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    Basic Set Theory
    Basic Set Theory
    Basic Set Theory
    Ebook797 pages16 hours

    Basic Set Theory

    By Azriel Levy

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    Although this book deals with basic set theory (in general, it stops short of areas where model-theoretic methods are used) on a rather advanced level, it does it at an unhurried pace. This enables the author to pay close attention to interesting and important aspects of the topic that might otherwise be skipped over. Written for upper-level undergraduate and graduate students, the book is divided into two parts. The first covers pure set theory, including the basic notions, order and well-foundedness, cardinal numbers, the ordinals, and the axiom of choice and some of its consequences. The second part deals with applications and advanced topics, among them a review of point set topology, the real spaces, Boolean algebras, and infinite combinatorics and large cardinals. A helpful appendix deals with eliminability and conservation theorems, while numerous exercises supply additional information on the subject matter and help students test their grasp of the material. 1979 edition. 20 figures.
    LanguageEnglish
    PublisherDover Publications
    Release dateJun 11, 2012
    ISBN9780486150734
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      Basic Set Theory - Azriel Levy

      DOVER BOOKS ON MATHEMATICS

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      POINT SET TOPOLOGY, STEVEN A. GAAL. (0-486-47222-1)

      DISCOVERING MATHEMATICS: THE ART OF INVESTIGATION, A. GARDINER. (0-486-45299-9)

      LATTICE THEORY: FIRST CONCEPTS AND DISTRIBUTIVE LATTICES, GEORGE GRÄTZER. (0-486-47173-X)

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      COMPANION TO CONCRETE MATHEMATICS: MATHEMATICAL TECHNIQUES AND VARIOUS APPLICATIONS, Z. A. MELZAK. (0-486-45781-8)

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      Copyright

      Copyright © 1979 by Springer-Verlag Berlin Heidelberg

      All rights reserved.

      Bibliographical Note

      This Dover edition, first published in 2002, is an unabridged republication of the work published by Springer-Verlag, Berlin, Heidelberg, and New York, in 1979 as part of its series Perspectives in Mathematical Logic. Corrections and Additions, Additional Bibliography and a Preface to the Dover Edition have been prepared especially for this edition by the author.

      AMS Subject Classification (1970): 04-02, 04-01, 04A15, 04A20, 04A25, 04A30, 02K35

      Library of Congress Cataloging-in-Publication Data

      Levy, Azriel.

      Basic set theory / Azriel Levy.

      p. cm.

      Originally published: Berlin : New York: 1979, in series:

      Perspectives in Mathematical Logic.

      Includes bibliographical references and indexes.

      9780486150734

      1. Set theory. I. Title.

      QA248 .L398 2002

      511.3’22—dc21

      2002022292

      Manufactured in the United States by Courier Corporation

      42079503

      www.doverpublications.com

      To the memory of my parents Sarah and Shlomo Levy

      The Lord by wisdom founded the earth, by understanding he established the heavens.

      Proverbs 3:19

      Preface to the Dover Edition

      Much has happened in set theory since the original edition of this book. Naturally, this has happened not in the basic areas covered by the book but in more advanced matters. Therefore this book is almost as up-to-date as it was originally, but the road from it to the current research has become much longer. Among the books which cover parts of this road are Jech 1978 and Kunen 1980 in general set theory, Kanamori 1994 in large cardinals, Moschovakis 1980 in descriptive set theory, and Shelah 1994 on cardinal numbers.

      The original edition contained its fair share of misprints and mistakes. All those of which I am aware are listed in the Corrections and Additions appendix, which also contains some additional historical references and a small number of exercises based on the material in the book. An Additional Bibliography also follows the original Bibliography.

      The author owes much to all his colleagues who pointed out the improvements needed. Since this new edition is a reprint and not a fully revised, newly edited edition, it has been possible to implement only some of the suggestions. The most useful were supplied by K. Carmodi, W Hodges, A. Kanamori, and G. Moran, to whom I owe a great debt.

      I hope this edition will reach more students and will help them to learn this beautiful area of mathematics.

      March 25, 2002

      A. Levy

      Jerusalem

      Preface

      Almost all the recently-published books on set theory are of one of the following two kinds. Books of the first kind treat set theory on an elementary level which is, roughly, the level needed for studying point set topology and Steinitz’s theorem on the existence of the algebraic closure of a general field. Books of the second kind are books which give a more or less detailed exposition of several areas of set theory that are subject to intensive current research, such as constructibility, forcing, large cardinals and determinacy. Books of the first kind may serve well as an introduction to the subject but are too elementary for the student or the mathematician who wants to gain a deeper understanding of set theory. The books of the second kind usually go hurriedly through the basic parts of set theory in their justified haste to get at the more advanced topics. One of the advantages of writing a book in a series such as the Perspectives in Mathematical Logic is that one is able to write a book on a rather advanced level covering the basic material in an unhurried pace. There is no need to reach the frontiers of the subject as one can leave this to other books in the series. This enables the author to pay close attention to interesting and important aspects of the subject which do not lie on the straight road to the very central topics of current research.

      I started writing this book in 1970. During the long period since that time I have been helped by so many people that I cannot name them all here. Several of my colleagues advised me on the material in the book, read parts of the manuscript and made very useful remarks, and taught me new theorems and better proofs of theorems I knew. Many typists typed the numerous versions of the manuscript and bore with admirable patience all my inconsistent instructions. I shall mention in name only Klaus Glöde and Uri Avraham to whom I am most grateful for diligently reading the galley proofs, correcting many misprints and mistakes. This book would not have been written without the initiative and encouragement of my colleagues in the Ω group. I enjoyed very much their company and collaboration.

      I shall be most grateful to any reader who will point out misprints, mistakes and omissions and who will supply me with additional bibliographical references. This will hopefully be incorporated in later printings of this book.

      Most of this book was written while I stayed as a visitor at Yale University in the academic year 1971–72 and at UCLA in 1976–77. I extend my thanks to the National Science Foundation of the United States for partially supporting me during those years.

      May 12, 1978

      Jerusalem

      A. Levy

      Table of Contents

      DOVER BOOKS ON MATHEMATICS

      Title Page

      Copyright Page

      Dedication

      Preface to the Dover Edition

      Preface

      Introduction

      Part A - Pure Set Theory

      Chapter I - The Basic Notions

      Chapter II - Order and Well-Foundedness

      Chapter III - Cardinal Numbers

      Chapter IV - The Ordinals

      Chapter V - The Axiom of Choice and Some of its Consequences

      Part B - Applications and Advanced Topics

      Chapter VI - A Review of Point Set Topology

      Chapter VII - The Real Spaces

      Chapter VIII - Boolean Algebras

      Chapter IX - Infinite Combinatorics and Large Cardinals

      Appendix X - The Eliminability and Conservation Theorems

      Bibliography

      Additional Bibliography

      Index of Notation

      Index

      Appendix: Corrections and Additions

      Introduction

      are introduced, but their respective consistencies are not proved.

      A basic tacit assumption for almost all the metamathematical results in this book is that the Zermelo-Fraenkel set theory ZF is consistent. For example, when we say that the axiom V = L of constructibility is consistent with ZF we mean that assuming ZF is consistent then ZF with V = L is consistent as well.

      This book contains few historical remarks, since the history of the subject alone is material enough for a whole book. Nevertheless, to give the reader some feeling of the history of the subject, the original authors and publications are quoted whenever this information was available. There are of course theorems which were known to their discoverers a long time before they were published, but in most cases all that is given here is the publication reference. Gödel 1939 refers to the bibliographical item listed as such in the bibliography, but Gödel in 1939 means that the theorem next to which this is written was proved by Gödel in 1939. Theorems are often given here not in their original form but in a form which comprises improvements by several mathematicians. In some of these cases later contributors are mentioned beside the original one, but no uniformity of reference has been achieved.

      When we use in Chapter V, for example, the reference 4.29(i) we refer to part (i) of the 29th subsection of Section 4 of Chapter V, which is the subsection marked 4.29. When we refer to II.2.4 we mean Subsection 2.4 in Chapter II. When we say inside III.2.35 by II.2.14(i) and (ii) we mean by II.2.14(i) and part (ii) of the present subsection (which is III.2.35(ii)) if we want to refer to parts (i) and (ii) of II.2.14 we say by II.2.14(i) and II.2.14(ii).

      .

      Most of the exercises supply additional information on the subject matter and are therefore an integral part of the discussion. The exercises are usually followed by hints so as to make their solution not too difficult.

      Part A

      Pure Set Theory

      Chapter I

      The Basic Notions

      All branches of mathematics are developed, consciously or unconsciously, in set theory or in some part of it. This gives the mathematician a very handy apparatus right from the beginning. The most he usually has to do in order to have his basic language ready is to describe the set theoretical notation he uses. In developing set theory itself we have no such advantage and we must go through the labor of setting up our set theoretical apparatus. This is a relatively long task. Even the question as to which objects to consider, only sets or also classes, is by no means trivial, and its implications will be discussed here in detail. In addition, we shall formulate the axioms of set theory; we shall show how the concepts of ordered pair, relation and function, which are so basic in mathematics, can be developed within set theory, and we shall study their most basic properties. By the end of this chapter we shall be just about ready to begin our real mathematical investigation of the universe of sets.

      1. The Basic Language of Set Theory

      In the present section and in Sections 3 and 4 we shall thoroughly discuss the language we are going to use for set theory. Usually when one studies a branch of mathematics one is not concerned much with the question as to which exactly is the language used in that branch. The reason why here we must look carefully at the language lies in the difference between set theory and most other branches of mathematics. Most mathematical fields use a relatively small fragment of set theory as their underlying theory, and rely on that fragment for the language, as well as for the set theoretical facts. The source of the difficulty we have in set theory with the language is the fact that not every collection of objects is a set (something which will be discussed in detail in the next few sections), but we still have to refer often to these collections, and we have to arrange the language so that we shall be able to do it handily. This difficulty does not come up in the fragments of set theory used for most mathematical theories, since those fragments deal only with a very restricted family of collections of objects, and all these collections are indeed sets.

      Our present aim is to obtain for set theory a language which is sufficiently rich and flexible for the practical development of set theory, and yet sufficiently simple so as not to stand in the way of metamathematical investigation of set theory. For this purpose we start by choosing for set theory a very simple basic language. The simplicity of this language will be a great advantage when we wish to discuss set theory from a metamathematical point of view. The only objects of our set theory will be sets. One could also consider atoms, i.e., objects which are not sets and which serve as building blocks for sets, but they are not essential to what we shall do and, therefore, will not be considered in the present book. As a consequence of this decision, we view the sets as follows. We start with the null set 0, from it we obtain the set {0}, from the two sets 0 and {0} we obtain the sets {0, {0}} and {{0}}, and so on. Much of set theory is concerned with what is meant by this and so on.

      The language which we shall use for set theory will be the first-order predicate calculus with equality. Why first-order? Because a second-order or a higher-order theory admits already a part of the set theory in using its higher order variables. To see, for example, that second-order variables are essentially set variables let us consider the following axiom of second-order logic: ∃A∀x(x ∈ A ↔ Φ(x)), where Φ(x) is, essentially, any formula. This axiom is read: there exists a set A such that for every x, x is a member of A if and only if Φ(x). It would, of course, change nothing if we would choose another term instead of set, since set is what we mean anyway. When we develop a formal system of set theory it does not seem right to handle sets in two or more parts of the language, i.e., by considering some sets as first-order objects, while having around also second-order objects which are sets. As a consequence of our decision we shall have, in principle, just one kind of variable, lower case letters, which will vary over sets.

      The reason why we take up first-order predicate calculus with equality is a matter of convenience; by this we save the labor of defining equality and proving all its properties; this burden is now assumed by the logic.

      Our basic language consists now of all the expressions obtained from x=y and x ∈ y, where x and y (or), ∧ (and), ↔ (if and only if), and the quantifiers ∃x (there exists an x) and ∀x (for all x). These expressions will be called formulas. For metamathematical purposes we can consider the connectives ¬ and v as the only primitive connectives, and the other connectives will be considered as obtained from the primitive connectives in the well known way (e.g., φ ∧ ψ is ¬ (¬φ∨ ¬ψ), φ → ψ is ¬φ ∨ ψ, etc.). For the same reason, we can consider ∃ as the only primitive quantifier and the quantifier ∀ as defined by means of ∃ by taking ∀x to be an abbreviation of ¬ ∃x . We shall also use the abbreviations x ≠ y and x ∉ y for ¬ x = y and ¬ x ∈ y. We shall write ∃ ! x , and read: there is exactly one x for the formula ∃y∀x(x = y ), where y . Finally, we shall write (∃x ∈ yand (∀x ∈ yfor ∃x(x ∈ y ) and ∀x(x ∈ y ), respectively, and read: "there is an x in y , and for all x in y".

      By a free variable of a formula we mean, informally, a variable occurring in that formula so that it can be given different values and the formula says something concerning the values of the variable. E.g., x is a free variable in each of the following formulas (which are not necessarily taken from set theory): x < 3x, x² = y, x , ∀y(z < y → x < y). x is not a free variable in the formulas ∀x(x² ≥ 0), ¬ ∃x(x ∈ y. In the latter three formulas x is an auxilliary variable which cannot be given a definite value and which can be replaced throughout each formula by another variable, say z, without changing the meaning of the formula. In these examples x is used as a bound variable. Note that ∀x(x² ≥ 0) says exactly what ∀z(z; in fact, for appropriate values of x and z may be false. A variable may have both free and bound occurrences in the same formula, even though one would usually try to avoid it; e.g., in 7 < z ∧ ∃z(z > x), the occurrences of z in ∃z(z > x) are bound, while the occurrence of z in 7 < z is free (since the quantifier ∃z applies only to z > x).

      A formula with free variables says something about the values of its free variables. A formula without free variables makes a statement not about the value of some particular variable, but about the universe which the language describes. A formula of the latter kind is called a sentence. We shall also refer, informally, to formulas and sentences as statements.

      Whenever we use a formula with free variables as an axiom or as a theorem we mean to say that the formula holds for all possible values given to its free variables. Thus, if we state a theorem ∃z(z = x∪y) we mean the same thing as ∀x∀y∃z(z = x∪y).

      By a theory we mean a set of formulas, which are called the axioms of the theory. If T is a theory, we shall write T φ for "φ is provable from T".

      When we refer to a formula as φ(x) this does not mean that x is necessarily a free variable of φ(x) nor does it mean that φ(x) has no free variables other than x; it means that the interesting cases of what we shall say are those where x is indeed a free variable of φ(x). When we shall mention φ(z) after we have first mentioned φ(x), then φ(z) denotes the formula obtained from φ(x) by substituting the variable z for the free occurrences of x. (z may also be a bound variable of φ(x), and then before we substitute z for the free occurrences of x we may have to replace the bound occurrences of z by some other variable.)

      2. The Axioms of Extensionality and Comprehension

      By a set we mean a completely structure-free set, and therefore a set is determined solely by its members. This leads us to the first axiom of set theory.

      2.1 Axiom of Extensionality (Frege 1893). ∀x(x ∈ y ↔ x ∈ z) → y = z.

      In words: if y and z have the same members they are equal. The converse, that equal objects have the same members, is a logical truth.

      2.2 The Existence of Sets. Now we face the question of finding or constructing the sets. We want any collection whatsoever of objects, i.e., sets, to be a set. This is not a precise idea and therefore we cannot translate it into our language. We must therefore be satisfied with a somewhat weaker stipulation. We shall require that every collection of sets which is specifiable in our language is a set; i.e., for every statement of our language the collection of all objects which satisfy it is a set. We shall by no means assume that it is necessarily true that all sets are specifiable ; moreover, by introducing the axiom of choice we shall require the existence of sets which are not necessarily specifiable. The requirement that all specifiable collections are indeed sets is the following one.

      2.3 Axiom of Comprehension (Frege 1893). ∃y∀x(x∈y (x(x) is any formula (of the language of set theory) in which the variable y (x(x) with the y (xdoes actually contain free occurrences of the variable x.

      The axiom of comprehension is an axiom schema, in 2.3 is said to be an instance of the axiom schema, and is also called an axiom of comprehension.

      Those readers who were convinced by the axiom schema of comprehension are now in for a shock; the axiom schema of comprehension is not consistent — Theorem 2.4 below is the negation of one of its instances.

      2.4 Theorem (Russell’s antinomy — Russell 1903). ¬ ∃y∀x(x ∈ y ↔ x ∉ x).

      Proof. Notice that this theorem is not just a theorem of set theory; it is a theorem of logic, since we do not use in its proof any axiom of set theory. We prove it by contradiction. Suppose y is a set such that ∀x(x ∈ y ↔ x ∉ x), then, since what holds for every x holds in particular for y, we have y ∈ y ↔ y ∉ y, which is a contradiction.

      Russell’s antinomy is the simplest possible refutation of an instance of the comprehension schema. We refer to a refutation of such an instance as an antinomy. The first antinomy to be discovered is the Burali-Forti paradox discovered by Cantor and by Burali-Forti in the 1890’s; it is given in II.3.6 and II.3.15. Some variants of Russell’s antinomy are given in 2.5.

      2.5 Exercise. Prove the negation of the instance of the axiom of comprehension where φ(x) is one of the following formulas:

      (a) ¬ ∃u(x ∈ u ∧ u ∈ x),

      (b) ¬∃u1 . . .∃un(x ∈ u1 ∧ u1 ∈ u2 ∧ · · · ∧ un–1 ∈ un ∈ un ∧ un ∈ x).

      2.6 How to Avoid the Antinomies. (x), and this cannot be done since we can collect only those sets which have been obtained at an earlier stage of the game. This point of view was suggested first by Russell 1903 as one of the ingredients of his theory of types. The other possible reaction to Russell’s antinomy is to continue believing in the essential truth of the axiom schema of comprehension, viewing the Russell antinomy as a mere practical joke played on mankind by the goddess of wisdom. According to this point of view the axiom schema of comprehension is only in need of some tinkering to avoid the antinomies; the guide on how to do it will be the doctrine of limitation of size. The doctrine says that we should use the axiom schema of comprehension only in order to obtain new sets which are not too large compared to the sets whose existence is assumed in the construction. Also this doctrine, which is already implicit in Cantor 1899, was formulated first by Russell 1906. In our framework of set theory both approaches lead to the same result, and therefore there is no mathematical need to go through the arguments in favor of each one of them. Motivations for the choice of the axioms, from both points of view, are presented in the literature (see, e.g., Fraenkel, Bar-Hillel and Levy 1973 and Scott 1974) and will hopefully be presented in a later book in this series devoted to the axiomatics of set theory. Here we shall mostly rely on the acceptance by the reader of the axioms which we shall introduce as intuitively reasonable axioms.

      , there is a set y such that ∀x(x ∈ y (x)). However, if there is such a y it is unique, as stated in the next theorem.

      2.7 Proposition. If there is a y such that

      ∀x(x ∈ y (x))

      then this y is unique.

      Proof. If y′ is also such, i.e., ∀x(x ∈ y′ ↔ φ(x)), then we have, obviously, ∀x(x ∈ y′ ↔ x ∈ y), and by the axiom of extensionality, y′ = y.

      3. Classes, Why and How

      As we shall come to see, the main act of generation of set theory is that objects are collected to become a set, which is again an object which can be collected into a new set. We saw above that, because of the antinomies, not every collection of objects which can be specified in our language can be collected to become a new object. This is by no means disastrous for mathematics, since, by means of appropriate axioms which we shall introduce, we shall be able to show that sufficiently many of the intuitive collections can indeed be taken as sets to satisfy the mathematical needs. This will enable us to obtain sets such as the set of all real numbers, the set of all countable ordinals, the set of all measures on some given set, etc.

      There are many things we can say about an intuitive collection of objects without assuming that the collection is an object itself. Let us see an example. Suppose we want to say

      (1)

      This can also be said as.

      (2)

      Notice that (2) does not mention the collection mentioned in (1). We could decide never to use (1) and always to use (2) instead; but, as we shall point out now, and as will become even clearer to the reader as he goes on reading this book, this would have required a considerable sacrifice of convenience. Sometimes we want to say about many or all specifiable collections what we said in (1) about one particular collection. We can proceed as in (2) but this requires using an infinite family of formulas. This is illustrated by the following example. Suppose we want to say that for every specifiable collection A

      (3)

      (x, x1,..., xn)

      (4)

      To see that (3) and (4) say the same thing notice that the specifiable collections A are exactly those given as the collection of all objects x (x, x1,..., x(x, x1,..., xn) and for some fixed values of x1,..., xn. Comparing (3) with (4) shows that (3) is not only shorter but also much easier to comprehend than (4). Thus we see that it is a great advantage to be able to talk about collections as if they were sets even though we know, as a result of Russell’s antinomy, that not all of them are sets. A uniform way of talking about sets and collections has also the following advantage. We often come across collections which, at a certain point in the discussion, we do not know whether they are sets or not. Speaking about them in the same way as we speak about sets puts what we say about them in a form which retains its convenience even after the collections turn out to be sets.

      We shall now introduce into our language additional notation which will enable us to make our discussion of general specifiable collections look similar to the discussion of sets, while everything which we say by means of the additional notation can also be said without it, possibly only in a clumsy way. Thus, we shall be able to talk about general specifiable collections, which we want to do for the sake of convenience, without introducing them as objects of one kind or another.

      A general specifiable collection, which may or may not be a set, will be called a class(x) as the class of all objects x (x) holds. Such a class will be denoted by {x(x)}. The expression {x(x)} will be called a class term. The formula φ(x) may also contain free variables other than x. These other variables are called parameters. Different values of the parameters may yield different classes. For example, the class {x| x is a natural number ∧ x < y} is a class with no members if y = 0, has a single member if y = 1, and so on. Note also that since we took all specifiable collections to be classes, sets are classes too. In fact, the set y is the class {x|x ∈ y}.

      3.1 Class Terms. We shall extend our language by incorporating class terms also, but in such a way that every statement containing a class term will stand for some other, possibly longer, statement which does not contain class terms. We shall now write out what kind of statements containing class terms we allow, and for which statements containing no class terms these will stand. Since {x(x)} is the class of all x(x) holds, we shall take the statement y ∈ {x(x(y) (where φ(y(x) by proper substitution of y for x), i.e., "y belongs to the class of all x(x(y). Since we consider two sets with the same members to be equal, we shall also consider two classes with the same members as equal. Therefore we shall admit the statement {x(x)} = {y|ψ(y)} and let it stand for ∀z(z) ↔ ψ(z)). Since also the sets are classes, we admit also the statements x = {y(y)} and {y(y)} = x and let them stand for ∀z(z ∈ x (z)). To say that one class is a member of the other will mean that the first class is equal to a set which is member of the other. Accordingly, we admit the statement {x| (x)} ∈ {y|ψ(y)} and let it stand for ∃z(z = {x| (x)} ∧ z ∈ {y|ψ(y)}), and similarly we let the statement {x|φ(x)} ∈ y stand for ∃z(z = {x| (x)} ∧ z ∈ y).

      3.2 The Extended Language. We are now on our way to extending the language by adding notation for classes. This is a rather special way of extending the language. We shall also extend our language later, routinely, by new definitions as is done commonly in the development of any mathematical theory. All the extensions will be such that while they increase the ease of the use of the language, they do not add anything to the absolute expressive power of the language, since everything which can be expressed in the extended language is also expressible in the basic language of set theory. We shall examine this point again later. Let us only decide now to refer at every point throughout our book, to the language, as extended by various agreements and definitions up to that point, as the extended language, while by the basic language we shall always mean the basic language of set theory as outlined in Section 1. Note that while the term the basic language has a fixed meaning, the term the extended language does not, since as we go along we enrich the latter by new definitions. Unless otherwise mentioned, or implied, we shall use lower-case Greek letters for formulas of the basic language and capital Greek letters for formulas of the extended language.

      (x) in the class terms {x| (x)} not only formulas of the basic language, but also formulas of the extended language, i.e., formulas which contain also class terms. We have to tell now what do the formulas which contain the more general class terms {x|Φ(x)} stand for. It is easy to see that we can still use the same way of interpreting class terms as above, only that now more than one step may be needed in passing from the formula of the extended language to the corresponding formula of the basic language. Let us consider a simple example. We start with the statement:

      (1)

      By what we said in 3.1 about what z = {x| (x)} stands for, (1) stands for.

      (2)

      In (2), u ∈ {x|x ∈ {y(y)} ∨ x ∈ {y|ψ(y)}} stands for u ∈ {y|φ(y)} ∨ u ∈ {y|ψ(y)}, hence (1) stands for

      (3)

      We still have to eliminate in (3) the statements u ∈ {y| (y)} and u ∈ {y| ψ(y(u) and ψ(u), respectively. Thus, (1) stands for

      (4)

      This example suffices to show how to interpret the statements of the extended language. Later we will discuss the formal proof that such an elimination exists for every statement. In practice, none of the statements which we will use in this book will have an essentially more complicated nesting of class terms than (1) above, and hence, the reader will have no trouble at all in finding the formula of the basic language for which a given formula in our extended language stands.

      It should be noted that even though we have added now new class terms by allowing all the statements of the extended language inside the class terms, we did not add any classes; we have only more class terms referring to the same classes. Since every formula Φ(x) of the extended language is equivalent to some formula Φ(x) of the basic language, we get that every class {x|Φ(x)} is equal to some class {x|Φ(x)} (since {x|Φ(x)} = {x|Φ(x)} stands for ∀x(Φ(x(x)), which holds since Φ(x(x) are equivalent).

      Our second step is to add also class variables to our language. We shall use capital Roman letters for the class variables. The class variables will stand for arbitrary class terms and will enable us to make statements about all classes, while until now we were able to make statements essentially only about one class at a time or about a single family of classes given by a single class term with parameters. For example, if we state the theorem A = B → B = A we mean by it that for all Φ(x) and Ψ(y) we have

      (5)

      As we mentioned in 3.1, (5) stands for ∀z(Φ(z) ↔ Ψ(z)) → ∀z(Ψ(z) ↔ Φ(z)), which, in turn, stand for the schema ∀z(φ(z) ↔ ψ(z)) → ∀z(ψ(z) ↔ φ(z)) of formulas in the basic language. Thus, we saw that A = B → B = A stands not for a single formula of the basic language, but for a schema of such formulas. Since we allow the occurrence of class terms inside the formula Φ(x) of a class term {x |Φ(x)} we shall also allow the occurrence of class variables inside such a Φ(x). The role of a class variable in the formula Φ(x) of a class term {x |Φ(x)} is similar to the role of a set variable y which is a parameter of {x |Φ(x)}.

      3.3 Quantification of Class Variables. Note that the class variables occur only as free variables, i.e., we shall not use quantifiers like ∃X or ∀X over these variables. If we were to admit such quantifiers, we would get formulas which do not stand for any formula of the basic language, and our extended language would become a new language, and not just a convenient way of handling the statements of the basic language. However, the way we interpret a statement which contains a class variable, say A, is such that the statement is assumed to hold for all classes A (or, for a general A). As a consequence, we shall not hesitate to read A = B → B = A as "for all A and B, if A = B then B = A".

      3.4 A Historical and General Remark. The informal idea of using classes in set theory occurred already to Cantor 1899, once he became aware of the antinomies. An axiomatic system, using functions instead of classes, was presented first by Fraenkel 1922. This approach was brought to maturity by von Neumann 1925. An axiomatic system of set theory with classes was given first by Bernays 1937. The particular way in which we look here at classes has been stated explicitly first by Quine 1963, but its ideas were known earlier from the writings of Bernays (particularly Bernays 1958). Our way of looking at classes is by no means the only one. Many mathematicians, beginning with von Neumann, view the classes as objects as real as the sets, except that they are not members of other classes (unless they are sets). Systems of set theory based on that point of view are discussed in the literature (see, e.g., Levy 1976). However, the differences between such systems and the system with which we deal here turned out to be, from a mathematical point of view, only of a minor nature. Thus, once one accepts the axioms of ZF about the existence of sets, there is really only one underlying set theory.

      4. Classes, the Formal Introduction

      In §1 we have explained how to introduce class terms and class variables, considering them as abbreviated notation for formulas of the basic language. While we shall retain, in principle, the attitude that the class notations are just abbreviations for formulas of the basic language, we shall, nevertheless, follow Bernays 1958 and introduce a formal system in which class terms and class variables are expressions in their own right and not abbreviations of other expressions. The reason for doing this is that once we are anyway using in practice an extended language which is not the basic language there is nothing to be gained by not introducing the system we actually use as a formal system. This does not conflict with our attitude of regarding the expressions of the extended language as abbreviations of expressions of the basic language; this attitude will be justified by rules which will tell us how to interpret the formulas of the extended language in the basic language. We shall now first introduce the formal system for the extended language, and then dwell on its relationship to the basic language.

      4.1 The Formal System of the extended language, which admits classes as well as sets, is formulated in a two-sorted predicate calculus with equality, having capital letters as variables for classes and lower case letters as variables for sets. Notice that the class variables are now formal variables and not metamathematical symbols standing for class terms as in the last section. Atomic formulas are formed from the class terms and the class and set variables by means of the equality and membership relation symbols, i.e., atomic formulas are of the form a ∈ b or a = b, where each of a and b is either a class term or a class variable or a set variable. All other formulas , ∧, → and ↔ and by quantification over set variables only. Class terms are expression {x |Φ(x)} where Φ(x) is an arbitrary formula. The logic of this language consists of the axioms and rules of inference of the first order predicate calculus with equality for the set variables, the axioms and rules of inference of the free variable calculus with equality for the class variables, as well as the following rule of inference.

      4.2 The Rule of Substitution of Sets for Classes. From Φ(A) infer Φ(x).

      This rule says, essentially, that all sets are classes (since it says that whatever holds for all classes holds also for all sets).

      It is not necessary that the reader should memorize the logical axioms and rules of inference of the extended language. All one has to do is to use ordinary mathematical reasoning in the extended language, remembering that all sets are classes (but classes are not necessarily sets).

      4.3 The Basic Extralogical Axioms of the Extended Theory, including the axiom of extensionality, are as follows.

      y ∈ {x |Φ(x)} ↔ Φ(y)

      A = B ↔ ∀x(x ∈ A ↔ x ∈ B)

      A ∈ B ↔ ∃x(x = A ∧ x ∈ B)

      The statements A = y ↔ ∀x(x ∈ A ↔ x ∈ y), y = z ↔ ∀x(x ∈ y ↔ x ∈ z) and A ∈ y ↔ ∃x(x = A ∧ x ∈ y) are consequences of the axioms 4.3 by the rule 4.2. The first and third axioms of 4.3 are not proper axioms of set theory since they convey no information concerning sets. The second axiom of 4.3 is partly a proper axiom of set theory since it includes the axiom of extensionality.

      4.4 Terminology. We shall use A is a set as an abbreviation for ∃x(x = A). We shall say "A is a proper class for the negation of A is a set", i.e., for ¬∃x(x=A). We shall also write {x ∈ A |Φ(x)} for {x| x ∈ A ∧ Φ(x)}.

      Let us denote by P the theory formulated in the basic language, with the axiom of extensionality as its only extralogical axiom. Let P* be the theory formulated here in the extended language with the axioms 4.3 in addition to the logical axioms. We claimed in §1, informally, that in P* we are able to do only what we can already do in P, with the difference that in the extended language we can handle by means of one formula infinitely many formulas of the basic language, using class variables. When we say that we are able to do in P* only what we can do already in P, we mean two things by the word do, namely, to express and to prove. Our claim is fully established by the following two theorems.

      4.5 The Eliminability Theorem. (Comparison of the expressive power of the extended language and the basic language.) Let Φ(A1,..., An) be a formula of the extended language with no class variables other than indicated. Let {x1| Ψ(x1)}, ..., {xn |Ψn(xn)} be class terms which do not contain class variables. Then there is a formula φ of the basic language such that

      PΦ({x1 | Ψ1(x1)},...,{xn| Ψn(xn.

      Such a formula is said to be a basic instance of Φ(A1, ..., An) corresponding to the class-terms {xi| Ψi(xi)}, i = 1, ..., n.

      Remark. We had to require in 4.5 that the class terms {xi |Ψi(x)} contain no class variables in order that the formula Φ({x1| Ψ1(x1)},..., {xn|Ψn(xn)}) contain no class variables, which it must if it is to be equivalent to a single formula of the basic language.

      The eliminability theorem will be proved in the appendix (X.1.3). Let us only note that it says that whatever Φ(A1,..., An) says about the particular classes Ai = {xi |Ψi(xi)}, i = 1,..., n, can be said by a basic instance φ of it, which is in the basic language.

      4.6 The Conservation Theorem. (Comparison of the deductive power of corresponding axiom systems in the basic language and in the extended language.)

      (i) Let be any formula of the basic language. P* iff P .

      (ii) Let T be a theory formulated in the basic language and T* a theory formulated in the extended language such that for every formula of the basic language T* iff T . Then, if Q is any set of axioms of the basic language we have, again, for every φ of the basic language T*∪Q iff T∪Q .

      (iii) If T and T* are as in (ii), Ψ(A) is a formula of the extended language having no class variables other than A, and R is a set of basic instances of Ψ(A) (or of formulas equivalent in T to basic instances of Ψ(A)) which contains for each class term {x |χ(x)} a corresponding basic instance of Ψ(A) (or a formula equivalent to it in T), then we have for every φ of the basic language T*∪{Ψ(Aiff T∪R φ.

      Partial proof. Parts (i) and (iii) of this theorem will be proved in the appendix (X.1.4 and X.1.5). Part (ii) is shown as follows. Suppose that T*∪Q , then for some finite subset {χ1, ..., χn} of Q we have T*∪{χ1,..., χ(since a proof can use only finitely many axioms). Therefore, we have T* χ1 ∧ ··· ∧ χn → φ. Since χ1 ∧ ··· ∧ χis a formula in the basic language (Q being a set of formulas in the basic language) we have, by the hypothesis of (ii), also T χ1 ∧ ···∧ χn and hence T∪{χ1, ..., χnand T∪Q . The other direction, namely, that if T∪Q also T*∪Q , is shown in the same way.

      4.7 Applying the Conservation Theorem. Altogether, the conservation theorem tells us that if we start with P and P* and keep extending them as in 4.6(ii) and 4.6(iii) we always get corresponding theories which are the same as far as the basic language is concerned. The correspondence between the extensions in 4.6(iii), of T* by {Ψ(A)} and of T by R, is indeed natural, since under the interpretation given in the last section the formula Ψ(A) stands for nothing more than what is obtained from it after substituting all class-terms {x | χ(x)} for A, and this gives exactly the basic instances of Ψ(A).

      4.8 Asserting the Existence of Classes. We shall sometimes formulate theorems of the kind

      (1)

      where a1, ..., an are class or set variables, and Ψ has no free variables other than indicated. This is to be understood as a promise, fulfilled explicitly or implicitly in the proof of the theorem, to provide a class term {x| Φ(x, a1, ..., an, y1, ..., yk)}, where y1, ..., yk are possible additional parameters, and to prove the statement

      (2)

      (1) is not a statement of the extended language, since the extended language does not admit (existential) quantifiers over class variables. (1) is just a way of telling the reader to expect (2). Note that, on the other hand, "For every class A, Ψ(A)" stands directly for the statement Ψ(A) of the extended language.

      4.9 Exercises. (i) Formulate the axiom of comprehension (2.3) as a single statement about classes.

      (ii) Prove that {x |x ∉x} is a proper class.

      4.10 Adding Relation and Function Symbols. In the development of set theory, we shall have to add to our extended language, as we should have done in the case of any other mathematical theory, symbols for defined relations and functions. For practical reasons, this is absolutely necessary for going beyond trivialities, since the original language contains only the simple relation-symbols of equality and membership, and to express even relatively simple relations between sets and classes we need very long formulas in the original language. (One can avoid extending the language by regarding definitions as rules for abbreviation, but once one introduces function symbols, and uses them to build up complicated terms, it is difficult to keep track of which formula is abbreviated by a given formula.)

      The new symbols thus introduced can be eliminated from the language, and they do not add anything to the expressive or deductive power of the language. Since, unlike the case of the introduction of classes, the introduction of new function and relation symbols is familiar to the reader, being used so often throughout all of mathematics, we shall not give here the corresponding eliminability and conservation theorems.

      Let us denote with lower case German letters variables which can be either class variables or set variables. A relation R(a1, . . . , an) is defined by means of the statement

      (1)

      where Φ is a formula with no free variables other than a1,..., an.

      A function whose values are sets is defined as follows. Let Φ(a1,...,an,y) be a formula with no free variables other than a1, . . . , an, y, such that in whatever theory T we are dealing with at that time we have

      T ∃!yΦ(a1, . . . , an , y).

      We define y as a function F(a1, ..., an) of a1 , . . . , an by the statement

      (2)

      4.11 More on Function Symbols. When we define functions in practice, we will sometimes not exhibit the formula Φ(a1 , . . . , an , y) explicitly, but it will be clear which is the appropriate formula Φ(a1, . . . , an, y). Also, in several cases, we will not be interested in all n-tuples a1, . . . , an but only in a restricted collection of such n-tuples. In such a case, we shall prove the existence and uniqueness of y only for the relevant a1, . . . , an. For other a1, . . . , an we shall assume, tacitly, that y has some fixed value, for example 0 (which is the null set, to be defined in the next section).

      Functions whose values are classes are given by the class terms. We can, of course, abbreviate an expression {x (x, a1,..., an)} by F(a1, . . . , an), but this will be regarded as a mere abbreviation.

      4.12 Discussion. When we defined relations and functions whose values are sets we allowed the arguments a1, . . . , an to be set or class variables. We shall see that, in principle, there is an advantage to using only class variables for a1, ..,an. This would not lead to any difficulty since we can always substitute set variables for the class variables. If we intend the variable a1 of F(a1, . . . , an), e.g., to vary only over sets, we can still use a class variable for a1 and set the value of F(a1, . . . , an) in the case where a1 is a proper class to be some arbitrary object, such as 0. The advantage of using class variables only is that we shall be able to substitute also class terms and class variables in the expression for the function or the relation, which is very useful since sets are often given by means of class terms. Therefore, we shall assume that whenever a set variable is used as an argument in a definition of a relation or a function, this definition will only be considered as an informal way of writing the true definition, in which all the arguments are class variables. This is embodied in the following agreement.

      4.13 Agreement on Notation. A relation defined as R(A1,..., Al, y1,..., ym) will in fact be R(A1,..., Al, B1,..., Bm), where it will be defined to be false when at least one of the B¡’s is a proper class. A function defined as F(A1,..., Al,y1,..., ym) is indeed F(A1,..., Al, B1, . . . , Bm) and is defined to have the value 0 when at least one of the B¡

      The reason why we still intend in practice to use also set variables when we define relations and functions is that often we are interested only in the values of the functions in the case where some fixed arguments are sets; defining the function explicitly also for the case where the arguments are classes may divert the reader’s attention from the main point.

      Later on we shall mostly use the terms relation and function for certain kinds of sets and classes. Those notions are somewhat related to the relations and functions mentioned here, but they are still different notions and one has to bear that in mind.

      Once we have symbols for defined functions F(a1,..., an), we get by means of these symbols and by means of the class terms, more complicated expressions which we call terms. Terms are the expressions obtained from expressions like F(a1...,an) and class terms by repeated substitution of such expressions for variables in such expressions. Examples of terms are

      F(G(H1(a), H2(a), b), c,I(a)), {{u}, {u, υ}}, log (sin(x)),

      H({x | (x)},y,A), {y |F(y,z) ∈ {u |χ(u)}}.

      Terms other than class terms and class variables, i.e., set variables or terms whose leftmost symbol is a function symbol, will be called set terms. The values of such terms must always be sets, by our definition of the functions. Note that a set term, while not a class term, may have a class term as a part, as in H({x| φ(x)}, y,A).

      It is now convenient to extend our notation for class terms.

      4.14 Agreement on Notation. Let τ be a term (which may be a set term or a class term). Let us pick some variables x1, . . . ,xn and call them the active variables of τ (for the purposes of the following syntactical construction), while the variables which occur free in τ other than x1,..., xn will be called parameters. We shall write τ as τ(x1,..., xn) to emphasize that x1,..., xn are the active variables. Let φ(x1,..., xn) be any formula which, again, may have free variables other than x1,..., xn. The class of all the sets which have the form τ(x1,..., xn) for x1, . . . ,xn such that φ(x1, . . . ,xn), i.e., the class

      (1)

      will be denoted by

      (2)

      The parameters of the class term (1) are the free variables of τ and φ other than x1, . . . ,xn.

      Note that the value of τ(x1,..., xn) for x1,..., xn such that φ(x1,..., xn) can also be a proper class. Such a class will, of course, not be a member of {τ(x1, . . . ,xn) |φ(x1, . . . ,xn)}. As a matter of practice, we will avoid using expressions of the form (2) unless we know that all the values of τ(x1, . . . ,xn) are sets for x1, . . . ,xn such that φ(x1,..., xn).

      We do not introduce a consistent notation to distinguish between the active variables and the parameters of (2); but, unless otherwise mentioned, the active variables will be those which occur free in both τ and φ, while all the variables occurring in only one of τ and φ are parameters. Examples of this notation are:

      {2n |n∈ w} — where w is a parameter,

      {〈xy〉 | x ∈ s ∧ y ∈ t} ∈ t} — where s and t are parameters,

      {{F(x, y)|x ∈ a} |y ∈ b} — here we have the term {F(x, y) | x ∈ a} inside the outer term; {F(x,y) |x ∈ a} has y and a as its parameters, {{F(x,y)| x ∈ a}| y ∈ b} has a and b as its parameters.

      4.15. Definition. .

      is the class of all members of the members of A, and is called the union of A.

      .

      is the class of all common members of all members of A and is called the intersection of A, where 0 is the class with no members (5.9) and V is the universal class (5.16(iv)), since every set belongs to each member of 0.

      (iii)

      .

      Read: The union of all sets τ(x1, . . . , xn) such that (x1,..., xn).

      (iv)

      Read: The intersection of all sets τ(x1, . . . ,xn)such that (x1,..., xn).

      5. The Axioms of Set Theory

      As we have already made clear, we shall consider set theory as a theory developed, in principle, in the basic language, while in practice we shall use the extended language which admits also

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