least one occurrence is bound; it is a free variable in a formula if and only if at least one occurrence is free. In formula (1) of this section all variables are bound; in (3) all variables are free; in (4) ‘x’ is bound and ‘y’ is free. By virtue of the convention stated in the preceding paragraph concerning omission of universal quantifiers in axioms and theorems, all variables occurring in axioms and theorems are bound.

      § 1.3 Axiom Schema of Abstraction and Russell’s Paradox. In his initial development of set theory, Cantor did not work explicitly from axioms. However, analysis of his proofs indicates that almost all of the theorems proved by him can be derived from three axioms: (i) The axiom of extensionality for sets, which asserts that two sets are identical if they have the same members; (ii) the axiom of abstraction, which states that given any property there exists a set whose members are just those entities having that property; (iii) the axiom of choice, which will not be formulated at this point and is not pertinent to our discussion of the paradoxes.

      The source of trouble is the axiom of abstraction. The first explicit formulation of it seems to be as Axiom V in Frege [1893]. In 1901 Bertrand Russell discovered that a contradiction could be derived from this axiom by considering the set of all things which have the property of not being members of themselves.* Because this paradox was historically important in motivating the development of new, restricted axioms for set theory, its derivation will be given here. For symbolic formulation we need to introduce the binary predicate ‘∈’ of set membership. The formula ‘x∈y’ is read ‘x is a member of y’, ‘x belongs to y’ or sometimes, ‘x is in y’ Thus, if A is the set of first five odd positive integers, the sentence ‘7∈A’ is true and ‘6∈A’ is false.

      Using ‘∈’ and the logical notation introduced in the previous section, we may give a precise formulation of the axiom of abstraction:

      where it is understood that φ(x) is a formula in which the variable ‘y’ is not free. To obtain Russell’s paradox, we want φ(x) to assert that x is not a member of itself. The appropriate formula is clearly:

      –(x ∈ x).

      We then have as an instance of the axiom of abstraction:

      Taking x = y in (2), we infer:

      which is logically equivalent to the contradiction:

      This simple derivation has far-reaching consequences for the axiomatic foundations of set theory. It plainly shows that in admitting (1) as an axiom we have granted too much. If we adhere to ordinary logic we cannot in a self-consistent manner claim that for every property there is a corresponding set of things having that property.

      In considering how to build anew the foundations of set theory, perhaps the first thing to notice is that the axiom of abstraction is really an infinite bundle of axioms rather than a single axiom: when we replace the expression ‘φ(x)’ in (1) by any formula in which ‘y’ is not a free variable, we have a new axiom. An axiom which permits this sort of blanket substitution of formulas is usually called an axiom schema. The reason for using the word ‘schema’ should be obvious. As it stands (1) is not a definite assertion, but a scheme for making many assertions. From the schema we obtain a definite assertion by substituting a definite formula for φ(x)’.

      The axiom schema which we shall use is due to Ernst Zermelo [1908], and is usually called the axiom schema of separation (Aussonderung Axiom) because it permits us to separate off the elements of a given set which satisfy some property and form the set consisting of just these elements. Thus if we know that the set of animals exists, we may use the axiom schema of separation to assert the existence of the set of animals which have the property of being men. That is, the property of being human enables us to separate men from other animals. Corresponding to (1) the precise form of the axiom is:

      The change from (1) to (5) is slight but potent. (1) asserted the existence of sets unconditionally. (5), on the other hand, is completely conditional; first we have to be given the set z, then we can assert the existence of the subset y.

      It should be clear that we cannot pass from (5) to a contradiction like (4). Using again the formula ‘−(x ∈ x)’ as an instance of (5) we have:

      and again taking x = y, we infer:

      which is not contradictory. To make the meaning of (7) a little clearer let z be the set A whose only two members are the set consisting of the number 1 and the set consisting of the number 2, that is,

      (In (8) we have informally introduced a familiar notation for describing sets: we describe a set by writing down names or descriptions of its members, separated by commas, and enclosing the whole in braces. In the next chapter this notation is formally defined.) Considering now the set A and Russell’s formula ‘−(x ∈ x)’, we have from the axiom schema of separation:

      The truth of (9) is seen by choosing A itself as an appropriate y, for A is not a member of itself. Thus the left-hand side is false, and the right-hand side is also false, since ‘A ∈ A & −(A ∈ A)’ is contradictory.

      Both the axiom schema of abstraction and the axiom schema of separation have been stated as though it were perfectly clear exactly what formulas may be substituted for ‘φ(x)’. In the next chapter a precise syntactical definition of formula is considered. What has been important historically is that it is by means of an exact definition of the formulas of a theory (here set theory) that application of an axiom schema like that of separation may be made precise. Zermelo [1908] originally formulated the axiom schema of separation in terms of questions or statements which have the property of being definite. Roughly speaking, he said that a statement is definite if it can be decided in a non-arbitrary manner whether or not any object satisfies the statement.* His formulation of the axiom schema is then (slightly paraphrased): If a statement φ(x) is definite for all elements of a set M, then there is always a subset Mφ of M which contains exactly those elements x of M for which φ(x) is true.

      The first real clarification of this notion of definiteness was given by Skolem [1922], who characterizes definite statements as just those which satisfy

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