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Infinite sets and numbers Bunn, Robert 1974

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INFINITE SETS AND NUMBERS by Robert Bunn B.A., University of North Dakota, 1967 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of Philosophy We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA November, 1974 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r a n a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l m a k e i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s m a y b e g r a n t e d b y t h e H e a d o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t b e a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f Philosophy T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 Wesbrook Place Vancouver, Canada V6T 1W5 D a t e A p r i l 24, 1975 ABSTRACT This dissertation is a conceptual history of transfinite set theory from the earliest results until the formulation of an axio-matic set theory of Cantorian extent which avoids the paradoxes; i t also contains some information on pre-Cantorian views concerning con-cepts important in Cantor's theory. I begin with explications of the concept of infinite set due to Dedekind, Pierce, Frege, and Russell. The i n i t i a l chapter features Dedekind's work, since he was the first to give rigorous demonstrations involving the concepts "finite" and "infinite." Chapter Two describes traditional views on infinite numbers, as well as on numbers in general. It emerges that the traditional objec-tions against infinite numbers were based merely on the fact that numbers had been defined to be finite. I also discuss the Frege-Russell definition of the cardinal numbers, which was the first precise defini-tion of numbers to accommodate infinite as well as finite numbers, and I analyze the proofs that there are infinite sets and numbers. Chapter Three deals with pre-Cantorian views on quantitative relations between infinite sets, and the associated old-fashioned "para-doxes of the infinite." I find that the errors of pre-Cantorian authors regarding quantitative relations between infinite sets were due to the mistaken belief that the relations of greater and less, when defined in the traditional way, are incompatible with numerical equality defined as one-one correspondence. - i i -The chapter on the Cantorian theory of the transfinite discusses the basic concepts and the main theorems which either were proved by Cantor or are generalizations of theorems proved by Cantor. Cantor himself did not present the concepts of the theory of the transfinite in the most precise way, and often formulated definitions (and theorems) for the ordinals of the first two number classes which can be extended to ordinals in general. I describe improvements and generalizations introduced by Russell, Hausdorff, von Neumann, and others. While developing the theory of the transfinite, Cantor came upon several paradoxes, and other mathematicians and logicians discovered more paradoxes later. My last two chapters deal with the analysis of these paradoxes which originated with Cantor, and with the corresponding way of avoiding them. According to Cantor's analysis of the paradoxes, the properties which do not determine classes are exactly those which belong to as many things as there are in some 'absolute totality' such as the totalities of all ordinals, all sets, or all entities. There are, for example, at least as many classes which do not belong to themselves as there are ordinals, as was shown by Russell. The way of avoiding the paradoxes corresponding to Cantor's analysis is a system of axioms which implies the theorems of the theory of the transfinite, but not the exis-tence of 'absolute totalities' or totalities of equal power. Cantor himself formulated several important axioms in correspondence with Dedekind. Later, Zermelo published a system of axioms in accordance with the idea that the 'paradoxical classes' are those which are 'too big,' and he showed that some of the main theorems of Cantor's theory of transfinite cardinals can be derived from these axioms. Von Neumann formulated a system of axioms based on the idea that in the case of properties (e.g. the property of being a class) which do not determine classes, some of the classes having such properties are not elements of other classes; therefore there can only be a class of all elements having such a property (e.g. a class of all classes which are elements), and such classes are not elements. Von Neumann's system included an axiomatic criterion for being a class which is an element, i.e. a set: A class is a set if and only if it is not of at least the power of the class of all elements. The axioms formulated by Cantor, as well as the axiom of choice, are theorems of the system containing this axiom. Later, von Neumann showed that his axiom of limitation of size follows from a system containing the axioms of choice, replacement, and foundation. It follows from the axiom of foundation and von Neumann's theory of limitation of size that the universe of elements decomposes into a sequence of (disjoint) strata containing sets of ever greater complexity, which sequence is similar to the sequence of ordinals. The classes that are not elements are the classes containing sets from these strata but not themselves belonging to any of the strata, since they contain ele-ments from "too many" of the strata in the sense that, for any stratum, they contain elements of higher strata. In general, my investigations show the subject of quantitative relations to be an important and pervasive factor in the history of the theory of the transfinite. A great deal of widespread erroneous reason-ing about the infinite concerned quantitative relations. Some of the principle mathematical problems of Cantor's theory concerned these relations: Are there unequal transfinite powers? Is there an increas-ing sequence of transfinite powers? Are any two transfinite powers comparable? The new difficulties in the theory of the transfinite were discovered by attempting to solve these problems and by reflection on the solutions. Analysis of the paradoxes seems to show that the classes involved in the paradoxes are those which would have the greatest cardin-al number. CONTENTS GENERAL INTRODUCTION 1 CHAPTER I. Concepts of Finite and Infinite 8 1. Introduction 8 2. The Mathematical Concept of Infinity 13 3. Dedekind's Definition 29 4. Potential Infinity 34 5. Cantor on the Actual and Potential Infinite 37 6. Absolute Infinity 41 CHAPTER II. Infinite Numbers 44 7. Traditional Objections to Infinite Numbers 44 8. Abstract Number and Numerical Equality 50 9. The Frege-Russell Definition of Numbers 54 10. Proofs that there are Infinite Sets and Numbers 61 CHAPTER III. Quantitative Relations Between Infinite Sets. 70 11. Traditional Definitions of Greater and Less 70 12. Leibniz and the Problem of the Infinite 73 13. Bolzano on Quantitative Relations... 79 CHAPTER IV. Cantor's Theory of the Transfinite 82 14. Cantor's Definitions 82 15. Cantor's Theorem -. 83 16. Relations in Extension 87 17. The Numbers of Transfinite Weil-Ordered Sets 93 18. Cardinal and Ordinal Numbers 106 19. Proof and Definition by Transfinite Induction 118 20. Number Classes and Alephs 127 CHAPTER V. The Theory of Limitation of Size 135 21. Introduction 135 22. The Cantor-von Neumann Analysis of the Paradoxes 141 23. Russell on the Theory of Limitation of Size 152 24. Paradoxes of the Ultrafinite: Hessenberg and Zermelo 160 25. Mirimanoff's Solution of "The Fundamental Problem of Set Theory." 164 26. Von Neumann's Axiom of Limitation of Size 172 - vi -v i i CHAPTER VI. The Axioms of Set Theory 176 27. The Extensional Concept of Set 176 28. Axioms Implicit in Dedekind and Cantor 184 29. The Axioms of Separation and Replacement 189 30. Zermelo's Strong Axiom of Infinity 197 31. The Axiom of Choice' 200 32. The Axiom of Foundation 206 BIBLIOGRAPHY 219 1 General Introduction This dissertation covers the Cantorian theory of the i n f i n i t e , the associated paradoxes, and the way of avoiding these paradoxes— by means of a system of axioms based on an analysis of the paradoxes— originating with Cantor himself. This way of avoiding the paradoxes Is distinguished from most others, and in particular from Russel l 's, in being a set theory of Cantorian extent. Thus, I provide a history of the main points in the development of the theory of i n f i n i te sets and numbers beginning with the great works of Cantor and Dedekind and continuing through the formulation of a set theory of Cantorian extent which avoids the known paradoxes. This is a period of about f i f t y years—from about 1880 to 1930; i t ends with the last works of Zermelo and von Neumann on the axiomatic theory of sets. In my exposition of the basic concepts of Cantorian set theory, I emphasize the efforts to make these more precise. In general my concern i s with the elements and the foundations of the theory, rather than with the r ich development of the theory which also occurred by 1930. For example, Hausdorff's theory of ordered sets appeared in the same year (1908) in which the systems of Russell and Zermelo were pub-l ished. Major works by Mahlo, Sierpinski, Tarski, Ulam and many others extending considerably the theory of the transf inite also appeared by 1930, but these are beyond the scope of this work. While i t i s not my intention to present a general history of thought about the in f in i te , I do discuss ear l ier ideas on certain topics 2 which are important in Cantor's theory. By far the most interesting of these topics are the traditional views on i n f i n i t e number and quanti-tative relations between i n f i n i t e sets. Perhaps the two most important advances making possible the theory of the i n f i n i t e are (1) the de f i n i -tion of the relation of greater in cardinality between sets and (2) the definitions of well-ordered sets and of isomorphism. So far as I know, no one before Cantor had any thought of well-ordering and the po s s i b i l i t y of transfinite ordinal numbers, and a l l reasoning that I have seen prior to Cantor concerning inequality relations between i n f i n i t e sets was incorrect. Since ancient times there have been c r i t i c a l evaluations of con-cepts of i n f i n i t y , and attempts to improve on previous definitions. Aristotle, Spinoza, Kant, and Bolzano a l l engaged in such work. The i n f i n i t e was already a subject of speculation for the presocratic philos-ophers. Their opinions were c r i t i c a l l y examined by Aristotle, who dis-cussed the concepts of i n f i n i t y and the mode of being of the i n f i n i t e . Likewise, in the writings of Kant we find several concepts of i n f i n i t y distinguished, as well as criticism of one of these. It was not, however, unti l Dedekind that various definitions of infinitude or distinctions between f i n i t e and i n f i n i t e were precisely formulated and proofs of their equivalence given. After giving his definition of an i n f i n i t e set, Dedekind remarks in a note that, " A l l other attempts that have come to my knowledge have met with so l i t t l e success that I think I may be permitted to forego any criticism of them."''' The mark of the real j Richard Dedekind, Essays on the Theory of Numbers, trans. W.W. Beman (New York, 1963), p. 63. 3 beginning of the theory of the i n f i n i t e i s the appearance of proofs of propositions concerning f i n i t e and i n f i n i t e . This f i r s t occurs in the work of Dedekind and Cantor. It was sometimes asserted that there could not be a science of the i n f i n i t e , or that the i n f i n i t e cannot be an object of human knowledge. The ancient arithmetician Nicomachus admitted the existence of i n f i n i t e s , but thought that they could not be subject to s c i e n t i f i c investigation. In his opinion, science can cope only with what i s limited. St. Thomas maintained that the i n f i n i t e is unknowable by f i n i t e minds; God, however, is able to know the i n f i n i t e , for He i s supposed to know things in a d i f -ferent way than we do: ...to know something by the numbering of i t s parts belongs to an intellect that knows one part after the other; i t does not belong to an intellect that comprehends the diverse parts together. Therefore, since the divine intellect knows a l l things together without succession, i t is no more prevented from knowing i n f i n i t e things than from knowing f i n i t e things. 2 Our f i n i t e intellects "can know i n f i n i t e things only successively by numbering them. This is not the case with the divine intellect which sees many things together grasped through one species" (Ibid., I, 69, xiv). It would seem, however, that there i s nothing to prevent us from grasping many things together "through one species" or thinking of an i n f i n i t e class of things by means of a formula satisfied by exactly the elements of that class. The difference between man and God, so far as knowledge of the i n f i n i t e i s concerned, i s that God would have a simul-_ On the Truth of the Catholic Faith, Summa Contra Gentiles, Book I: God, trans. Anton Pegis (New York, 1955), I, 69, x i . 4 taneous consciousness of each of the objects satisfying a concept, while a "finite mind" knows the infinite only by means of class concepts. But it is not true, as St. Thomas asserts, that we know multitudes only through enumerating (others have liked to say "synthesizing") their elements. One of the points Bernard Bolzano made in refutation of those who said that nothing infinite can exist because a whole is made by the mind's uniting its parts (elements) in thought, is that it is not at all the case that a set is apprehended by apprehending individually each of its elements: I can think of the set, of the aggregate, or if you prefer i t , the totality of the inhabitants of Prague or of Pekin without forming a separate representation of each separate inhabitant. So indeed do I act every time that I speak of this set and put forward my estimate, in the case of Prague, that the population lies between 100,000 and 120,000. As soon, in fact, as we pos-sess a representation A which represents the objects a,b,c,d,... and no others, it is extremely easy to arrive at a representa-tion which represents the aggregate of all these objects taken together. Nothing more is needed than to combine the idea de-noted by the words 'the aggregate of all A.' This single re-mark, whose correctness I trust will be evident to all, removes all the difficulties made against the idea of a set comprising infinitely many members: provided only that a specific concept is present, under which every member, and no non-member, can be subsumed.3 We form an idea of a set by means of a general idea corresponding to the general description satisfied by the elements or parts and only these; this is the idea of the set corresponding to the class abstract. Collec-tion, synthesis, or enumeration of the elements one by one in thought (or any other way) is necessary neither for the existence nor for the Paradoxes of the Infinite, trans. Dr. Fr. Prihonsky' (London:-Routledge and Kegan Paul, 1950), sect. 14. Henceforth Paradoxes. 5 r e c o g n i t i o n o f a s e t , a c c o r d i n g to B o l z a n o . L i k e w i s e , R u s s e l l s ay s i n c r i t i c i s m o f Kant t h a t " c l a s s e s w h i c h a r e i n f i n i t e a r e g i v e n a l l a t o n c e by the d e f i n i n g p r o p e r t y o f t h e i r members, so t h a t t h e r e i s no q u e s t i o n o f ' c o m p l e t i o n ' o r o f ' s u c c e s s i v e s y n t h e s i s . ' " ^ The r e c o g n i t i o n o f t h e s i g n i f i c a n c e o f p r o p o s i t i o n a l f u n c t i o n s f o r s p e c i f y i n g i n f i n i t e s e t s was e s s e n t i a l f o r a s c i e n c e o f t h e i n f i n i t e . But R u s s e l l ' s p a r a d o x (as w e l l as many o t h e r s ) showed t h a t t h i n g s a r e no t as s i m p l e and ea sy as B o l z a n o t h o u g h t . I n f a c t t h e who le p r o b l e m p r e s e n t e d by t h e p a r a d o x e s w h i c h were d i s c o v e r e d i n d e v e l o p i n g t h e t h e o r y o f t h e i n f i n i t e i s t h a t o f m o d i f y i n g the ax iom o f s e t e x i s t e n c e i m p l i c i t i n B o l z a n o ' s remarks i n s u c h a way t h a t t h e c o n t r a d i c t i o n s ( pa radoxes ) cannot be d e r i v e d . Some have t a k e n t h e p a r a d o x e s a s d i s c r e d i t i n g t h e e f f o r t s o f C a n t o r . Even F r e g e came t o r e g a r d s e t t h e o r y as d e s t r o y e d ( v e r n i c h t e t ) by the paradoxes . " * But o t h e r s d i d n o t g i v e up so e a s i l y . T h e i r p o i n t o f v i e w was w e l l e x p r e s s e d by R u s s e l l : "I h o l d . . . t h a t m a t h e m a t i c a l t r u t h i s w h o l l y and u l t i m a t e l y t r u e , and t h a t t h e c o n t r a d i c t i o n s w i t h w h i c h i t has a p p e a r -ed to be i n f e c t e d can a l l be removed by p a t i e n c e i n d i s t i n g u i s h i n g and d e f i n i n g ; " * * and by H i l b e r t : "No one s h a l l be a b l e to d r i v e us f r om t h e p a r a d i s e t h a t C a n t o r has c r e a t e d f o r us . " ' ' ' _ B e r t r a n d R u s s e l l , Our Knowledge o f t h e E x t e r n a l W o r l d , 2nd e d . (1929; r p t . , New Y o r k , 1 9 6 0 ) , p. 123n. 5 G o t t l o b F r e g e , N a c h g e l a s s n e S c h r i f t e n , e d . Hans Hermes, F r i e d r i c h K a m b a r t e l , and F r i e d r i c h K a u l b a c h , V o l . I (Hamburg, 1 9 6 9 ) , p. 289. 6 " S o m e E x p l a n a t i o n s i n R e p l y to Mr. B r a d l e y , " M ind 19 ( 1910 ) , p. 373. ^Dav id H i l b e r t , "On the I n f i n i t e , " i n From F r e g e to G o d e l : A S o u r c e  Book i n M a t h e m a t i c a l L o g i c , 1879 -1931 , e d . J e a n v a n H e i j e n o o r t (Cambr idge , Ma s s . , 1 9 6 7 ) , p. 376. 6 Today, after the paradoxes have been known for about 80 years, there s t i l l exists a diversity of opinion about their significance. Some believe the paradoxes have almost no importance for mathematics, while others believe that they are completely destructive of set theory and of classical mathematics in general. It has been said that the g "antinomies should be relegated to the history of mathematics" and that "the paradoxes are at present little more than historical curios-9 ities." Perhaps the paradoxes are of only historical interest, but to the historian they are very interesting indeed.' Bernays and Godel have contended that the paradoxes are not a serious problem for mathematics, including Cantorian set theory. As Bernays puts i t : It must certainly be admitted that mathematics does not suffer any real distress from [the paradoxes of set theory]. The standpoint of Cantorian set theory, which beginning from the set of integers ascends without limit to higher 'powers' by processes of forming sets, is not troubled by the paradoxes 10 And Godel has said that, "The set-theoretical paradoxes are hardly any more troublesome for mathematics than the deceptions of the senses are for physics."^ These opinions may be contrasted with Gentzen's statement: "The Helena Rasiowa and Roman Sikorski, Mathematics of Metamathe- matics (Warsaw, 1963), p. 9. 9 Brian Rotman and G.T. Kneebone, The Theory of Sets and Trans- finite Numbers (London, 1966), p. 66. "^Paul Bernays, "Bermerkungen zur Grundlagenfragen" in Philosophie  Mathematique by F. Gonseth (Paris, 1939), p.85 (my translation). "^What is Cantor's Continuum Problem?" in Philosophy of Mathematics: Selected Readings, eds. P. Benacerraf and H. Putnam (Englewood Cliffs, N.J., 1964), p. 271. 7 actualist (an sich) interpretation has led to contradictions in set theory; who knows whether one day contradictions could not also occur 12 in analysis." It is suggested that if one takes the view that sets have an independent reality, the paradox of the set of all sets will naturally follow. On the constructivist view, "New sets may, as a mat-ter of principle, be formed only constructively one by one, on the basis of already constructed sets"(Ibid., p. 225). This understanding of sets makes the set of all sets an impossibility, but it also makes the Cantorian theory of the transfinite an impossibility. Whatever their import, the paradoxes of set theory are of a totally different character from the old-fashioned paradoxes of the infinite. The latter were not contradictions, but only the results of mistaken reasoning. By contrast, the paradoxes of set theory are real contra-dictions validly derived from certain assumptions. It may be that some would want to reject some of the principles of reasoning used in the derivations of the contradictions, but the reasoning is correct on the basis of those principles; there is no mistake in the reasoning given the premisses and rules of inference. Furthermore, the new paradoxes do not really deserve to be called "paradoxes of the infinite" in the sense of being paradoxes involving the general concept of an infinite set or number. Nevertheless, they certainly are paradoxes which must be dealt with in order to preserve the theory of the infinite. The Collected Papers of Gerhard Gentzen, ed. M.E. Szabo (Amsterdam, 1969), p. 227. CHAPTER I CONCEPTS OF FINITE AND INFINITE 1) Introduction In 1901 Bertrand Russell was so contemptuous of definitions of infinity given by philosophers who preceded Cantor, or who, like him-self several years earlier, had a pre-Cantorian point of view, that he went so far as to represent these philosophers as having never even asked what infinity was. He added that, "If any philosopher had been asked for a definition of infinity, he might have produced some unin-telligible rigmarole, but he would certainly not have been able to give a definition that had any meaning at all."''" Two years later, in his Principles of Mathematics, Russell said that, "Of all the philosophers who have inveighed against infinite number, I doubt whether there is 2 one who has known the difference between finite and infinite numbers." Now Russell knew very well that philosophers had actually given definitions of infinity; he had once used the traditional formulations, himself. After appreciating the work of Cantor and Peano, however, Russell apparently came to think the old definitions so imprecise as to be meaningless. The description "unintelligible rigmarole" would Mysticism and Logic (New York, n.d.), p. 80. 2nd ed. (New York, 1964), p. 192. Henceforth Principles. - 8 -9 ' no doubt have been bestowed, for instance, upon Hegel's efforts in The  Science of Logic. As a matter of fact, three principal definitions of infinitude can be distinguished in the traditional literature on i n f i n i t y . They may be stated roughly as follows. ( 1 ) A quantity (plurality, totality) i s i n f i n i t e i f no f i n i t e part of i t comprises a l l of i t , or, more simply, i f i t i s not f i n i t e . ( 2 ) A quantity is i n f i n i t e i f i t has no limits, or i f i t s parts or elements form an endless series. These two definitions were hardly distinguished, and both were associated with ( 3 ) , the idea that the i n f i n i t e i s the uncountable. If a quantity is unlimited, i t cannot be counted, and i f no f i n i t e part contains a l l elements of a plurality, i t cannot be counted. The f i r s t two definitions, though in use since antiquity, were not made precise u n t i l the modern period of foundational studies. The pre-cise formulations (explications) due to Frege, Russell, and Peirce w i l l be given later in this chapter, and I w i l l also discuss Dedekind's defin-i t i o n , which is not found in the older literature, though the property used in the definition was recognized centuries earlier. While the definitions mentioned relate to so-called "actual" quan-titat i v e i n f i n i t y , tradition also distinguished two other sorts of i n f i n -i t y , "absolute," and "potential." Absolute i n f i n i t y , the i n f i n i t y which was attributed to God, i s worth brief consideration (in section 6) because Cantor, who was well acquainted with speculations in natural theology, characterized the series of transfinite ordinals as absolutely i n f i n i t e , and also referred to the inconsistent m u l t i p l i c i t i e s as absolutely i n f i n i t e m u l t i p l i c i t i e s . 10 Though the traditional definitions were certainly not notable for precision, philosophers, beginning at least with Aristotle, had made some efforts to formulate and c r i t i c i z e definitions of i n f i n i t y . Kant, for example, distinguished between the "true transcendental concept" and the "mathematical concept" of infinitude. According to the former, "the successive synthesis of units required for the enumeration of a quantum 3 can never be completed," while according to the la t t e r , an i n f i n i t e "quantum...contains a quantity (of given units) which is greater than any number (Critique, A 433, note a). -Although, as Hermann Lotze has remarked, the former i s only a definition by a consequence and "not by 4 the proper nature" which i s the reason for that consequence, essenti-a l l y this same concept has been employed at least since Aristotle, and was also accepted by Russell in an early manuscript. Kant not only distinguished two concepts of i n f i n i t y , he also c r i t i c i z e d what he called a "defective" definition of i n f i n i t y , accord-ing to which a quantity is i n f i n i t e i f no greater multiplicity of units i s possible. It seems that even before Aristotle the i n f i n i t e had been defined as the maximum, for Aristotle says that, "The i n f i n i t e turns out to be the contrary of what i t is said to be. It i s not what has nothing outside i t that is i n f i n i t e . . . . " ^ This concept of the i n f i n i t e -Critique of Pure Reason, trans. Norman Kemp Smith (New York, 1965), A 433; cp. A 426 and B 111. 4 Metaphysic, trans. T.H. Green et a l . , ed. B. Bosanquet, 2nd ed. (Oxford, 1887), I, 327. 5 h Physics, trans. R.P. Hardie and R.K. Gaye (Oxford, 1930 ), 206 33; see also 207a15-20. as the maximum may have been used by Spinoza in his Short Treatise, where he says that "...Nature is infinite and...all is contained therein; the negative of this we call Nothing."*' What Kant called the defective concept of infinity seems to be related to the tradition-al ideas of the absolute infinity attributed to God: the goodness of the infinitely perfect Being is infinite, and would not be infinite if one could conceive of a_ good- ness greater than this. This characteristic of infinity is proper also to all his other perfections...they must be the greatest one can imagine.7 The "defective" definition of infinity probably resulted from defining the infinite in general in accordance with ideas of God's unsurpassability. What Kant called the true transcendental concept of infinity was used by Aristotle in characterizing the infinite as "what admits of g being gone through, the process however having no termination." I assume that counting would be one way of going through, so that the infinite is the uncountable or innumerable. Both of Kant's definitions were given by Russell during his ideal-ist period, but he called the infinitude of the true transcendental concept "absolute infinity:" For absolute infinity is merely the negation of possible syn-thesis, and thus the negation of number. All number is syn-thesis of parts into a whole, and infinity denies the whole, as zero denies the parts. Mathematical infinity, on the other hand, is quantity larger than any assignable quantity, Spinoza Selections, ed. John Wild (New York, 1930), p. 60. ^Pierre Bayle, quoted in Leibniz' Theodicy, trans. E.M. Huggard; ed. and abridged Diogenes Allen (Don Mills, Ont., 1966), section 177; my underlining. 8Physics, 204a4, 204a14. r i 9 and therefore one [ofj which...we must remain ignorant. The true transcendental concept of infinity is the one involved in the first part of the thesis of Kant's first antinomy. Though the argument is an excellent illustration of confused thinking about the infinite which is not uncommon in the traditional literature, Kant prided himself on his logical reasoning in his antinomies.^ Kant's thesis in the first antinomy states that the world has a beginning in time. The indirect proof is as follows:1"1" (1) "The world has no beginning in time" (assumption). (2) An infinite series of events have preceded any particular event. (This is supposed to follow from the assumption.) (3) "Now the infinity of a series consists in the fact that it can never be completed through successive synthesis" (true transcendental concept of infinity). (4) "It thus follows that it is impossible for an infinite world-series to have passed away...." Hence, the thesis. The basic defect in the proof is that if the definition (3) is interpreted so that the conclusion does follow from it, then, with this meaning of "infinite," the second proposition will not follow from the initial assumption. If (3) means that a sequence is infinite if it is impossible to finish counting i t , i.e. if it is "innumerable," then a 9 "On Some Difficulties of Continuous Quantity," unpublished MS. from the Bertrand Russell Archives, Hamilton, Ontario (1896), p. 5. See his footnote on the certainty of his demonstration in Prolegomena to any Future Metaphysics, trans, and ed. Paul Carus, reprint edn. (La Salle, Illinois, 1961). ^Critique, A 426. 13 beginningless world is infinite in that sense, and (4) does not follow. But if (3) is interpreted so that it is true as Kant says that it follows from his definition that an infinite series cannot "have passed away," and this is taken as bringing out the force of "completed" in the defini-tion, then (2) would not follow from (1). Kant had indicated that an "infinite" series is one "without limits or beginning" (Critique, A 418). It is doubtless with this concept of the infinite in mind that Kant made the transition from (1) to (2). But, in order to get (4) he had to bring in the true transcendental concept of an infinite series, understood in such a way that it is not coexten-sive with the concept of an infinite series as one "without limits or beginning." 2) The Mathematical Concept of  Infinity The definition which Kant called the "mathematical concept" has proved more fruitful. This definition did not originate with Kant but was stated much earlier, for example, by Spinoza, who characterized the infinite in one sense as "that whose parts we cannot equate with or ex-12 plain by any number...." Mathematicians, he said, have discovered "many things which cannot be equated with any number but exceed any number that can be given" (Ibid., p. 415). This mathematical definition of infinity says essentially that a quantity is infinite i f it is not finite, and that is the formulation given it by St. Thomas. He says that "a thing is called infinite because — Spinoza Selections, p. 411. 13 i t i s not f i n i t e . " The defect i n such a definition is that i t was not preceded by a satisfactory definition of finitude. In Bolzano's Paradoxes of the Infinite, we find an improvement in the formulation of the "mathematical" definition, in that i t is based on a prior definition of finitude. Bolzano defined the f i n i t e 14 sets—which he identified with concrete numbers — i n terms of series of successive additions: Let us imagine a series whose f i r s t term is an individual of the species A, and whose every subsequent term is derived from i t s predecessor by adjoining a fresh individual of the species A to form a sum with that predecessor or i t s equal. Then clearly w i l l a l l the terms of this series, with the ex-ception of the f i r s t , which i s a mere individual of the species A, be multitudes of the species A. Such multitudes I c a l l f i n i t e or countable multitudes, or quite boldly: numbers (Paradoxes, sect. 8). According to this, a concrete number (in the species A) i s either an individual (a number one) or a f i n i t e set (of elements of the species A). Although he followed tradition in making numbers f i n i t e by d e f i n i -tion, Bolzano argued vigorously for the existence of i n f i n i t e sets. Bolzano defined an i n f i n i t e multitude as "one so constituted that every single f i n i t e [sub]multitude represents only a part of i t . " 1 ^ This means that a set i s i n f i n i t e i f i t is not f i n i t e . Moreover, an i n f i n i t e manifold i s one for which "no number can be given..., which is why we c a l l i t uncountable."''"*' 13 Summa Theologica, trans. Frs. of the English Dominican Province (London), I, 7,i. 14 See Jan Berg, Bolzano's Logic (Stockholm Studies in Philosophy, 2 Stockholm, n.d. , pp. 165-167, on Bolzano's distinction between con-crete, abstract, and absolute numbers. ^Paradoxes, sect. 9; cp. sect. 16. ^Theory of Science, ed. and trans. Rolf George (Berkeley, 1972), p. 1 3 1 . 15 Cantor gave a definition of a finite set which is quite similar to the one given by Bolzano. A finite set is one "welche aus einem ursprungliche Element durch sukzessive Hinzufugung neuer Elemente derartig hervorgeht....An infinite set or an actual infinite quantity, unlike the potential infinite, is "a constant Quantum fixed in itself, which is greater than any finite quantity of the same kind" (GA, p. 409). The set of all natural numbers, "which is conceptually determin-ate by a law and which contains all finite whole numbers n_ presents the simplest example of an actual-Infinite quantum" (GA, p. 409; cp. p. 401). Such definitions of finite and infinite sets as those given by Bolzano and Cantor are, however, s t i l l not precise, at least not by the standards of classical logicians like Peirce, Frege, Dedekind, and Russell. As Peirce put i t , if we say that a finite multitude (or, in his terminology, an "enumerable" multitude) is one which can be reached by starting at 0...and successively increasing it by one, we shall express the right idea. The difficulty is that this is not a clear and distinct state-ment. ... An enumerable multitude is said to be one which can be constructed from zero [i.e. the empty class, for which Peirce used the symbol '0'] by 'successive additions of unity.' What does 'successive,' here, mean? Does it allow us to make innumerable additions of unity? ... But if we say that by 'successive' additions we mean an enumerable multitude of additions, we fall into a circulus in definiendo.x^ Peirce gave the following explication of the idea of finite multitude: Gesammelte Abhandlungen, Mathematischen und Philosophischen Inhalts, ed. Ernst Zermelo (1932; rpt. Hildesheim, 1962), p. 415. Henceforth GA. 18 Collected Papers of Charles Sanders Peirce, ed. Charles Hartshorne and Paul Weiss (Cambridge, Mass., 1933), IV, p. 154. the enumerable multitudes are those multitudes every one of which possesses any character whatsoever which is, in the first place, possessed by zero and, in the second place, if it is possessed by any multitude, whatsoever, is likewise possessed by the multitude next greater than M.^ The mathematical concept of infinity was closely associated tradi-tionally with the idea of the infinite (or the unlimited) as an endless sequence of things. In his paper "On the Logic of Number" published in 1881, Peirce gave what is, so far as I know, the first precise formula-tion of this traditional idea of the infinite. His definition is pre-ceded by a number of interesting preliminary definitions (Collected  Papers, III, 158-160). A quantitative relation is one which is transi-tive, reflexive, and asymmetrical between distinct terms. This last property is equivalent to antisymmetry; the relations which Peirce is calling "quantitative" are therefore those now called "partial order-ings." The elements of the field of a quantitative relation are called quantities. This class together with the relation is called a system  of quantity, which is the same as what we would call "a partially ordered set." A simple system of quantity is one with a connected quantitative relation. Hence, a simple system of quantity is what is now called "a totally ordered set." "A discrete system is a simple system in which every quantity greater than another is next greater than some quantity...." A limited system is a discrete system having an absolute maximum and minimum; such a system is finite. An unlimited system has _ — Ibid., p. 155. It is of course possible to improve on this explication as regards details; see the definition of the inductive (i.e. finite) classes, explained below, which was given in Principia Mathematica. neither, while a semi-limited system has one. A semi-limited system is infinite in the direction (let us con-sider only the direction of increase) in which it is umlimited, if any subclass M of the system which contains the (immediate) successor of every quantity which belongs to M contains every quantity greater than any particular quantity belonging to that subsystem. What is defined here is denumerable infinity. Other semi-limited, simple, discrete systems of quantity are called super-infinite by Peirce. Peirce's definition of a (denumerably) infinite system employs the same method used by Peano to axiomatically characterize the (stand-ard denumerably infinite system of) finite numbers: any class which contains 1 as well as the successor of every number which belongs to 20 it contains the class of positive integers as a subclass. In his Principles of Mathematics, Russell explicitly uses this "axiom" of mathematical induction as a definition of the finite numbers: "the class of finite numbers is the class of numbers which is contained in every class s^  to which belongs 0 and the successor of every number be-longing to js." The finite numbers and sets are defined In Principia Mathematica by means of the concept of ancestral relations which was first formulated by Frege (1879), who used it to define the finite numbers in his 21 Foundations of Arithmetic (1884). Frege, like Whitehead and Russell, 20 The Principles of Arithmetic, Presented by a New Method (1889), in van Heijenoort, p. 94. 21 Foundations of Arithmetic: A Logico-Mathematical Enquiry into the  Concept of Number, trans and d. J.L. Aus in (New York, 1950), p. 96. 18 was particularly interested in the concept of the ancestral of a relation (or the relation of following in a sequence corresponding to a relation) in connection with the program of deducing arithmetic from logic. As Frege put it: "Only by means of this definition of following in a series is it possible to reduce the argument from n to (n+1), which on the face of it is peculiar to mathematics, to the general laws of logic" (Ibid., P. 93). The class A is an R-hereditary class if R"A = A, i.e. R-successors 22 of members of A are also in A. The ancestral relation R^  corresponding to the relation R is the relation which holds between x and y_ when _x belongs to the field of R and y_ belongs to the intersection of all R-hereditary classes containing x: xR^ y xeF(R) & yefl{A: R"A - A & xeA}.23 The intersection of all R-hereditary classes containing x is itself an R-hereditary class containing x; it is the class of R-descendents of x. or the R-posterity of i.e. the class R^ "{x}. Let M be the relation which holds between a class A and a class B_ if there is an (element) x such that A = Bu{x}. The class of inductive (i.e. finite) classes is the M-posterity of the empty set: els induct = MA"{0}.24 Essentially the same definition of the finite numbers was formulated 22 " -The converse of R, R, is defined by: xRy yRx. S A is the class of things having the relation S_ to something in the class A. 23 In more recent terminology, an R-hereditary class is called closed  with respect to R, and the ancestral R^  corresponding to a relation R_ is called the transitive closure of R. 24 Bertrand Russell and Alfred North Whitehead, Principia Mathematica, 2nd ed. (Cambridge, 1968) II, 182; henceforth PM; see also p. 207. 1 9 by Dedekind using his concept of a chain. A set K i s a <j>-chain i f for every x in K, <j>(x)eK, i.e. K i s a <J>-chain i f i t contains the ^-correlates of any of i t s elements. It can also be said that K is a <J>-chain i f i t includes i t s 4>-image <j> [K] , the class of values of <f> for arguments in For any particular function the intersection of a set of 4>-chains is again a <j>-chain. The 4>-chain of the set A is the intersection of a l l <{>-chains of 25 which A is a subset. Dedekind used the notation "<J>Q(A)" to refer to the <J>-chain of A. <{>Q(A) i s the smallest set including A and the <f>-26 correlate of any of i t s elements. The <f>-image of A i s a subclass of <J> 0(A), i.e., <|>[A] £ <t>0(A), and i f A i s a (j>-chain, then A = <J> 0(A). <l>0(a) is the <()-chain of the object a_, i.e. the smallest <J>-chain containing a as an element. A general theorem of complete induction i s a consequence of the definition of the (f-chain of A: ASM & <fr[* (A)»M]&M -»- <j>0(A) M, i.e. i f A i s a subset of M and the <J>-image of the intersection of the <J>-chain of A and the class M i s also a subclass of M, then the <j>-chain of A is a subclass of M. The proof is as follows: let G = ^ ( A ) " M . Then, G £ 't'o^ and, therefore, <|>[G] £ <()Q(A). From the second conjunct of the antecedent of the theorem i t follows that G is a <j>-chain, for i f that conjunct is true, then ^[G] i s a subset of both ^ ( A ) and M and, hence, 25 Using the terminology of Godel's monograph "The Consistency of the Continuum Hypothesis," a <}>-chain is called a class closed with respect to <)>, and the <{>-chain of A is called the closure of A with  respect to jj>. 26 In general, when we speak of the smallest class X having the property *, we of course mean that whenever *(Y), X £ Y. 20 of their intersection G. Since A is a subset of both M and <}>Q(A), and therefore of G, and G £ <j>g(A), i t follows that G = <|>Q(A), because a chain containing A which is a subset of <CQ(A), the smallest such chain, must be identical to (J>Q(A). Dedekind defines denumerably infinite systems, or as he says simply infinite systems, by means of the concept of the chain of a set. A set N is simply infinite (denumerable) i f f there exists a one-one function <j) mapping N into itself such that the following conditions are satisfied: 1 ) <|>[N] £ N (N is a <j>-chain), 2) N =n{X: lex & <j>[x] s X}, 3 ) 1 ^ [ N ] . According to (2), N is the <j>-chain of the element 1 of N. The symbol "1" is, of course, not used to denote the number one, but refers to a designated element of a simply infinite system. Dedekind seems to have maintained that any simply infinite system might be taken as the system 27 of finite ordinal numbers. He understood the primary object of the science of arithmetic to be the system of "relations or laws" derivable from the defining conditons of a simply infinite system; these are the laws which hold in a l l simply infinite systems (Ibid.). . Dedekind specified that "the" system of numbers is to be the chain of its first element 1 because he wanted to exclude "alien intruders." To accomplish this exclusion without presupposing "knowledge of the sequence" and "the use of the language of arithmetic," he says, "was 27 Essays on the Theory of Numbers, p. 6 8 ; cp. p. 9 5 . 21 one of the most difficult points of my analysis and its mastery required lengthy reflection." With the presupposition of knowledge of the sequence one may say that a number is anything reached by counting from 1, i.e., by "a finite number of iterations" of the successor function <f>. But this method "would, after a l l , contain the most pernicious and obvious kind of vicious circle. In addition to proving the principle of proof by complete induction, Dedekind also established a very important theorem justifying definitions by induction. This theorem on definition by induction is a consequence of the following theorem of complete induction: Pm & vh(nemg & Pn -*• Pn') -»• Vn(n£m0 •+ Pn) , where "m0" is an abbreviation for "<)>0(m)" and "n"' for "(j>(n)". This theorem is an immediate consequence of the general induction theorem for chains proved above. The theorem on the definition of functions whose argument domains are simply infinite systems states that there is always one and only one function satisfying certain conditions. Before establishing this theorem Dedekind proves the following preliminary result. Let 0 be any function mapping the system K into itself and k be any element of K. For every element n of the number chain N, there is one and only one function ^ — — n with domain Z q satisfying the conditions (1) <J*n(l) = k, and (2) 'J' ( O = 0 ^ n ( t ) , for every _t < n. The relation less than between finite numbers is defined by the condition: m < n «->- £ m^'. The i n i t i a l segment Z R of the number chain 1J is the set of a l l elements of N_ which are not great-er than n: Z = {m: 1 < m i n } . Z is also defined by the condition: — n n 28 "Letter to Keferstein" (1890), in van Heijenoort, p. 100. 22 m£Z -«-»• n_ £ m_. n 0 0 Proof. I will give Dedekind's proof only of the part of the theorem asserting existence. The induction hypothesis is that for the number n, there is a unique function ty satisfying the conditions (1) and (2). We have to show for p = n', that i t follows from this hypothesis that there is a function ty satisfying the conditions (1") 'r'p(l) = k a n d" (2') for every m < p, ty (m") = 0i|> (m). The function ty is explicitly defined as follows: ty (m) = ty (m), for m £ n (m) P n * p(p) = G^n(n) (p). We must bear In mind that the concept of a function (Abbildung) Is a primitive idea for Dedekind, and a (naive) comprehension axiom for func-tions is implicitly used. It only remains to verify the recursive conditions (1') and (2') for ty . (1') follows from (m), (1), and the fact that leZ . (2') follows from (m), (2), and (p). Thus, i t is est-n ablished that for every number n, there exists a function f_ whose domain is the i n i t i a l segment satisfying the recursive conditions (1) and (2). The theorem justifying definition by finite induction states that for any function 0 mapping any system K into itself and any element k in K, there exists exactly one function ty satisfying the conditions: (1) ty(l) = k, and (2) ty(n') = 0i|/(n), for every neN. This is an immedi-ate consequence of the theorem just proved. The desired function ty is defined by the equation: >(n) = i|> (ti). We may note that the function tyn is the restriction of ty to the subset 23 Z of N : ij> = i|)|z . n n 1 n Dedekind also proves that for such functions 9 and <J>[N] = G ^ C k ) . That is, the value domain of ij; is identical with the smallest subclass of K containing k and the ©-correlate of anything belonging to i t . Furthermore, Dedekind uses the theorem on definition by induction to show that i f y:S -»• S and A£S, then P 0(A) = U A » where the classes n = <j)(n)[A] with the function <J> defined as follows: +(D = y, <j)(n') = y<|>(n). Controversy Concerning the Explica- tion of the Mathematical Concept It is well known that such definitions as the definition of a sim-ply infinite system and the definition of the R-posterity of an object met with criticism. Indeed, though one of the reasons for formulating these definitions was to avoid circularity, they were themselves accused of being circular. Such definitions came to be called "impredicative," and were thought by some to be the source of the paradoxes of logic and set theory. Russell himself became the foremost exponent of this view. In general, a definition is impredicative i f the definiens contains a bound variable, one of whose values is the thing defined. Typically, a subclass A of a class B_ is specified by means of an expression contain-ing a variable whose values are a l l the subclasses of B_. More generally, a set may be specified by an expression containing a bound variable whose values are a l l classes. It is, however, not evident that such definitions deserve to be called "circular." They are certainly not 24 29 circular in the usual meaning of "circular as applied to definitions. Russell did not at first consider impredicative definitions to be circular. His reply to the early criticism of B. Kerry is particularly noteworthy in this connection. Kerry had apparently criticized Frege's definition on the grounds that (1) there is no catalogue of R-hereditary properties, and (2) i t is a vicious circle that the property of belong-ing to the R-posterity of x is an R-hereditary property. Russell's reply to these criticisms was that: the proposition concerning every does not necessarily result from enumeration of the entries in a catalogue. ...general propositions can often be established where no means exist of cataloguing the terms of the class for which they hold (Principles, p. 522; PM, I, p. 45). The meaning of a definiens containing a universal quantifier is not the same as the conjunction of a l l its instances. The denial that impredicative definitions are circular was made in a particularly convincing way by Ramsey. We may use impredicative definitions to, e.g., specify a class, "just as we may refer-to a man as the tallest in a group, thus identifying him by means of a totality 30 of which he is himself a member without there being any vicious circle." Furthermore, Zermelo pointed out in answer to criticisms made by Poincare that impredicative definitions had been used for a long time in analysis 31 without being considered circular. 29 See Willard Van Orman Quine, Set Theory and Its Logic (Cambridge, Mass., 1963) p. 242. 30 Frank P. Ramsey, Foundations of Mathematics and Other Logical Essays, The International Library of Psychology, Philosophy, and Scientific Method, ed. C.K. Ogden (Paterson, N.J., 1960) p. 41. 31"A New Proof of the Possibility of a Well-Ordering" (1908) , in van Heijenoort, p. 191. It is true that impredicative definitions are indeed inadmissable i f sets and properties are understood in a certain way. If these are thought to be "constructed" entities—definitions then being descrip-tions of constructions—then, as Godel puts i t , "the construction of a thing can certainly not be based on a totality of things to which the 32 thing to be constructed itself belongs." Some writers had objected to impredicative definitions such as that of the finite numbers because in order to determine that a number 11 has a l l hereditary properties belonging to 0, and thus is finite, i t must be determined in particular that n has the property of having a l l 33 inductive properties. Is this property hereditary? To determine this 34 it must be ascertained whether i t belongs to n. Thus i t is alleged that there is a sort of circularity in verification, which makes veri-fication impossible. Carnap refutes this objection in a way similar to Russell's reply to Kerry: If we had to examine every single property, an unbreakable circle would indeed result, for then we would run headlong against the property 'inductive.' Establishing whether something had i t would then be impossible in principle, and the concept would therefore be meaningless. But the verification of a universal logical or mathematical sentence does not consist in running through a series of individual cases.... (Ibid., p. 189). Besides being subjected to allegations of circularity and unverifi-ability, the classical methods of definition have been criticized for 32 "Russell's Mathematical Logic," in The Philosophy of Bertrand  Russell, ed. Paul A. Schilpp (New York, 1963), p. 136. 33 Perhaps Felix Kaufmann, Das Unendliche in der Mathematik und  seine Ausschaltung, Eine Untersuchung liber die Grundlagen der Mathe- matik (1930, rpt. Darmstadt, 1968). "^See Rudolf Carnap, "The Logicist Foundations of Mathematics," in Bertrand Russell: A Collection of Critical Essays, ed. D.F. Pears (New York, 1972), pp. 185-186. 26 employing the concepts of a l l sets or a l l properties, or even the concept of a l l subsets of an infinite set. It is claimed by critics that such notions are vague or not sufficiently intelligible. So far I have explained the definitions for finite sets and numbers that were given by classical authors, the various criticisms of these, and the rebuttals of these criticisms. Of course, the critics were not satisfied with the classical point of view in general, and proceeded to develop their own systems. What did they put in place of the classical definitions? Did they avoid that circularity which was one of the main reasons for those definitions? Poincare opposed a l l attempts to define the ^finite) integers and prove the principle of mathematical induction, which he considered to be a synthetic a priori truth. Indeed, he thought a l l definitions of number circular. Weyl and Skolem seem to have been in basic agreement with Poincare. It was Weyl's view (in 1921 at least) that: Die Reihe der naturlichen Zahlen und die in ihr liegende Anschauung der Iteration ist ein letztes Fundament des mathe- . matischen Denkens. In unserm Iterationsprinzip kommt diese ihre grundsatzliche Bedeutung flif den Aufbau aller Mathematik zum Ausdruck. -*5 Apparently, on the basis of this "intuition" of the number series one recognizes the truth of the principle of induction: Auf dem Umstand...dass man, von 1 ausgehend und von jeder Zahl zur nachstfolgenden fortschreitend, schliesslich zu jeder beliebigen Zahl gelangen kann, beruht der wichtige Schlus der vollstandigen Induktion. The classical authors sought to avoid a l l use of "etc." and "...", — Hermann Weyl, Gesammelte Abhandlungen (Berlin, 1968), II, 149. 36 Das Kontinuum und andere Monographien (New York, n.d.), p. 17. 37 for they considered these to be entirely vague and unsatisfactory. Their opponents, on the other hand, are ready to set up the words "and so on" as the mark of essentially arithmetical thought. Weyl, for one, complained of the procedure of replacing "etwas spezifisch Arithmetischem, dem 'immer noch eins,' der Wiederholung in infinitum" by general logical 38 and set-theoretical concepts. In a 1923 work on the foundations of arithmetic, Skolem did not use " a l l " and "there i s ; " he took "the recursive mode of thought as a 39 basis." This means that the concepts of natural number and the suc-cessor function are not defined, and the principle of induction is assumed (Ibid., p. 305; cp. p. 300). New predicates and functions are introduced by recursive definitions. Skolem remarks that "a proof by mathematical induction represents an infinite process" (Ibid., p. 308). Moreover, i t seems that Skolem preferred to leave the concept "finite" undefined (Ibid., p. 299). Not a l l critics of classical methods, however, seem to have been completely content with the doctrines of Poincare, Weyl, and Skolem. 37 For example Russell said: "What are the numbers that can be reached given the terms '0' and 'successor?1 Is there any way by which we can define the whole class of such numbers? We. reach 1, as the successor of 0; 2 as the successor of 1; 3, as the successor of 2; and so on. It is this 'and so on' that we wish to replace by something less vague and indefinite. We might be tempted to say that 'and so on' means that the process of proceeding to the succes-sor may be repeated any finite number of times; but the problem upon which we are engaged is the problem of defining 'finite number,' and therefore we must not use this notion in our definition. Our defini-tion must not assume that we know what a finite number i s " (Intro- duction to Mathematical Philosophy [London, 1919], p. 21). 38 Gesammelte Abhandlungen, II, 522. 39 "The Foundations of Elementary Arithmetic Established by Means of the Recursive Mode of Thought, without the Use of Apparent Variables Ranging over Infinite Domains" (1923) in van Heijenoort, p. 304. Some wanted to define the finite numbers and to prove the principle of induction on the basis of that definition. For example, Hilbert and Bernays explained the numbers (numerals) as figures constructed accord-ing to a rule: Die Dinge, die wir, ausgehend von der Ziffer 1 , durch Anwendung des Fortschreitungsprozesses erhalten...sind Figuren von f o l -gender Art: sie beginnen mit 1 , sie enden mit 1; auf jede 1, die nicht schon das Ende der Figur bildet, folgt eine ange-hagte 1. Sie werden...durch einen konkret zum Abschluss kommenden Aufbau erhalten....^0 The principle of complete induction is then claimed to be not a self-evident principle, but a consequence "die wir aus dem konkreten Aufbau der Ziffern entnehmen" (Ibid., p. 23). We have seen that the classical authors (Peirce, Dedekind, Frege, Russell) formulated their definitions of finite sets and numbers in order to avoid circularity and vagueness. They seem to have thought the explanations of the natural numbers to be unacceptably vague and the obvious way of making them more explicit circular. Have those who have refused to use impredicative definitions been able to avoid circularity? It is not entirely obvious that they have. In the classical systems the principle of induction is an immediate con-sequence of the definitions of the finite numbers; i t seems that the constructivists s t i l l want this result, but i t is not clear that they succeed in getting i t . Van Heijenoort wrote: The repeated iteration of the successor operation...either is circular (to obtain any number, we take the successor of 0 a certain number of times) or rests upon hidden and rather com-plex set-theoretic assumptions ('finitely many times'). The 4<^David Hilbert and Paul Bernays, Grundlagen der Mathematik (Berlin, 1934), I, 21. 29 intuitive characterization is so clear because, in fact, no definition at a l l has been given.41 I believe that the classical authors would agree. 3) Dedekind's Definition Bolzano is the first that I know of to state that every infinite set contains a proper infinite part to which i t stands in a one-one correspondence. Though Bolzano did not use this property as the defin-ing characteristic of infinitude, he thought i t "a very remarkable peculiarity" which "to the disadvantage of our insight into many a 42 truth of metaphysics and physics...has hitherto been overlooked." This property of infinite sets was also mentioned by Cantor in 1878, but it was not used for the definition of the infinite sets. Cantor defined a finite set as one consisting of a finite number (Anzahl) of elements and then noted that "Ein Bestandteil einer endlichen Mannigfaltigkeit hat' immer eine kleinere Machtigkeit als die Mannigfaltigkeit selbst; dieses Verhaltnis ho'rt ganzlich auf bei den unendlichen, d.i. aus einer unendlichen Anzahl von Elementen bestehenden Mannigfaltigkeiten" (GA, p. 119). Dedekind remarks in a letter to Heinrich Weber in 1888 that Cantor "questioned the possibility of a simple definition in 1882 and was very surprised, when, motivated by his doubt, and at his wish, I communicated my definition to him; sometimes one possesses something, without appre-43 elating the worth and importance belonging to i t . " 41 "Logical Paradoxes," Encyclopedia of Philosophy, V, 356. 42 Paradoxes, sect. 20 (about 1850). Of course many earlier authors had noted particular examples of infinite sets which are equivalent to some proper part of themselves. 43 Gesammelte Mathematische Werke, ed. R. Fricke, E. Noether, and 0. Ore (Braunschweig, 1932), III, 488. 30 Later Cantor said that the property of not being equivalent to any of its proper subsets "must be regarded as an absolutely essential Merkmal of a finite set" (GA, p. 415) and attempted a proof that every finite set has this property. This theorem has been called "the fundamental theorem of finite arithmetic." Although Bolzano and Cantor were well aware of the fact that a l l and only infinite sets are in one-one correspondence with one of their proper subsets, Dedekind was the first to use this property for a defini-44 tion of the infinite sets, and, what is of real significance, he was the first to prove theorems on the basis of this definition. He did not, however, isolate a l l principles used in his reasoning, and his proof that there exists an infinite set cannot be reproduced in revisions of 45 set theory made necessary by the discovery of the paradoxes. A definition of the infinite sets would hardly be satisfactory unless i t had the consequence that a set is infinite i f and only i f i t is not equivalent to any i n i t i a l segment of the series of finite integers. Thus, we might say that any satisfactory definition of the infinite sets must be coextensive with the mathematical concept; on this Essays on the Theory of Numbers, p. 63. 45 There are of course systems having theorems of infinity. In the systems of Quine and Ackermann i t can be proved that there is an infin-ite set; see H. Scholz and G. Hasenjaeger, Grundziige der Mathematischen  Logik (Berlin, 1961), pp. 435-436. See also P. Bernays, "Zur Frage der Unendlichkeitsschemata in der Axiomatischen Mengenlehre," in Essays on  the Foundations of Mathematics, ed. Y. Bar-Hillel et al. (Amsterdam, 1962), pp. 11-12, and A. Tarski, "tiber unerreichbare Kardinalzahlen," Fundamenta Mathematica, 30 (1938), p. 85. 31 ground we may agree with Russell's remark that "Mathematical induction affords, more than anything else, the essential characteristic by which the finite is distinguished from the infinite" (Intro. Math. Phil., p. 27). 46 It seems likely that Dedekind would also have agreed with this. The "mathematical" definition of the infinite sets can be conveni-ently formulated in Dedekind's system as follows: A set is finite i f f for some natural number n, i t is equipollent to Z^; a set is infinite i f f i t is not finite. In view of the primacy attributed to the mathe-matical concept of infinity, we shall simply use the term "infinite" for infinity in this sense. For infinity in Dedekind's sense, we will use the term "D-infinite." Dedekind proved that a_ set is D-infinite i f f i t is infinite. He showed that a set equivalent to a D-finite set is also D-finite and that for every natural number n, is D-finite. Therefore any set equivalent to some Z is D-finite. Hence, i f a set A is not D-finite, then A is n — — not equivalent to any Z^ ; that is, i f A is D-infinite, then A is infinite. Furthermore, i t is easy to prove that every D-infinite set contains a simply infinite (denumerable) subset. Suppose that the set S^  is D-infinite, then there is a one-one function $ such that for some proper subset S' of j[, <J>[s] = S'. S is a <{>-chain, for <f>[s] £ S. Let s_ be an element of which does not belong to S'. ^ Q ^ ) ^ s a denumerable subset of S_. In order to prove that every infinite set is D-infinite, Dedekind showed that every set containing for each n_ a subset equivalent to Z^  46 See Werke, III, 459-460. is D-infinite, i.e. (VneN) GS'sS^^Z + D-inf(S). The proof of this assertion begins with an unacknowledged assumption. According to the hypothesis of the theorem, for each n. there is a one-one function mapping Z q into S_, i.e. for each n, the set A r of one-one mappings from Z r into S_ is not the empty set. Dedekind goes beyond the hypothesis of the theorem when he assumes that there is a sequence of functions a such that a belongs to A . Thus the proof of the theorem n n ° n needs the axiom of choice. That is, in addition to the hypothesis of the theorem, Dedekind must assume that there is a function f_ such that f(A ) = a e A in order to derive the consequent of the theorem, n n n The proof proceeds as follows. A sequence of one-one functions ty :Z •> S satisfying the condition n n J e> m £ n -*• ty (m) = ty (m) m n is defined by complete induction in terms of the functions a . Several explicit definitions are also needed. Let A = U A and 0 be a mapping n n of A into A such that for each argument 3, 0(3) is the element 6 of A such that meZ -»• 6(m) = 3(m), (where Z is the domain of 3), n n 6(n') = the least p such that aft,(p)^3(Zn). If functions were being considered as classes or ordered pairs, then 0(3) = 3u{<n',p». The function ty may now be inductively defined by the conditions: * i = V * „ ' = e ( * n > . n n 33 It follows by complete induction that for each n_, ty is one-one and that for all m < n, ty (m) = ty (m). m n The function u whose value for each n is ty (n) is a one-one map-— n r I 4 7 ping of N into .S. Thus, S_ contains the denumerable subset y [NJ . It is evident that every simply infinite (denumerable) set is D-infinite. ' Dedekind showed that every set containing a D-infinite subset is D-infinite. It follows that S is a D-infinite set. Dedekind also proved that a set not having a subset equivalent to some is finite. Therefore, if a set is infinite, then for each ri, it contains some subset equivalent to Z r . Thus, every infinite set is D-infinite. The authors of Principia Mathematica did not consider the equiva-lence of Dedekind1s and the "mathematical" definition of infinitude established, because they did not wish to commit themselves regarding the truth-value of the axiom of choice. All that is really established from their point of view is a conditional theorem whose antecedent is the axiom of choice and whose consequent is the equivalence in question. For this reason, and because Whitehead and Russell thought that neither way of defining "finite" and "infinite" can be regarded as "giving more 4 8 exactly than the other what is usually meant by" these words, the terms "inductive" (meaning finite in the sense of the "mathematical" 4 7 We saw above that it is easily proved without the axiom of choice that every D-infinite set contains a denumerable subset. But the axiom of choice is needed for the proof that every infinite set has a denum-erable subset. 4 8 But see the later remark by Russell quoted above (p. 31) with which we have agreed (looking at things from the historical point of view.) concept) and "reflexive" (meaning the same as "D-infinite") are intro-duced in Principia. In his Introduction to Mathematical Philosophy, Russell said that, "At present, i t is not known whether there are classes and cardinals which are neither reflexive nor inductive" (p. 88). It can be established without the axiom of choice that every inductive class is non-reflexive (i.e. every finite class is D-finite); indeed, the axiom of choice can be proved for inductive (finite) classes. But, in the opinion of Whitehead and Russell, the possibility is left open that there are non-inductive, non-reflexive classes (i.e. infinite classes which are D-finite.) 4) Potential Infinity Aristotle, to whom the term "potential infinity" is due, defined a (potentially) infinite quantity thus: "A quantity is infinite i f i t is such that we can always take a part outside what has been taken already" (Physics, 207a28). Each thing that is taken is finite but different from what has been previously taken. The idea is that what is potentially infinite is finite at each particular time but is never complete. In regard to the being of potential infinities, Aristotle said: "the infinite has this mode of existence: one thing is always being taken after another, and each thing is always finite, but always different" (206a28). The infinite, he says, does not exist as a sub-stance; its being "consists in a process of coming to be or passing away; definite i f you like at each stage yet always different" (206a35). Since Aristotle there have been many who have meant by infinite just potentially infinite. Thus Poincare says: "If the Menge M 35 possesses an infinite number of elements, this means not that these elements can be conceived of as existing beforehand a l l at once, but 49 that i t is possible for new ones to arise constantly...." Poincare illustrates his meaning with the case of the integers: "When I speak of a l l the integers, I mean a l l the integers which have been invented and a l l those which could be invented some day. ... And i t is this 'that can' which is the infinity" (Ibid., p. 60). Traditionally those who have admitted the reality of potential infinity have justified their views by the consideration of possibility. Thus, Aristotle supposed that matter is always divisible: "For division, being limitless, warrants the potential being of this infinite activity, but not the separate being of the i n f i n i t e . Y e t a skeptic (finitist) can raise the question of how i t can be known that i t is possible to divide any bit of matter whatever, so that the activity of dividing is potentially infinite. Aristotle himself says that " i t is evident that one discovers what is potential by performing an operation" (1051a30). But this does not apply in the case of potential infinity. Whereas those who maintain the truth of the axiom of i n f i n i t y ^ believe i t is evident that an infinite set exists, those who accept potential infinity must maintain the possibility of an endless process. In neither case can the grounds for the judgment be empirical; in both cases an a. priori recognition must be alleged. As Leibniz put i t regard-49 Henri Poincare, Mathematics and Science: Last Essays, trans. John W. Bolduc (New York, 1963), p. 59 50 51 °Metaphysics, trans. Richard Hope (Ann Arbor, 1960), 1048b14. See sect.22 on the axiom of infinity. ing potential infinity: "It is this rational consideration [that the same reason always exists for going farther] which achieves the notion of the infinite or the indefinite in possible progress. Thus the senses alone cannot suffice to cause the formation of these notions. Today, a large school of mathematicians engaging in foundational research maintains that potential infinity alone is mathematically admis sable. They will assume what they call "the abstraction of potential realizability," but not "the abstraction of actual infinity." Frege seems to have held just the opposite—that potential infinity is not sufficiently clear to belong to mathematics. He quoted Thomae's definition according to which a sequence is infinite i f "in accordance with a given perscription new terms and more new terms can always be 53 constructed." A sequence is (potentially) infinite i f i t is always possible to construct further terms. Does the possibility exist? For an almighty God, yes; for a human being no. We encounter here the difficult concept of the possible, but we can see in any case that the answer to our question is independent of the character of the terms composing the sequence. For sequences are not thereby divided into finite and infinite ones; a l l become finite or else a l l . infinite, depending upon the sense which is attached to 'can' (Ibid.). If the construction of sequences depends upon men, then a l l sequences are finite: "we can see in advance that the possibility of continuing will some time cease" (Ibid., p. 221). "Numerical sequences are in no more danger of being continued indefinitely than trees are of growing — New Essays Concerning Human Understanding, trans. A. G. Langley, 2nd ed. (Chicago, 1916), p. 158. 53 Translations from the Philosophical Writings of Gottlob Frege, ed. Peter Geach and Max Black (Oxford, 1970), p. 220. 37 up to heaven" (Ibid., p. 224). I suppose we could say that an endless succession of men could keep an infinite sequence going, or an immortal man. But here we only become more deeply involved with "the difficult concept of the possible." 5) Cantor on the Actual and  Potential Infinite Potential infinity as i t was conceived by Cantor was the infinity associated with the limit concept. Cantor explained the potential infin-ite as "an indeterminate, variable quantity which always remains finite but whose value either becomes smaller than any finite limit however small, or larger than any, however large" (GA, p. 409). An actual infinite is, however, "a constant Quantum fixed'in itself, which is greater than any finite quantity of the same kind" (GA, p. 409). Unter einem A.-U. [actual infinite] ist...ein Quantum zu vers-tehen, das einerseits nicht veranderlich, sondern vielmehr in alien seinen Teilen fest und bestimmt, eine richtige Konstante 1st, zugleich aber anderseits jede endliche Grosse derselben Art an Grosse Ubertrifft. Als Beispiel fiihre ich die Gesam-theit, den Inbegriff aller endlichen ganzen Zahlen an; diese Menge ist ein Ding fur sich und bildet, ganz abgesehen von der natiirlichen Folge der dazu gehorigen Zahlen, ein in alien Teilen festes, bestimmtes Quantum...das offenbar grosser zu nennen ist als jede endliche Anzahl (GA, p. 401). Cantor maintained that "every potential infinite presupposes an actual-infinite, i f i t is supposed to be rigorously utilizable mathe-matically." He explains why this is so as follows: If it admits of no doubt...that we cannot do without variable quantities in the sense of the potential infinite, then the necessity of the actual-infinite can also be thoroughly proved from this in the following way: In order that such a variable quantity be utilizable in a mathematical consideration, the domain of its variability must strictly speaking be known in advance through a definition; this domain cannot however be 38 again something variable, since otherwise every fixed basis of the consideration would f a i l ; therefore this domain is a definite actual-infinite set of values (Wertmenge) (GA, p. 410). Frege was in agreement with Cantor that the potential infinite presuppos-ed the actual infinite. It was, of course, the case that the classical method of limits was based on actual infinite sets: "To be sure, the infinitely small and the infinitely large were eliminated from analysis, as established by Weierstrass.... But the infinite s t i l l appears in the infinite number sequences that define the real numbers, and, further, in the notion of the real number system, which we conceive to be an 54 actually given totality, complete and closed." Cantor once said that One can say unconditionally: the transfinite numbers stand  or f a l l with the finite irrational numbers; they resemble each other, in their innermost essence; for the former as well as the latter are definite delimited formations or modifica-tions...of the actual infinite (GA, p. 396). As Zermelo pointed out, the assumption of the existence of infinite sets is not needed for the theory of finite numbers or of rational num-bers, but this is not so "for higher analysis and function theory in which the limit concept and irrational numbers play the leading role." The irrational numbers are infinite sets or sequences of rational num-bers, "and likewise a limit can only be defined by an infinite set of argument and function values." Zermelo concludes that "whoever wants to treat seriously the rejection of the 'actual infinite' in mathematics, must renounce the whole of modern analysis."^^ Later developments have 5 4David Hilbert, "On the Infinite" (1925), in van Heijenoort, p. 369. ^"Uber die Grundlagen der Arithmetik," Proceedings of the 5th  International Congress of Mathematicians, 2 (1909), 11. shown, of course, that not all of modern analysis has to be given up along with the actual infinite. Moreover, the intuitionists have been content to relinquish what cannot be established from their point of view. In Cantor's opinion the potential or indefinite infinite has significance only "als Beziehungs-begriff, als Hilfsvorstellung unseres Denken;" it does not constitute an idea by itself (GA, p. 373). Cantor says that he can ascribe "no being" to the "indeterminate, improper, or variable infinite." The reason is that in this form the infinite is only a "pure subjective idea or intuition," and not an "adequate idea." The improper or potential infinite belongs to the imagination, not to the understanding (GA, p. 205). The concepts of Cantor's set theory relate only to actual infinite sets, since only such sets have an interest for us, which are deter-minate in themselves and of which therefore all elements must be conceived as ready made (fertig) existing together. The potential infinite does not come into account here, because according to its concept it can be related only to indeter-minate or variable things (GA, p. 422). Traditional authors often maintained that the infinite is incom-prehensible; for example, Arnauld said that "our minds are finite; and blinded by the infinite, they are lost in it—forever overwhelmed by 56 the multitude of conflicting thoughts the infinite furnishes." Cantor discussed the claim that the actual infinite is incomprehensible on several occasions. In particular, he was concerned with the contention that actual infinite numbers are beyond human understanding. ^The Art of Thinking; Port-Royal Logic, trans. James Dickoff and Patricia James, Library of Liberal Arts (Indianapolis, 1964), p. 297. 40 The finitude of the human understanding is so often cited as the reason why only finite numbers are conceivable; yet I see in this assertion...a circular inference. It is tacitly sup-posed in the 'finitude of the understanding' that its capacity of forming numbers is limited to finite numbers. If we see, however, that the understanding is also in a determinate sense infinite, i.e. i t can define transfinite (uberendliche) num-bers and distinguish them from each other, then either an extended meaning must be given to the words 'finite under-standing,' whereupon that conclusion can no longer be drawn from them; or the predicate 'infinite' must also be granted to human understanding in a certain respect, which in my opinion is the only correct conclusion (GA, p. 176). Cantor stated emphatically that the transfinite "may not be designated as 'transcendent' (i.e. exceeding the human powers of understanding)" (GA, p. 391). DeMorgan also combated the doctrine of the incomprehensi-bility of the infinite: Those who affirm that infinity is inconceivable, ought to mean that i t is not adequately imageable as a quantity in relation to finite quantities. This is true: but reasoning requires only such conceptions as gives propositions. ... Let there be two sects: and let them be distinguished by the acceptance and rejection of unimaged concepts. The rejectors, i f consis-tent, will decline a l l magnitudes which cannot be placed before the mind's eye.57 DeMorgan pointed out that we need no more have an infinite mind in order to apprehend the concept of infinity than we need have a blue mind to apprehend the concept blue (Ibid., p. 157). Cantor also dealt with the objection that even i f the set of natural numbers must be taken as an actual infinite set, we, unlike God, are not able to grasp the cardinal or ordinal number of this set because "wir bei der Beschrankheit unsers Wesens nicht imstande sind, alle die unendlich vielen zur Menge (n) gehorigen Zhalindividuen ii uno intuitu "On Infinity; and on the Sign of Equality." Cambridge Philosophical  Society, Transactions, 11 (1866), 155-156. 41 aktuel zu denken" (GA, p. 402). Cantor's answer is that the same is true in the case of large finite numbers, and nevertheless there can be knowledge of these numbers. Furthermore, it is almost never the case that something is known by intuitively taking it in as a whole as it is in itself without the use of negation, symbols, and examples. On the contrary, our way of conceiving enables us "to determine a thing from general predicates and with the aid of comparisons, exclus-ions, symbols or examples in such a way that it is well-differentiated from every other thing." As concerns infinite numbers he says: I now go so far as to unconditionally assert that this second kind of determination and delimitation of things is incompar-ably more simple, convenient, and easy for the smaller trans-finite numbers (e.g. for w or to+l or o)n for small finite num-bers 11) than for very great finite numbers for which we also are dependent on the same resources corresponding to our in-complete nature (GA, p. 403). A similar point has been made recently by Gerog Kreisel: ...finiteness and counting seem to be more difficult (for humans, not mechanical devices) than abstract operations. Isn't the plain fact simply this: whatever special role finiteness may have in our thinking it's not simply that the objects we think about are finite. 6) Absolute Infinity Besides infinity in quantity (continuous or discrete) tradition distinguished absolute infinity, which was the infinity of God or of His perfections (attributes). The idea was—at least Leibniz took this view—that the perfections of God are attributes which are also present in His creatures: "The divine perfections are concealed in all things." "Two Notes on the Foundations of Set-Theory," Dialectia (1969), p. 96. 59 Philosophic Papers and Letters, selected, trans., and ed. Leroy E. Loemker (2nd edn.; Dordrecht-Holland, 1969), p. 608. Henceforth PPL. 42 But in God perfections are infinite in degree. In creatures these perfections occur in limited degree, while in God they are supposed to be absolutely unbounded, so that there cannot be a higher degree. Thus, "God is an absolutely perfect being;" He has all perfections and each in a degree that has "no limits."^ ...God is absolutely perfect, perfection being understood as the magnitude of positive reality in the strict sense, when the limitations or the bounds of those things which have them are removed. There where there are no limits, that is to say, in God, perfection is absolutely infinite.^ This is also connected with the claim—often debated in olden times— that there is a positive idea of infinity: "ultimately it may be said that the idea of the absolute is anterior in the nature of things to that of the limits which are added, but we notice the former only as 62 we commence with what is limited and strikes our senses." To think of "an infinite in perfection" beginning from the limited, "we need only to think of the absolute, by setting aside all limitations" (PPL, p. 626) The absolute infinite is an absolute maximum. The concept of absolute infinity is also explained by Spinoza. A thing is absolutely infinite in its own kind if it cannot be limited (or equaled) by anything else of that kind. The attributes of God are supposed to be infinite in this sense; each of God's infinitely many 63 attributes is "in the highest degree perfect in its kind." 60Discourse on Metaphysics, in Leibniz Selections, ed. Philip P. Wiener (New York, 1951), sect. 1, p. 290. 61Monadology_, in Wiener, sect. 41, p. 541. 62 New Essays, p. 158. 63 Spinoza Selections, p. 404. Cantor too explained absolute infinity; he distinguished two kinds of actual infinite—the transfinite and the absolute infinite. Something which is transfinite (the infinite numbers) can always be enlarged (Vermehrbares), while the absolute infinite is conceived "wesentlich als unvermehrbar und daher mathematisch undeterminierbar" (GA, p. 375). Cantor also refers to the absolute infinite as the "true" infinite (GA, p. 175) and says that as concerns the absolute or true infinite (God) he is in agreement with tradition. Though each transfinite number is surpassed by another trans-finite number, Cantor thought that the whole set of transfinite numbers deserved to be called "absolutely infinite." The whole series of numbers is even In a sense incomprehensible, like God. We are not able to attain "even an approximate comprehension (Erfassung) of the absolute." CHAPTER II INFINITE NUMBERS 7) Traditional Objections to  Infinite Numbers Until the modern period of foundational studies, a number was usually defined as a plurality (multitude or aggregate) of units. Here are a few sample definitions from some old-time arithmetic books. "Number is a multitude brought together or assembled from several units, and always from two at least."^ "...number...containeth a multitude of 2 unities: And is nothying els but a collection of unities." In both of these definitions a number is said to be a plurality of units, but 3 we can also find a number defined as "that whereby any thing is numbered." These are in fact the two sorts of number distinguished by Aristotle. As a rule Aristotle defined a number as a plurality of units, though not just a plurality, but "a limited plurality" (Metaphysics, 1020a13). Aristotle also says that a number is "a measured plurality or a plurality of measures" (1088a4). Thus, numbers (in this sense) are finite by '''The unknown author of the first printed arithmetic (1478) partially trans. David Eugene Smith, "The First Printed Arithmetic," in Isis (1924), p. 314. 2 Robert Record, The Whet-Stone of Witte (1557), f a c , The English Experience Series (New York, n.d.). 3 William Bedwell, The Principles of Arithmetic (1616), p. 1. - 44 -definition. A number is not just a plurality, i t is a limited or count-able (measurable) plurality. Likewise, Nicomachus of Gerasa defined number as "limited multitude," but this is not present in Euclid's version: "A number is a multitude composed of units." It seems that Plato, like Aristotle, also understood a number to be a limited multi-tude,^ and Plotinus states that "limitlessness and number are in contra-diction," no doubt because a number is countable. A limited quantity is a definite quantity. Socrates speaks of "definite and measured quantity,"^ and says that "definite quantity is something that has stop-ped going and is fixed." But Plato also recognized that there are also unlimited multitudes, in particular, the multitude of a l l numbers: "now, i f number is, there must be many things that are, for we must admit that number, unlimited in plurality, also proves to have being. Kant also defined number in terms of counting, but may have meant, unlike Aristotle, that a number is an aggregate which has been actually counted; for he said: "number...is itself only an aggregate distinctly apprehended by numeration, i.e., by the process of adding one to one successively in a given time."^ DeMorgan certainly thought of a number as an actually counted multitude, and was definitely of the opinion that not a l l multitudes are numbers. Number, he said, _ Plato would no doubt have wanted to say that a number is a limited plurality of pure units or a form in which they participate. 5"Philebus," trans. R. Hackforth, in The Collected Dialogues of  Plato, ed. Edith Hamilton and Huntington Cairns (New York, 1961), 24c. 6Ibid., "Theaetetus," trans. Francis M. Cornford, 144a. ^Kant's Inaugural Dissertation and Early Writings on Space, trans. John Handyside (Chicago, 1929), p. 65. 46 is a distinction made by thought between one plurality and another.... Common language, which in general confounds the two ideas, knows how to distinguish: there is no number without multitude, but multitude without number is intel-ligible use. 8 Among the multitudes which are not numbers are the infinite multitudes; these are the innumerable multitudes. In our ideas of external things there is infinite multi- tude; but we cannot subjoin, and we have counted i t . And number, except as counted, is not number: i t is vague multi-tude under the name of number. Multitude may be counted: number has been counted (Ibid., p. 162). Although Aristotle, and many others before and after him, usually said that a number is a plurality of units, he actually distinguished two senses of number; "Number, we must note, is used in two senses— both of what is counted or the countable and also of that with which we count" (Physics, 219^5). I do not know of any passage in which he explains what he means by "that with which we count," but i t may be the number in abstraction mentioned in the following argument against the infinite: Nor can number taken in abstraction be infinite, for number or that which has number is numerable. If then the numerable can be numbered, it would also be possible to go through the infinite (Ibid. , 204b7-10, cp. Metaphysics, 1066t>26). That which has number is evidently the plurality which is usually said to be a number. A plurality which in one sense is a number, but which in the other sense has a number, is countable. Evidently Aristotle means that number in abstraction is always the number of some plurality which is countable; hence, number taken in abstraction is always finite. "On Infinity; and on the Sign of Equality," p. 158. 47 It would seem that Aristotle might admit the possibility of infin-ite pluralities, even i f there are no infinite numbers in any sense. He says that "...plurality is, as i t were, a kind including number; for number is plurality measurable by one" (Metaphysics, 1057a3). This sug-gests that there might be pluralities which are not measurable by one (i.e. not countable). Yet he also says that "A quantity...is a plurality i f i t can be counted..." (Metaphysics, 1020a9). Moreover, Aristotle seems to object to infinite quantity in general when he says that "the infinite cannot be quantity—that would imply that i t has a particular quantity, e.g. two or three cubits; quantity just means these" (Physics, 206a3). This is actually just another expression of the doctrine that 9 a l l numbers are finite. A number is either one or two or three, or any number which can be reached by counting;"^ this is what i t meant to say that a number must be some definite number.^ If Aristotle intended the objections quoted above to hold against infinite multitude, i t is obvious that they are insufficient, as was remarked by St. Thomas: "anyone who would claim that there is an infinite multitude would not hold that i t is a number, because number 9 Aristotle did not really regard one as a number. ^As Hobbes put i t , "Number is one and one, or one one and one, and so forwards" (Hobbes Selections, ed. Frederick J.E. Woodbridge [1930; rpt., New York, 1958], p. 72). ^Even in early manuscripts of Russell (c. 1900) we find the opinion expressed that what is infinite "does not contain an assign-able number of simple parts." The collection of numbers is one "of which no definite number can be asserted. Thus when a collection is given, i t must always remain a question whether or not i t has a number." "A collection is infinite when no number N that can be specified is the number of its terms." 48 12 is multitude measured by one.... But nothing measured is infinite." St. Thomas upholds Aristotle's definition of numbers as measurable pluralities, but points out that nobody holding that there are infinite pluralities would say that they are numerable. He apparently does not think that anyone would object to the definition of number. It may even be said that tradition did not really have objections to or arguments against infinite numbers; numbers were simply defined to be finite. Aristotle's argument was criticized in a very different way by Cantor. He pointed out that infinite sets can also be numbered in a way that does not involve the "going through" of which Aristotle spoke. It is only necessary to formulate a law defining a relation well-ordering the set whose elements are to be counted and to determine the number which is the type of that order. Cantor remarks: "Dass ohne eine solche gesetzmassige Sukzession der Elemente einer Menge keine Zahlung mit ihr vorgenommen werden kann—dies liegt in der Nature des Begriffes Zahlung..." (GA, p. 174). If numbers are not required to be limited or countable multitudes, then, following the traditional definition of number in the most common sense, we would define an infinite number as an infinite plurality of units. One of Newton's criticisms of Bently is interesting in this con-nection. Bently apparently had said that a positively existing infinite arithmetical sum or number is an absurdity and a contradiction in terms. But Newton replied that Bently did not __ Commentary on The Metaphysics of Aristotle, sect. 2329. Cp. his Commentary on Aristotle's Physics, trans. Richard J. Blackwell and W.E. Thirlkel (New Haven, 1963), sect. 352. 49 prove, that what Men mean by an infinite Sum or Number, is a Contradiction in Nature; for a Contradiction in Terminis implies no more than an Impropriety of Speech. Those Things which Men understand by improper and contradictious Phrases, may be sometimes really in Nature without any Contradiction at a l l : Silver Inkhorn, a Paper Lanthorn, an Iron Whetstone, are a l l absurd Phrases, yet the Things signified thereby, are really in Nature. If any Man should say, that a Number and a Sum, to speak properly, is that which may be numbered and sum-med, but Things infinite are numberless, or, as we usually speak, innumerable and sumless, or insumable, and therefore ought not to be called a Number or Sum, he will speak proper-ly enough.... And yet i f any Man shall take the Words, Number and Sum, in a larger sense, so as to understand thereby Things, which in the proper way of speaking are numberless and sumless (as you seem to do when you allow an infinite Number of Points in a Line) I could readily allow him the Use of the contra-dictious Phrases of innumerable Number, or sumless Sum, with-out inferring from thence any Absurdity in the Things he means by those Phrases.13 Newton in fact was definitely in favour of a different modification of the traditional definition of number. There was a tendency, beginning apparently with Wallis, to define the number concept in such a way as to include rational and irrational numbers, as well as whole numbers. Here is the definition given by Newton: By a Number we understand not so much a Multitude of Unities, as the abstracted Ratio of any Quantity, to another Quantity of the same kind, which we take for Unity. And this is three-fold: integer, fraction, and surd: An Integer is what is measured by Unity, a Fraction that which a submultiple Part of Unity measures, and a Surd, to which Unity is incommensur-able. 1 4 13 Isaac Newton's Papers and Letters on Natural Philosophy, ed. I.B. Cohen (Cambridge, 1958), p. 304. 1 4Universal Arithmetick (1728, first published 1722), p. 2. 50 8) Abstract Number and  Numerical Equality Although the most common definition of number said that a number is a plurality of units, number in a more abstract sense was also some-times recognized. According to this more abstract conception of number, a number is something which different sets (i.e. numbers in the concrete sense) may have in common: thus, we speak of an abstract number as some-thing belonging to a concrete number of things and of two sets having the same (abstract) number. Plato actually says very l i t t l e about the number concept in his dialogues, but i t would seem that he must have con-sidered an abstract number to be a Form.''""' There are some passages in Aristotle which are relevant to the con-cept of abstract number. In the Posterior Analytics, he says that number is a genus whose species are dyad, triad nad.... What belongs to a species of number is a number in the sense of a plurality of units, i.e. what I have been calling a "concrete" number.^ The same idea also is expressed in the Physics: It is said rightly, too, that the number of the sheep and of the dogs is the same number i f the two numbers are equal, but not the same decad or the same ten.... For things are called the same so-and-so i f they do not differ by a differ-entia of that thing, but not i f they do Therefore the number of two groups also is the same number (for their num-ber does not differ by a differentia of number), but i t is not the same decad (224a3-13). As Anders Wedberg explains i t : "From the point of view of Plato's theory of Ideas both the common Greek and Pythagorean conception of number appear very imperfect. Since numbers are pre-dicated of many particular collections of objects, the numbers them-selves must—according to the theory of Ideas—be certain ideal entities above those collections" (Plato's Philosophy of Mathematics [1955], p. 74). 1 6Trans. G.R.G. Mure (Oxford, 1928), 96b. 51 In his commentary on the Physics, St. Thomas explains that Aristotle means in this passage that a number is a genus having the various numbers as species: Number is divided into diverse species, one of which is ten. Therefore, a l l things which are ten are said to have one number, because they do not differ from each other in res-pect to the species of number, but are contained under the one species of number.17 St. Thomas accepted this doctrine of abstract number and even had an argument against infinite multitude which makes use of i t . It was mentioned above that St. Thomas criticized an argument of Aristotle's insofar as i t was directed against infinite multitude, but Thomas seems to have thought that he had a good argument against this, not merely against infinite number. Here It i s : every kind of multitude must belong to a species of multitude. Now the species of multitude are to be reckoned by the species of number. But no species of number is infinite; for every number is multitude measured by one. Hence i t is impossible for there to be an actually infinite multitude....1^ We have seen that Newton believed that there is no difficulty in expanding the conception of number in the sense of a plurality of units (or that which has number) to include infinite numbers. It is evident that the same thing can be done in the case of numbers in the second sense distinguished by Aristotle, or In the sense of number in which different pluralities of units may have the same number. We will simply say that a number is a species of pluralities; an infinite number (in this sense) is a species of infinite pluralities. ^Commentary on Aristotle's Physics, p. 284. 18 Summa Theologica, I, 7, iv. 52 But one very important thing is s t i l l lacking. Aristotle spoke of two numbers (=pluralities, in his example, decads) as being equal, but he does not define under what condition two pluralities are equal in number or when they belong to the same species of number. He would, no doubt, have given the counting criterion. Two pluralities have the same number when the result of counting the members of the one is the same as the result of counting the members of the other. This criterion works only for finite sets, but there is another criterion of equality in (cardinal) number which works for infinite as well as finite pluralities. This criterion was explicitly formulated by Hume: We are possest of a precise standard, by which we can judge of the equality and proportion of numbers.... When two num-bers are so combin'd, as that the one has always an unite answering to every unite of the other, we pronounce them equal.... 19 Now we can define a number as a species of pluralities which are equal in number in the sense explained by Hume, i.e. a number is a species of pluralities which are in one-one correspondence with each other. Of course, Hume did not himself use this criterion of numerical equality as a basis for a definition of infinite numbers. He knew nothing at a l l about infinite numbers, and, in fact, he even said that "infinite num-bers, properly speaking, can neither be equal nor unequal with respect to each other" (Ibid., p. 46). This was a common medieval doctrine, which goes back to Aristotle. So far as I know, about a l l Aristotle ever says about equality is that things are "equal when they have one quantity" (1021a12). We have — A Treatise of Human Nature, ed. L.A. Selby-Bigge (Oxford, 1964), p. 71. 53 supposed, however, that Aristotle would have allowed that two pluralities have the same quantity (abstract number) when the result of counting them is the same. But the criterion stated by Hume is not only of more general applicability, i t is also more fundamental. The reason is that one-one correspondence is involved in counting. As Russell put i t , "the notion of [one-one correspondence] is logically presupposed in the operation of counting, and is logically simpler though less familiar" (Intro. Math. Phil., p. 17). We may also note that besides determination of the number of things in a set (counting), there is also the determination that two sets have the same number of elements. Sometimes this can be done without counting by apprehending that there is a one-one corres-pondence between the elements of the two sets. Thus i t is possible to know that two sets have the same number of elements without knowing what their number is. It seems appropriate to close this section with a remark on "extensions" of the number system. The Cantorist maintains that the system of transfinite numbers constitutes a "natural" extension of the concept of whole, positive number (Cantor, Godel). The key to this ex-tension is the explication of "numerical equality" in terms of the existence of a one-one correspondence. The concept of a correspondence is of course itself in need of explication. This extension is as legitimate as the extension of the number concept to include irrational and complex numbers. In fact, these "numbers" are much further removed from the original concept than the transfinite numbers. One may well ask, what is the common feature which is the basis of the application of the t i t l e "number" to the original integers 2,3,4,...and the entities of the extended systems? The answer seems to be the. existence of ordering relations of certain properties between the "numbers." It is particularly desirable that any two "numbers" should be comparable with respect to the relation of order, i.e. the ordering relation should be connected as well as irreflexive and transitive. 9) The Frege-Russell Definition  of Numbers We have seen that tradition (sometimes) distinguished between two senses of number, though favoring the definition of number as a count-able or limited plurality of units. Newton pointed out that there is no serious objection to defining a number as any plurality of units, thus allowing for infinite numbers. We then noted the possibility of likewise extending the concept of number In the abstract sense, accord-ing to which a number is something common to various different multitudes It was not, however, until Frege (1884) and Russell (1901) that an explication of the concept of abstract number, which would accommodate infinite as well as finite numbers, was advanced and justified on the basis of logical principles. There are only two requisites for such an explication: (1) Every set has one and only one cardinal number, and (2) equipollent sets have the same cardinal number. If the explication is logically satisfactory, these assertions should follow from i t and the principles of logic and set theory. According to the Frege-Russell definition, a cardinal number is a 55 class of equipollent classes; the cardinal number of the class A is the class of a l l classes equipollent to A. Aristotle and St. Thomas defined an abstract number as a species of number (in the concrete sense), i.e. a species of multitudes; but they do not seem to have understood by a species a class or a set. Moreover, they did not say that a number is a species of equipollent multitudes, and, of course, St. Thomas, and no doubt Aristotle too, recognized only species of finite multitudes as numbers. Russell formulated his definition of the cardinal number of a class during the time—after becoming acquainted with Peano's symbolic l o g i c — when he was setting up his logic of relations. In fact Russell f i r s t published his definition in his 1901 paper "The Logic of Relations, with 20 Some Applications to the Theory of Series." This paper contains the proof of an important theorem in the logic of relations, which Russell called "the principle of abstraction." The theorem provided the basis of Russell's definition of the cardinals, as well as other kinds of numbers. This theorem also appears in Principia Mathematica, where i t is said that the principle of abstraction (*76.66) "embodies a great part of the reasons for our definitions of the various kinds of numbers" (PM, I, 442). The relation of being equi-numerous or of equivalence has the properties of being reflexive, symmetrical, and transitive. Any relation which is symmetrical and transitive may be called an equality relation — Logic and Knowledge: Essays 1901-1950, ed. Robert C. Marsh (New York, 1956). 56 or an equivalence relation. If a relation has these two characteristics, then i t follows that it is reflexive in its field, i.e. i f x belongs to the field of R, then xRx (Principles, p. 219). The field of a symmet-rical relation = its domain = its converse domain. Possession of a common property is an equivalence relation. That is, the relation E defined by the formula: xEy P(x) & P(y), is an equivalence relation. The relation of being the same in some res-pect P_ is an equivalence relation. On the other hand, as Russell noted, if there is an equivalence relation holding between any two of the things a,b,c,... there is a tendency to attribute a common property to these 21 things. Russell found i t advantageous for his purposes to explicate the concept common property so that a common property need not be a property in the usual sense of being a universal, i.e. a common property in his sense could just as well be an object (substance, class) which is the common possession (in a precisely definable sense) of the things said to have the common property. Russell says that "the most general way in which two terms can have something in common is by both having a given relation to a given term" (Principles, p. 51). He maintained that "The so-called predicates of a term are mostly derived from relations to other terms" (Ibid., p. 471; cp. pp. 166, 226). This could also be expressed by saying that most properties or attributes of things are relational properties. — See especially Our Knowledge of the External World, pp. 101-102. 57 The logical theorem which Russell called the principle of abstraction states that if is an equivalence relation, then there is a many-one relation R_ such that J5 is identical to the relative 21 product of R. into R. converse. This means (roughly) that whenever an equivalence relation holds between certain entities, there is a common property which they a l l have. The proof of the principle shows that the common property which always will exist is the class of a l l those things 22 each two of which have the equivalence relation to each other. That i s , for any equivalence relation S_, any things x and y_ which have j3 to each other have at least this common property (in the sense of Russell's explication): {y:xSy} = {x:ySx} . As Russell says, the class of terms that have the given transitive symmetrical relation to a given term will f u l f i l a l l the formal requisites of a common property of a l l the members of the class. Since there certainly is the class, while any other common property The relative product of the relation R. into the relation S_, R | S is the relation which holds between x and y_ i f f there is a z_ such that xRz & zSy. Therefore, xR|Ry i f f there is a z to which x and y_ both have the relation R.. In the case of transitive; sym-metrical relations ;S, like the relation of numerical equality, there is always a many-one relation R such that for any terms x. a n a X i n t n e field of S_, xSy i f f x and y_ both have R to a unique term z_. 22 "fc The theorem is proved by showing that the relation Conv[SfF(S)] is such a relation R. By definition: GS"x -*-»-G = {y:xSy}. Thus, i f x does not belong to the domain of S^, 0t>x; hence, x may belong to the converse domain of S~ but not to the domain of S. The converse domain of S~ restricted to the field of S_, i.e. the converse domain of ? T F(S) , is identical to the domain of S_, which is identical to the field of jS since J5 is an equivalence relation. The converse of S" f F(S) holds between any element of the domain of S_ and the class of its S-successors: x{Cnv(SPF(S)) }G «-»• xeD(S) & G = {y:xSy} Thus R = Cnv(S>F(S)) is a many-one relation holding between terms of S^  and their respective equivalence classes. may be illusory, i t is prudent, in order to avoid needless assumptions, to substitute the class for the common property which would be ordinarily assumed. ... In the absence of special knowledge...the method we have adopted is the only one which is safe, and which avoids the risk of introducing fictitious metaphysical entities.23 From this point of view, the principle of abstraction could be called "the principle of dispensing with abstractions," and Russell does so refer to i t in later writings. When S_ is an equivalence relation, such classes as {y: xSy) may be called equivalence classes of the relation S_. The things x. and y_ are in the relation S_ i f and only i f they have the common property of belonging to the same equivalence class. An equivalence relation parti-tions its field into non-empty disjoint equivalence classes whose union is that field. This is easily shown: If x and y_ are not equivalent, their respective equivalence classes are disjoint, for anything equival-ent to the one will not be equivalent to the other. Indeed i f x and y_, are not equivalent, then i f we suppose that there is some z_ in both {y: xSy} and {x: ySx}, such a z_ will be equivalent to both x and y_, so (by the transitivity of an equivalence relation) x. and y_ will have to be equivalent, in contradiction to the assumption. Hence, the relation of equivalence between sets partitions the class of a l l sets into dis-joint, non-empty equivalence classes; these classes have the requisites 24 for being the cardinal numbers. Our Knowledge of the External World, p. 102. This definition will be seen to require some modification. 59 Remarks on the Justification of the  Frege-Russell Definition Since ancient times there seems to have been substantial agreement on what is meant by "number" in the concrete sense, i.e. that which has number in the abstract sense. What has a number is a set or a plurality 25 of things. There was, however, not such agreement concerning numbers in the abstract sense, i.e. the numbers which may belong to different sets (which are equal in number). Furthermore, the various definitions of abstract number which were frequently given by philosophers and mathe-maticians are imprecise and worthless as bases of proofs. Therefore, there was ample justification for introducing some new precise definition, 26 usage being indeterminate and the existing alternatives unsatisfactory. The Frege-Russell definition of the cardinal numbers is an example 27 of what Carnap has called explication. This is the procedure of formu-lating a definition of a term already in use but which is either not defined or has only an imprecise or otherwise unsatisfactory definition. In explication the aim is not to try to formulate what the term really means or what people have always actually meant by i t . Hempel states 2 5 It should be noted, however, that Frege was inclined to att r i -bute numbers primarily to concepts. 26 It is easy to find a variety of obscure and logically worthless definitions of numbers in the abstract sense. See, for example, Descarts, The Philosophical Works of Descartes, trans. E.S. Haldane and G.R.T. Ross (New York, 1955), I, 242; Leibniz, PPL, pp. 76, 100; Peirce, Collected Papers, IV, 555; Cantor, GA, p. 418; Russell, Critical Exposition of the Philosophy of Leibniz (New ed.; London, 1937), p. 112; Arend Heyting, Intuitionism: An Introduction (Amsterdam, 1956), p. 13. 27 Logical Foundations of Probability, (2nd ed.; Chicago, 1962), esp. sects. 2,3 and 6. 60 the aim of explication very well in saying that i t is to give the expressions to be explicated "a new and precisely determined meaning, so as to render them more suitable for clear and rigorous discourse 28 on the subject matter at hand." This is the point of Frege's remark: To those who feel inclined to criticize my definitions as unnatural, I would suggest that the point here is not whether they are natural, but whether they go to the root of the matter and are logically beyond criticism.29 Similarly, Russell said of his definition of cardinal number that i t is sure to produce, at first sight, a feeling of oddity, which is liable to cause a certain dissatisfaction. It defines 2, for instance, as the class of a l l couples, and the number 3 as the class of a l l triads. This does not seem to be what we have hitherto been meaning when we spoke of 2 and 3, though i t would be difficult to say what we had been meaning.30 A feature of Russell's definition which especially appealed to him was that "new indefinables and indemonstrables are wholly avoided" (Principles, p. 127). In Principia Mathematica, i t is implied that Russell's definition of the cardinal numbers is the simplest among many otherwise satisfactory definitions; Whitehead and Russell say that unless we adopt this definition or some more complicated and practically equivalent definition, i t is necessary to regard the cardinal number of a class as an indefinable. Hence the above definition avoids a useless indefinable with its atten-dant primitive propositions (PM, II, 4). Thus, the authors justify their definition as being the simplest defini-tion satisfying the requisite that "the formal properties which we Fundamentals of Concept Formation in Empirical Science (Inter-national Encyclopedia of Unified Science, ed. Otto Neurath, II, 7 [Chicago, 1952]), p. 11. 29 Foundations of Arithmetic, p. x x i i i . 30 Our Knowledge of the External World, p. 159. 61 expect cardinal numbers to have result from i t " (Ibid.). The number of a class A cannot be defined as the property (in the unual sense) common to A and a l l equivalent classes, but no other classes. We cannot speak of the common property of A and a l l equivalent 31 classes, because there are a great many such common properties. Russell remarked concerning the possibility of defining numbers as properties that philosophically we may admit that every collection of similar classes has some common predicate applicable to no entities except the classes in question, and i f we can find, by inspec-tion, that there is a certain class of such common predicates, of which one and only one applies to each collection of similar classes, then we may if we see f i t , call this particular class of predicates the class bf numbers. For my part, I do not know whether there is any such class of predicates, and I do know that, i f there be such a class, i t is wholly irrelevant to Mathematics (Principles, p. 116; cp. 167n). It should be clear, especially from this passage, that Russell did not have the absurd idea that he was finally discovering what numbers really are. 10) Proofs that There Are Infinite Sets and  Numbers There have been many philosophers who have supposed the existence of infinitely many things, and many who have denied this. The question of the size of the universe, for instance, has been given more answers than one would expect. It has been held to be finite by, among others, Aristotle, St. Thomas, and Kepler. Writers such as Lucretius, Crescas, Bruno, Spinoza, Cantor, and Russell (1903) have, on the other hand, See, however, Carnap's method of explicating the number concept in Meaning and Necessity, 2nd ed. (Chicago, 1956), pp. 115-117. 62 thought i t infinite. Leibniz believed that there are infinitely many substances, but no infinite whole containing them a l l as parts or elements. Newton, Clarke, and Locke thought the universe to be spati-ally and temporally infinite, but to contain only a finite amount of matter. Nicolas of Cusa and Kant maintained that the universe is neither finite nor infinite, but somehow indefinite. Galileo, Descartes, Russell, and Lukasiewicz thought that the universe might well be either finite or infinite, but that we cannot know which. Some positivists seem to have held that the question of the infinitude or finitude of the universe is meaningless. A priori arguments have been common. Some have thought that the world ought to be finite since i t was created by God, while others have thought i t ought to be infinite for the same reason. All these beliefs relate to the question of a concrete infinite. There can of course be no correct logical argument that there are, say, 32 infinitely many stars or atoms. Thus, I am going to deal in this section with a priori proofs that there does exist an infinite set or number of abstract entities, i.e. proofs that there is an infinite in the case of such entities as propositions, properties, sets, and numbers. Such proofs have been given by Bolzano, Dedekind, Frege, and Russell. Bolzano proposed to show that there is some species A whose number 32 There is however a theorem setting a limit to the number of finite particles there can be. There can be at most the smallest infinite number of particles of finite magnitude. This follows from the theorem proved by Cantor in 1882 stating that space can contain at most denumerably many disjoint finite sub-volumes (GA, p, 153). 63 series is endless, i.e. that an infinite set has "objectivity" (Gegenstandlichkeit). Bolzano finds what he is looking for in the set of absolute propositions or truths (Satze and Wahrheiten an sich). For any proposition A, there is another proposition A' = the proposition that A JLS_ true. Bolzano adds that "such further propositions exist whether we construct them or not" (Paradoxes, sect. 13). For criticisms of Bolzano's proof, as well as a more detailed exposition of it, see the fine article by Heinrich Scholz, "Die Wissenschaftslehre Bolzanos," in Mathesis Universalis (ed. H. Hermes, F. Kambartel, and J. Ritter [Darmstadt, 196l]). Dedekind's proof was somewhat similar to Bolzano's, but it is based on a different definition of the infinite sets. Dedekind must prove that there is a D-infinite system; it would, of course, follow from this that there are simply infinite systems, i.e. that there is at least one number system. Indeed, this was apparently Dedekind's motive for constructing his proof; Dedekind explains in his letter to Keferstein that: After the essential nature of the simply infinite system, whose abstract type is the number sequence N, had been recognized in my analysis...the question arose: does such a system exist at all in the realm of our ideas? Without a logical proof of existence it would always remain doubtful whether the notion of such a system might not perhaps con-tain internal contradictions (van Heijenoort, p. 101). It is curious that, while no contradiction can reasonably be supposed to be hidden in the concept of a simply infinite system, the concept on which Dedekind's proof is based is involved in contradictions. The example which Dedekind uses to prove the existence of infinite systems is "the totality S_ of all things, which can be objects of my thought." In order to show that S_ is D-infinite, Dedekind must define 64 a proper subsystem S' of S which is equivalent to S.. Dedekind maintains that for each object s in S_ there is also in the thought that s^  can be an object of thought. But let us replace this "thought" by the proposition that s^  belongs to S^, thus eliminating idealistic and modal factors in favor of logical concepts. We may say, then, that for every element £ in S the proposition that s_ belongs to is also in j3. Now i f s^  and t_ are two dif-ferent elements of J3, the corresponding propositions are different. Clearly not a l l things js are propositions, hence £ is D-infinite. Furthermore, i f <j> is the function which correlates with each thing s. the proposition that seS, then i f t_ is any element of S. which is not a proposition, then the subsystem ^ ( t ) of S is a simply infinite system, which can be taken as the system of the finite ordinal (natural) numbers. (Thus, we have Dedekind's 'consistency proof for number theory!) The concept of a proposition is an extra-set-theoretical (though not an extra-logical) primitive idea, and Dedekind's proof, as I have formulat-ed i t , must use assumptions about propositions, e.g. that for each entity s^  there exists the proposition that s^  belongs to the class S^  of a l l enti-ties. It is, of course, easy to replace propositions by sets, and to replace the supposition about propositions with the postulate that for every object x there is a set {x} containing only x as an element, and 33 x 4 {x}. But i t must be noted that Dedekind thought x = {x}, and _ Dedekind said that "every element s of a system can itself be regarded as a system" (Essays on the Theory of Numbers, p. 46). Later Dedekind explained very clearly the errors which arise from the identi-fication x = {x}; see the manuscript from Dedekind's Nachlass published by M.-A. Sinaceur in his "Appartenance et Inclusion: Un Inedit de Richard Dedekind," Revue D'Historie des Sciences, 24 (1971), 247-254. Bolzano considered couples to be the smallest sets 65 34 35 Frege seems to have approved of Dedekind's proof, and Russell also found Dedekind's reasoning convincing for some time, though he eventually became critical of it (see Intro. Math. Phil., pp. 139-140). Indeed even after he had come to entertain theories needing "an assump-tion to give Kg entities," Russell s t i l l thought that the existence of at least C(Q entities "may be easily shown by Dedekind's case of the series _x, idea'jc, idea'idea'x.... 36 This series cannot be constructed by logic, but seems undeniable." Thus, Russell apparently believed at the time, that, while i t is necessary to formulate an "essentially arithmetical" axiom stating "that the NC [cardinal number] of entities is not finite" which "can be expressed, but not proved, in terms of logic" (Ibid.), he nevertheless could base his belief in the existence of infinitely many entities on a convincing extra-logical argument. Dedekind's proof is logically correct given the assumptions about propositions and also a principle of set existence from which i t follows 34 For example, Bolzano said that " i f A were identical with JB, i t would of course be absurd to speak of an aggregate composed of the objects A and B_" (Paradoxes, sect. 3). This is in accordance with the traditional idea that 2 is the smallest number (in the concrete sense of a plurality of units); a single thing is not a plurality or number of things. 35 Nachgelassene Schriften, pp. 147-148. Frege emphasized the necessity of not understanding the "thoughts" referred to in the proof to be psychological entities. 3 6Russell Archives MS. 230.030890, pp. 27-28. 66 that there is a set of a l l things. The proof is not circular as was claimed by C.J. Keyser, who criticized i t for the wrong reasons and did not even mention the really problematic point: the paradoxes of set theory, and in particular, the paradox of the greatest number (see section 37 22 below). It should be noted that there are systems of set theory which apparently avoid the paradoxes while preserving Dedekind's proof (with sets replacing propositions). What Russell called "zig-zag" theor-ies admit a class of a l l things; the systems formulated by Quine are such, 38 and have a theorem of infinity. But, though Quine's systems preserve Dedekind's existence theorem for infinite sets, they do not preserve Cantor's theorem (see section 15 below). In axiomatic set theories which avoid the paradoxes by "limitation of size," as well as in the theory of types, Dedekind's proof cannot be reproduced. Let us consider, finally, the Frege-Russell proof that there is an infinite number. When Russell first became acquainted with the writings of Cantor in 1896, he thought that Cantor's transition to the infinite ordinal numbers was illegitimate: "The method of genesis of his second class is very ingenious, but its philosophical validity is, to say the — See Keyser's papers: "Concerning the Axiom of Infinity and Mathematical Induction," Bulletin of the American Mathematical Society, 9 (1902-1903), 424-434; "The Axiom of Infinity: A New Presupposition of Thought," The Hibbert Journal, 2 (1903-1904), 532-552; "Concerning the Concept and Existence-Proofs of the Infinite," The Journal of Phil- osophy, 1 (1904), 29-36. The second of these was criticized by Russell in his article "The Axiom of Infinity," The Hibbert Journal, 2 (1903-1904), 809-812. Keyser replied to this with "The Axiom of Infinity," The  Hibbert Journal, 3 (1904-1905), 380-383. 38 See his "New Foundations for Mathematical Logic," From a. Logical  Point of View, 2nd ed. (New York, 1963), pp. 93-94, and Mathematical  Logic, 2nd ed. (1951; rpt. New York, 1962), p. 252. 67 39 least, in the highest degree doubtful." "But how," Russell asked, "is a) ever to arise?" (Ibid., p. 8). Since there is_ no number larger than any of the natural numbers— so the argument may be stated—let us invent one which shall be larger, and we are sure of getting something new. Sure indeed! but at a price which most would not care to pay (Ibid.). Russell's conclusion in 1896 was that "Cantor's transfinite numbers, then, are impossible and self-contradictory" (Ibid., p. 9). When Russell eventually accepted Cantor's theory, he made a partic-ular point of the existence theorems so that skeptical philosophers like his former self would have nothing to complain about. Moreover, he con-tinued to think Cantor's work was deficient as regards existence theorems (see Principles, p. 322). The proof that infinite numbers exist which Russell Outlined in his Principles and in his article "The Axiom of Infinity" (1904) is essentially the same as the one given in Frege's Foundations of Arithmetic (1884). It also reminds one of Cantor's way of introducing in his 1895 paper; but Cantor did not present a proof. It would be nearer the mark to say that the existence of Kg was axiomatic with Cantor (cp. sect. 28 below). The principle steps of the proof that there exists an infinite card-40 inal number are the following. (1) A successor relation between the —— "On Some Difficulties of Continuous Quantity" Russell Archives TS 220.0105040 (1896), p. 7. When this was written, Russell was acquaint-ed only with the earlier works of Cantor and did not know Cantor's 1895 paper. 40 More complete outlines of the proof may be found in Principles, pp. 356-357, "The Axiom of Infinity," and Frege's Foundations of  Arithmetic, sects. 74-84. 68 41 cardinals is defined, and i t is proved that every number has a successor. (2) The finite cardinals are defined in such a way that the principle of complete induction is a consequence of the definition (see section 2 above). This principle is then used to prove that for every finite num-ber n, n 4- n+1. Thus, every finite number has a successor differing from itself. (3) Finally, it is shown that the number of the class of finite numbers is not a finite number, i.e. i t is an infinite number. No finite number is its own successor, but K Q, the cardinal belonging to any class equipollent to the class N of finite numbers, has the property of being its own successor, for N ^  Nu{x} whatever object x ^ y be. After giving a sketch of such a proof in his article "The Axiom of Infinity," Russell concluded that "from the abstract principles of logic alone, the existence of infinite numbers is rigidly demonstrated" (p. 810). As in the case of Dedekind's proof, the Frege-Russell proof cannot be reproduced either in Principia Mathematica or in systems of axiomatic set theory such as were formulated by Zermelo and von Neumann (minus their respective axioms of infinity). The reason is essentially the same in both cases: It is not possible to prove that there is a set contain-ing 0 as well as the successor of every number i t contains. In the case of Principia Mathematica, without resorting to the axiom of infinity i t For example, Frege defined the successor relation so that: nSm *-*• 3A3x(xeA & A = n & A — {xT = m). This relation is , of course, really suitable only as the successor relation for finite numbers. The successor of the infinite cardinal X should be .., not X +1 = K . i s only p o s s i b l e to show that every f i n i t e c a r d i n a l y has a successor u+^ 1 / y by taking V+ Cl as the c a r d i n a l of the c l a s s of car d i n a l s {0,...,y}. But then "'u+ l 1 i s n e c e s s a r i l y i n a higher type than 'y, because i t applies to a class of which y i s a member" (PM, I I , 221). Thus, according to the doctrine of types, which i s the means of avoid ing the paradoxes i n PM, no clas s can contain both y and i t s higher type successor as elements. CHAPTER III QUANTITATIVE RELATIONS BETWEEN INFINITE SETS 11) Traditional Definitions of  Greater and Less It i s a remarkable fact that while a number of philosophers who wrote about the in f i n i t e before Cantor took the existence of a one-one correspondence between sets as the relation of numerical equality, none of them formulated the definitions of the corresponding inequality relations. 1" They continued to use the traditional definition of the relations of greater and less. This produced many errors in connection with i n f i n i t e sets: some scholars were led to the belief that quantita-tive relations do not hold between i n f i n i t e s ; Leibniz thought i t would involve a contradiction to suppose the existence of i n f i n i t e wholes; Bolzano restricted the explication of the concept of numerical equality as one-one correspondence to the case of f i n i t e sets. In every case, the under-lying error was the belief that i t would be a contradiction to say that sets were both equal and the one greater than the other. But, using the definitions of equality as one-one correspondence and the traditional definition of greater, there is no contradiction. The relations are not incompatible. The traditional authors apparently did not see the need to prove the incompatibility of relations they called "equality" ^By corresponding relations i s meant relations having the expected properties which are incompatible with equality. - 70 -71 and "greater than." The traditional definitions of greater and less, which have the consequence that the whole is greater than the part, were formulated very clearly by Aristotle: "the greater is, in relation to what is less, 'so much and something more"' (Metaphysics, 1021a5). Kant had this concept in mind when he represented the proposition that the whole is greater than the part thus: "(a+b) > b" (Critique, B 17). Essentially the same definitions are given by Leibniz. Thus, he defined the relation less than as follows: A is less than JB i f and only if A is equal to some part B_" of B_. He remarked in justification of his definition that: This definition is very easily understood and is consistent with the general practice of men, when they compare things with each other and measure the excess by subtracting an amount equal to the smaller from the greater (PPL, p. 267). On the basis of this definition of "less than," Leibniz demonstrates the theorem that the whole is greater than the part. If A is a part of B, then A is equal (in the sense of having the same quantity) to a part 2 of B_ because A is equal to A. The proposition follows by the definition. Thus, the proposition that the whole is greater than the part is not an axiom in Leibniz' system, but rather a theorem which is an immediate consequence of his definition of less than. This was, moreover, the only way that the proposition could be satisfactorily justified. As Leibniz explains: By induction alone we should never perfectly know the propo-sition that the whole is greater than its part. For someone PPL, pp. 226 and 267. Leibniz Selections, p. 205. 7 2 would soon appear and for some reason deny that i t is true in cases not yet observed'. We know this from the fact that Gregory of St. Vincent denied that the whole is greater than its part, at least of angles of contact, and that others have denied that i t is true of infinity (PPL, p. 130). By means of an invalid argument, some traditional authors came to the conclusion that the quantitative relations (equal, greater, and less) do not hold between infinite multitudes. Among the best known of these is Galileo. He noted the one-one correspondence between roots and squares, and also the fact that every integer is a root, though only some are perfect squares. There are "more" roots than squares. This Galileo apparently took to be incompatible with the one-one correspondence, which meant to him that there are as_ many squares as_ numbers altogether. The one quantity cannot be both greater than and equal to the other. Galileo did not, however, prove that the relation greater than, as he understood i t , is incompatible with equality in the sense of one-one correspondence. Therefore, Galileo was not justified in concluding that, "the attributes 'equal,' 'greater,' 'less' are not applicable to the 3 infinite, but only to the finite." Newton too maintained that, from one point of view at least, quantitative relations do not hold between infinites: "The generality of Mankind consider Infinites no other ways than indefinitely; and in this Sense, they say a l l infinites are equal: tho' they would speak more truly i f they should say, they are neither equal nor unequal, nor have any certain Difference or Proportion one to another" (Papers and _ Galilei Galileo, Dialogues Concerning Two New Sciences, trans. Henry Crew and Alfonso de Salvio (New York, 1914), p. 32. 73 Letters, pp. 293-294). Newton did not bring up the matter of one-one correspondence, and he seems to allow that one i n f i n i t e could be greater than another i f i t contained the other as a proper part; but "Infinites when considered absolutely without any Restriction or Limitation, are neither equal nor unequal..." (Ibid. , pp. 295-296; p. 299). 12) Leibniz and the Problem of  the Infinite We have seen that Galileo fallaciously reasoned from the traditional definition of "greater" and "less" and the insight that one-one correspon-dence is the criterion of equality in number to the conclusion that quantitative relations do not hold between i n f i n i t e quantities. Leibniz concluded from consideration of a similar example of one-one correspondence between i n f i n i t e p l u r a l i t i e s , that there cannot be an i n f i n i t e number (i.e. an i n f i n i t e plurality of units) or an i n f i n i t e whole. The propos-ition that the whole is greater than the part plays an essential role in Leibniz' reasoning. Galileo does not mention this proposition, and explic i t l y states that the plurality of a l l numbers is i n f i n i t e . Leibniz believed that he could show that there i s no greatest number, because the greatest number is the same as the number of a l l numbers, and the supposition that there i s a number of a l l numbers implies a contradiction. As he explains: ...the greatest number is the same as the number of a l l units. But the number of a l l units i s the same as the number of a l l numbers (for any unit added to the previous ones always makes a new number). But the number of a l l num-bers implies a contradiction, which I show thus: To any number there is a corresponding number equal to i t s double. Therefore the number of a l l numbers i s not greater than the 74 number of even numbers, i.e. the whole is not greater than its part.^ It is interesting that Leibniz defines the greatest number to be the number of a l l units, but he hardly establishes that the number of a l l units is the same as the number of a l l pluralities of units. We may note that Kant held, "no multiplicity is the greatest, since one or more units can always be added to i t " (Critique, A 413). I assume Leibniz meant that the totality of a l l monads or simple substances is the greatest number. Kant, of course, did not believe in the existence of simple substances. Leibniz' Error. Leibniz thinks that the supposition of the number of (or whole containing) a l l numbers implies the negation-of his theorem that the whole is greater than the part."* He wants to say that "the number of a l l numbers is not greater than the number of a l l even numbers, i.e., the whole is not greater than the part" because there is a one-one correspondence between the whole (set of integers) and one of its proper parts. But he is not entitled to say any such thing on the basis of his definitions—the definition of less than on which his proof of "the whole is greater than the part" is based. Leibniz can say that the whole, in the case in point, has the same number of elements as the part, but he cannot say that the whole is not _ Quoted by Russell, A Critical Exposition of the Philosophy of  Leibniz, 2nd ed. (London, 1937), p. 244. ^Leibniz used the traditional definition of number: integers are "multitudes of units" (New Essays, p. 160). He also says that "Plurality is...contained in number" (PPL, p. 516); this means that the concept of plurality is a constituent of the concept of number, hence, that any number is a plurality. 75 greater than the part. From the fact that the whole N has the same number of elements as the part E (in the sense of one-one correspondence) i t does not follow that there does not exist a part X of N which has the same number of elements as the part E of N (Leibniz* definition).* 5 Leibniz does not prove, and I dare say never realized the need to prove that E^ N -y -i(E<LN)7 If Leibniz could prove this, then, since he really has proved both E<LN E^ N he could derive the contradiction: (E<LN) & -i(E<LN). Leibniz must not have stopped to think of his definition at a l l while expounding his 'contradiction;' he must have merely jumped to his conclu-sion. If he had reflected on the matter he might have been led to a different definition of greater than—the definition of being greater than in cardinal number, which was first formulated by Cantor. Leibniz automatically took 1-1 correspondence as the criterion of equality in number, and then he must have thought: equal, therefore, not greater— without ever giving his definition a thought or considering the possibility of proving the conditional formulated above. Those who have commented on Leibniz' views on infinity, in particular 6Indeed i t is plainly false that ~>(E<LN), for in the example i t is the case that E ^  E c N. A'E", i i Nn^ M ^ I I ^ •>< »' denote respectively the set of even numbers, the set of (finite) numbers, the relation of equivalence (numerical equality), and the relation of being less than in Leibniz' sense. 76 g Russell and Rescher, have said hardly anything which seems quite right. Rescher even says that "on his premises Leibniz' reasoning is quite 9 valid; we have demoted one of his 'axioms' of logic from that status." The so-called axiom meant here is the proposition that the whole is greater than the part. We do not reject this statement today when we understand by "greater" the relation that Leibniz defined, and I believe I have shown beyond any possibility of doubt that Leibniz' reasoning was quite incorrect from his own point of view. Leibniz' denial of the existence of infinite wholes or numbers was based entirely on his belief that the negation of the proposition that the whole is greater than the part would follow i f i t was supposed that an infinite whole can exist. Leibniz at no time had any "horror of the infinite," and in fact in an early writing, he enthusiastically considers the possibility of a magnitude which is infinite, "yet bounded." Leibniz even concludes there that " i t is clear that the infinite is other than the unbounded, as we surely assume popularly." He also says that from several considerations i t follows that "there must be an infin-ite number" (PPL, p. 169). If he had avoided the logical error in his reasoning, which I have explained, he might have made some positive con-tribution to the theory of the infinite. Leibniz was of course very far from finitism, despite his supposed contradiction concerning the existence of infinite wholes. The following — A Critical Exposition, sect. 58; Our Knowledge, p. 149; Intro. Math. Phil., p. 80; A History of Western Philosophy (New York, 1945), p. 830. 9 Nicholas Rescher, "Leibniz' Conception of Quantity, Number, and Infinity," The Philosophical Review, 64 (1955), 112. 77 passage illustrates this especially well: the wisdom of God...does not exceed the possibles extensively . . . i t exceeds them intensively, by reason of the infinitely infinite combinations i t makes thereof.... The wisdom of God...goes even beyond the finite combinations, it makes of them an infinity of infinites, that is to say, an infinity of possible sequences of the universe, each of which contains an infinity of creatures.10 The contradiction Leibniz thinks he finds in the supposition that an infinite whole exists does not make him conclude that there are not infinitely many different things, but only that they do not form a whole. Leibniz could s t i l l say that "...the multitude of things surpasses every finite number, or rather every number.""^ Thus, Leibniz holds that there are infinitely many things and numbers, but that there are only finite 12 numbers or finite discrete wholes. Leibniz also has to conclude that a l l things do not form a whole (even though this seems to be inconsistent with his statement that creation is an ordered whole). "I concede the infinite plurality of terms, but this plurality itself does not constitute a number or a single whole.... Just so there is a plurality or a complex of a l l numbers, but this plurality is not a number or a single whole" (PPL, p. 514). Leibniz Theodicy, p. 225. See also Selections, p. 99, for what is perhaps Leibniz' best known affirmation of the "actual infinite." ^Quoted in Russell, A Critical Exposition, p. 244. 12 However great a particular (finite) number may be, "there is always another larger than i t , ad infinitum" (The Philosophical  Works of Leibniz, trans. G.M. Duncan [New Haven, 1908], p. 103). "There is no need of many worlds to increase the multitude of things, for there is no number which is not contained in this one world, indeed even in any one of its parts" (PPL, p. 168). 78 makes the point even better in a statement quoted by Russell: There is an actual infinite in the mode of a distributive whole, not of a "collective whole. Thus something can be enunciated concerning a l l numbers, but not collectively. So i t can be said that to every even number corresponds its odd number, and vice versa; but i t cannot therefore be accurately said that the multitude of odd and even numbers are equal.13 Leibniz' doctrine that there are infinitely many substances must, then, be formulated as: For any set (multitude, collective whole) M of monads, there is a monad not belonging to M. Leibniz' views here have some similarity to those that Cantor arrived at after he became aware of the paradoxes of set theory. Leibniz does not come to believe that there are not infinitely many things, he only holds that infinitely many things never form a whole. Cantor also concluded that in certain cases there are many sets of a certain character, but no set which contains them a l l as elements. The source of Leibniz' interest in the question of the greatest number and the number of a l l numbers is worth noting. These examples, together with that of the greatest speed, provided (or so he thought) illustrations of the fact that definitions may be formed to which nothing can correspond. Leibniz was especially interested in the matter of consistency in connection with the traditional ontological argument for the existence of God. Leibniz criticized this argument on the ground that a proof that God is possible was lacking. Now the contradiction of the greatest number could only make the desirability of a consistency proof for the concept of God more evident, since God is characterized — A Critical Exposition, p. 244. Cp. Russell, Principles, p. 143. 79 as an infinite being, as having a l l perfections, and as having each of infinitely many perfections in the maximum degree. Leibniz puts the point strongly: We seem to think many things (confusedly i f you like) which imply a contradiction, for example the number of a l l numbers. The notion of infinity ought to be highly suspect to us, and the notion of minima and maxima, of the most perfect and of totality i t s e l f . ^ 13) Bolzano on Quantitative  Relations Bolzano called the things belonging to a certain kind "quantities" i f for each two of these objects the one is either equal to or greater than the other. According to the definition given in his Theory of  Science, two manifolds are equal (in quantity) "whenever we are able to transform one of the manifolds into the other by merely exchanging each unit that is contained in i t for a unit that is contained in the other" (sect. 87). If the manifolds are not equal in this sense, then, as he thought at that time, "there are units left over in one of the manifolds after a l l the units that are contained in the second have been exchanged for the units that are contained in the f i r s t " (Ibid.). Now this would hold only for finite manifolds, but i t would seem that Bolzano intended to be stating the case for quantities in general. Moreover, here equality is not specifically said to be one-one correspondence between the two manifolds. Bolzano does much better in his later Paradoxes of the  Infinite, but we shall see that he makes the same error as Leibniz, _ Quoted in Gottfried Martin, Leibniz: Logic and Metaphysics, trans. K.J. Northcott and P.G. Lucas (Manchester, 1964), p. 28. 80 though his conclusions are quite different from those both of Leibniz and of Galileo. Bolzano defined the relation greater than in the same way as Leibniz, but he held that infinite sets exist and that one infinite set may be greater than another "in the sense that the one includes the other as a part of itself..." (Paradoxes, sect. 19). Apparently in order to avoid an imagined paradox concerning the proposition that the whole is greater than the part, Bolzano held that one-one correspondence is the criterion of equality in quantity only for finite sets. Bolzano seems to have been the first to recognize that a l l infinite sets are in one-one correspondence with a proper part of themselves.^ He had not been aware of this property at the time of his Theory of Science, as is clear from the errors in section 87, noted above. In the Paradoxes of the Infinite, however, Bolzano unfortunately concludes that the existence of a one-one correspondence between two sets is not the criterion of equality in the case of infinite sets. For even when there is such a correspondence, "the two sets can s t i l l stand in a relation of inequality, in the sense that the one is found to be a whole and the other a part of that whole." Bolzano does, however, maintain that equivalence is the criterion of "equimultiplicity" for finite sets (sects. 21 and 22). This conclusion is, of course, much more moderate than Galileo's. Bolzano maintains that relations of inequality and equal-ity hold between infinite sets, but he had no intelligible general concept ^Paradoxes, sect. 20. See also Cantor, GA, p. 119, and Dedekind, Essays on the Theory of Numbers, pp. 41 and 63. 81 of quantitative equality. Bolzano makes essentially the same error as Leibniz, and this is why he denies that equivalence is the criterion of equality for infinite (as well as finite) sets. Neither of them recognized that being greater in his sense is compatible with being equal in the sense of one-one correspondence (equivalence).1** Leibniz thought he had a contradiction; Bolzano thought that the obvious criterion of equality fails for infinite sets. Leibniz thought: one-one correspondence, hence equal; therefore not greater, but greater—contradiction. Bolzano, in contrast, thought: greater, yet one-one correspondence; hence the latter doesn't mean equal. """"The critical commentary of Hans Hahn can hardly be improved upon: "Dass zwei Mengen, von denen eine die andere als echte Teilmenge enthalt und somit nach [Bolzano's] Terminologie grosser ist als die andere hinsichtlich der Vielheit ihrer Telle, doch gleiche Machtichkeit haben kbnnen, bedeutet natiirlich keinerlei Widerspruch, ebensowenig wie etwa die Tatsache, dass zwei Menschen, von denen der eine grosser ist als der andere, doch gleiches Gewicht haben konnen. Insbesondere ist darin auch kein Widerspruch enthalten gegen das 'Axiom:' das Ganze ist gro'sser als der Teil" ["Anmerkungen" to Paradoxien des Unendlichen, ed. Alois Hofler (Hamburg, 1955), p. 141. CHAPTER IV CANTOR'S THEORY OF THE TRANSFINITE 14) Cantor's Definitions Once i t is recognized that the criterion of equality in number is one-one correspondence, i t seems as though i t would be evident that in-equality in number is non-existence of a one-one correspondence. But, in fact, this was first realized by Cantor. Once what is essential for inequality is seen, the definition of greater and less in number is clear. Cantor first published his definitions in 1878: If two well-defined manifolds M and N can be coordinated with each other uniquely and completely, element for element...these manifolds have equal power...they are equivalent. By a compon-ent (Bestandeil) of a manifold M, we understand any other mani-fold M', whose elements are also elements of M. If the two manifolds M and N are not of equal power, then either M has the same power as a component of N or N has the same power as a com-ponent of M; in the first case, we call the power of M smaller, in the second case we call i t greater than the power of N. ... From the circumstance alone, that an infinite manifold M is a component of another N or can be uniquely and completely coor-dinated with such, i t is by no means permissible to conclude that its power is smaller than that of N; this conclusion is only justified i f one knows that the power of M is not equal to that of N (GA, p. 119). In my opinion this one passage is worth more than a l l previous writings on the infinite. Cantor later gave another (equivalent) definition of the relation of less than in cardinality. The set A is less in cardinality than the set B_— A < B — i f and only i f (i) A is equivalent to a proper subset - 82 -83 of B_, and (ii) no subset A' of A is equivalent to B_. It follows at once that the relation < is irreflexive and transitive. It will be conveni-c ent to use this second definition in subsequent sections. The set A is less in cardinality or equipollent to the set IJ, A < B, i f and only i f A is equipollent to some subset of B_. It is a fact that A < B or B < A (comparability) and that A < B & B < A —> A ~ B (Bernstein's equivalence theorem). It is also the case that i f JB is a finite set, then A < B i f and only i f A < B. c L Once the quantitative relations of equality in number and greater (and less) in number have been satisfactorily defined, it is possible to raise the question about different magnitudes of infinite sets: Are a l l infinite multitudes equal (i.e. in one-one correspondence with each other) or not? The question may then be answered by proof. -In fact, Cantor was able to demonstrate that for any infinite set, there exists another set of greater cardinality. 15) Cantor's Theorem In 1874, Cantor proved that for every denumerable set of real numbers M, there is in any interval of real numbers a real number not belonging to the set M (GA, p. 117; cp. p. 143). That is, no denumer-able set of real numbers contains a l l real numbers in an interval of real . 1 That the relation is connected, which seems to be assumed in the above quotation, is not so easily established. 84 numbers. It was not until 1892 that Cantor found the method for prov-ing a more general theorem which has come to be called the "diagonal" argument. Cantor's theorem is the assertion that for every set there exists a set of greater cardinality, so that there is no set of greatest cardin-ality and no greatest cardinal number. In fact, the class of a l l sub-classes P(A) of a set A is greater in cardinality that A: for every set A» A <c P(A). The proof of this theorem has two parts corresponding to the two clauses of the definition of the relation < . (i) It is proved that A c — is equipollent to a subset B_ of P(A). (ii) It is proved that no subset A' of A is equipollent to P(A). (i) P(A) is at least as big as A, i.e. A < P(A), for A is equipollent to the set of unit subsets of A. (ii) To prove that P(A) is not equivalent to any subset A' of A, it is shown that no subclass 15 of P(A) which is equipollent to any subset A' of A contains a l l subsets of A. This is accomplished by defining a subset K of A which differs by at least one element from each member of any such B_. Let f be a one-one relation whose domain B_ is a subset of P(A) and whose converse domain'(argument domain) A' is a subset of A. I shall refer to the subset f(a) (a e A') of A as "the f-correlate of the element a of A'." What is wanted now is a definition of a subset of A which is not any one of the f-correlates of elements of the subset A' of A, i.e. not an element of B. 85 If a ^ f(a), then no set of which a is element can be = f(a). So the set K which contains a l l and only those things a. in A' which do 2 not belong to their f-correlates cannot be an f-correlate of any a £ A.". For, K cannot be the f-correlate of any a. which is not in f(a), because that a is in K, but not in f (a). And, i f a_ is in f(a), then that a^  is not in K, so K 4 f(a). In fact, i f it is supposed that K is an f_-correlate of an element z_ of A', a contradiction is derivable. (i) If z^  belongs to K = f(z), then j5 does not belong to K. (ii) If z^  does not belong to f(z) = K, then z_ does not belong to K. Hence, z e K i f f z I K. Therefore, B_ 4 P(A), so that, combining the two parts of the proof, A <c P(A). As a matter of = = AT fact, the cardinal number P(A) of the set P(A) is 2 . Thus, Cantor's theorem can also be formulated as: A < PlAT = 2A-. This means that there is no greatest cardinal number. There is also a generalization of Cantor's theorem which was formu-lated by Zermelo: For every set of sets T_, there exists a set of greater cardinality than any set belonging to T_. This generalization is estab-lished by noting that the union of T_ is greater in cardinality or equiva-lent to each element M of T, and the power set of the union of T_ is greater in cardinality than T_. What does not follow in Zermelo's system is that the union of a set of sets of increasing cardinality, none of which is the greatest, is a set of greater cardinality than any member of that set of sets. The reason is that the axiom of replacement is required to 2 It will be convenient to refer to K as the "diagonal class." 86 prove that there is such a set of sets. I should like to make a few remarks from the "Cantorist" point of view on Skolem's relativism here. The critics of the theory of the infinite are pleased to have the theorem of Lowenheim and Skolem, accord-ing to which every consistent first order theory has a denumerably infin-ite model, i f i t has an infinite model at a l l . It is not, however, possible to say that every model of ZF is denumerable, for ZF has models of every infinite cardinal number. That there should be denumerable models of axiomatic set theory is not very surprising when the definition of a non-denumerable set is con-sidered. A set A is non-denumerable i f f there does not exist any one-one relation whose domain is A and whose converse domain is the set of finite numbers N. The theorem of set theory stating that a non-denumerable set exists can be satisfied in a denumerable model p_ i f there is no. such  relation in D. However, i t is evident that such a p_ cannot be the class of a l l sets, for there must exist such a relation i f a set in p_ which is 'non-denumerable in D' is to be really denumerable. From the Cantorian point of view, we may say that there are 'absolutely' non-denumerable sets, i.e. sets for which there do not exist one-one relations between their elements and the set of finite numbers. The denumerable models are only special subsets of the class of a l l sets, and are, so to speak, miniature representations of the absolute totality of sets (which are elements). When theorems are formulated and proved about sets i t is not  intended that they refer to some limited portion of the universe of sets; rather, i t is intended that the variables have a l l sets as values. 87 I can see no reason to hold that cardinality is "relative," that denumerability is relative, and that really or for a l l that anyone knows there are only denumerably many entities. There are some, however, who would like to say that there is no reason to believe that there are non-denumerably many sets because a l l 'appearances' would be the same even if there are only denumerably many of them. This seems to be essentially the same argument that is used by phenomenalists and instrumentalists, and skeptics in general: everything in our 'experience' would be the same i f such and such objects did not exist. Should a Cantorist be any more concerned about such arguments than any other person who is reason-3 able enough not to be a solipsist of the present moment? H. Wang has said that we do not have a clear and distinct Idea of the non-denumerable. Yet there is a precise definition of the non-denumerable sets, and there are many theorems concerning such sets. What more can be desired? Leibniz laid down the following criterion for clear and distinct ideas: "The true mark of a clear and distinct idea is the means we have of knowing many truths of i t by a priori proofs" (Leibniz  Selections, p. 441). 16) Relations in Extension The concept of a relation is one of the very most important concepts in the theory of the infinite as well as in mathematics in general. In almost a l l cases, i t seems best to understand the relations dealt with in -See Bernays, "Die hohen Unendlichkeiten und die Axiomatik der Mengenlehre," in Infinitistic Methods (Warsaw, 1961), pp. 19-20, for further criticism of Skolem's relativism. 88 the theory of the transfinite to be relations in extension, i.e. classes of ordered couples. This was, however, not done by Cantor or Dedekind. Moreover, Russell did not favor relations in extension in his Principles  of Mathematics; the primitive idea of his general theory of relations was that of "relation in intension." He explained a binary relation (in intension) as "a concept which occurs in a proposition in which there are two terms not occurring as concepts" (Principles, p. 95). What is characteristic of relations in intension is "that two relations may have the same extension without being identical" (Ibid., p. 24). The case for the superiority of the concept of relations in exten-sion for the theory of the infinite is perhaps best made out In terms of a particularly significant example—the concept of equipollence. Cantor explained this as follows: Two sets M and N are equipollent "wenn es moglich ist, dieselben gesetzmassig in eine derartige Beziehung zueinander zu setzen, dass jedem Element der einen von ihnen ein und nur ein Element der andern entspricht" (GA, p. 283). This explanation brings in possibil-ity and laws, and there is the suggestion of an agent who might effect a correlation. Now i t does not seem proper to make the equivalence of sets depend either upon what correlations can be set up by means of laws or upon the existence or possibility of laws. Moreover, it is not entirely clear what those (e.g. Dedekind and Cantor) who use the term "law" mean by i t . Is a law a formula or is i t supposed to be something (perhaps a relation in intension) which is (or may be) defined or expressed by a formula? Certainly, i t would not be suitable to make equipollence depen-dent on the existence or possibility of formulas in the sense of certain 89 finite combinations of symbols. It seems clear that there are not enough (possible) formulas to define the equivalent sets as those whose elements could be correlated one-to-one by means of formulas, but there is also a suspicion that there may not be enough relations in intension either. For let us consider what relations in intension there are. We might think it natural to suppose that for every formula of two free variables there is a corresponding relation in intension. But what other relations in intension are there? The only other possibility of explaining what relations there are seems to be in terms of relations in extension. At the basis of the theory of relations in extension is the concept of an ordered pair. For any two elements ji and jb, there are two different ordered couples whose components or coordinates are the objects a. and b_, namely (a,b) and (b,a). Ordered couples satisfy the following axiom: <x,y> = (x',y'> -«-*- x = x ' & y = y'. A relation in extension is a class of ordered couples. For any two classes A and B_ there is a class AxB—the Cartesian product of A into j5—which consists of a l l ordered pairs (x,y) such that xeA and yeB. A (binary) relation R among the members of the class A is any subclass of AxA, i.e. such an R is a class of ordered couples of elements of A. Fortunately, the concept of an ordered couple does not need to 4 be taken as a primitive idea as i t was by Peano. In his excellent treatise Grundzlige der Mengenlehre (1914), Felix Hausdorf f noted that The concept of the Verbindung of two things is also a primitive idea for Cantor: "Jedes Element m einer Menge M lasst sich mit jedem Elemente ii einer andern Menge N zu einem neuen Elemente (m,n) verbin-den" (GA, p. 286). 90 "Ubrigens lMsst sich, wenn man will, der Begriff des geordneten Paares auf den Mengenbegriff zuriickf iihren. Sind, 1,2 zwei von einander wie von a und b_ verschiedene Elemente, so hat das Paar von Parren {{a,l}, {b,2}} genau die formalen Eigenschaften des geordneten Paares (a,b)...."^ This "reduction" seems to me a better policy than that (favored by Hausdorff) of taking ordered pairs as undefined entities. I do not recognize any primitive factor of order. A set is determined by its elements, but an ordered pair is not determined by its corrdinates. I cannot discern any additional factor determining an ordered pair which is not psychological or dependent upon relations in intension. Russell has the following remark on the "sense" of ordered couples in his Principles: It would seem, viewing the matter philosophically, that sense can only be derived from some relational proposition, and that the assertion that a^  is referent and b_ relatum already involves a purely relational proposition in which a. and b_ are terms, though the relation asserted is only the general one of refer-ent to relatum (p. 99). Russell did not think of Hausdorffs way of providing for the "sense" of an ordered pair of things without relying on relations in intension. Of course, there are other definitions of ordered pairs as sets of sets which provide the desired sense; the most common takes the set of sets {{a},{a,b}} as the ordered pair (a,b).^ In Principia Mathematica, the notation "x+y" is used for what the authors call "ordinal couples." These ordinal couples are conceived by Whitehead and Russell as the smallest relations, and, of course, corres-5(Rpt; New York, 1949), p. 32. 6 This definition is due to N. Wiener and K. Kuratowski. 91 pond to ordered pairs. In Principia Mathematica, the relation corres-ponding to the Cartesian product AXB is represented by "A+B." Ordinal couples x+y are defined in PM as {x}t{y}. The notation 'for' relations in extension is introduced in PM by means of contextual definition, for extensions are regarded as "logical fictions." But the essential proper- . ties of relations in extension are provable. One of Ramsey's criticisms of Principia Mathematica was that there relations were taken as relations in intension, not, as Ramsey thought they should have been, as "possible correlations, or 'relations in extension'" (The Foundations of Mathematics, p. 14). In Ramsey's view, "a function of a real variable is a relation in extension, which need not be given by any real relation or formula" (Ibid., p. 15). Russell was much inclined to accept Ramsey's criticisms of Principia and his proposals for reinterpreting Principia. Yet Russell s t i l l had doubts, in particular, about Ramsey's understanding of relations: If a valid objection exists—as to which I feel uncertain—it must be derived from inquiry into the meaning of 'correlation. 1 A correlation, interpreted in a purely extensional manner, means a collection of ordered pairs. Now such a collection exists i f somebody collects i t , or i f something either empirical or logical brings i t about. But, i f not, in what sense is there such a collection? I am not sure whether this question means anything, but i f i t does i t seems as i f the answer must be unfavourable to Ramsey.7 But a relation in extension is not to be understood as a "collection" in the sense of something which must be brought about, but as a class in the extensional sense (see sect. 27) of ordered pairs, i.e. as a certain Rev. of Foundations of Mathematics by Ramsey, Philosophy, 7 (1932) 85. 92 class of classes. A relation in extension so understood exists when 'the things in the relation' exist, and does not depend on anyone's activity (actual or possible) of correlating or collecting. Only this concept of relation in extension seems suitable for the definition of equipollence in transfinite set theory. We must note that with the analysis of the paradoxes underlying axiomatic set theory, i t is not possible to understand some relations as relations in extension. The fundamental relation e, for example, is not a class of ordered couples. There is an associated relation E. which is a class of ordered pairs, but the two relations are by no means the same. E is the class of a l l ordered pairs (x,y) of elements x,y such that xey, while e is a relation which holds not only between elements, but also between elements and proper classes, i.e. classes which are not elements of classes. The converse domain of IS is the class of a l l sets which are elements, but e does not even have a converse domain. The domain of both relations is the class of a l l elements (individuals and sets which are elements). It might even be said that relations in extension, while being the entities which are wanted for most purposes in set theory, are not strict-ly speaking relations at a l l , but classes which may not even correspond to (or be the extension of) any proper relation. Moreover, relations proper would seem to be what Russell called "relations in intension," for i t seems evident that, on the extra-set-theoretical concept of a relation, any number of different relations might hold between the same pairs of entities. Just as any number of proper relations may hold 93 between the same pairs, so there are proper relations—in particular e— to which no classes of pairs correspond as extensions. 17) The Numbers of Transfinite  Well-Ordered Sets In a letter to Dedekind of 1882 and in the fifth installment of his paper on infinite linear point manifolds (1883), Cantor gave the first descriptions of "a natural extension or continuation" of the g series of positive integers. Cantor did not at first use the term "ordinal numbers" or define the numbers of the extended system in terms of the well-ordered sets. He 'defined' the new numbers by means of two "principles of generation." The first principle of generation is that "of the addition of a unit to a number which has already been formed (einer vorhandenen schon gebildeten Zahl)" (GA, p. 195). According to the second principle, i f there is a definite succession of already defin-ed "real whole numbers" among which there is no greatest, then the limit (Grenze) of this succession of numbers is defined as the number next greater than them a l l (GA, p. 196). The system of finite and transfinite numbers (Zahlen) was thought of as generated by combined application of these two principles. The endless succession of finite numbers is generat-ed by means of the first principle, the first transfinite number w by the second, the next transfinite numbers to+l, oH-2,..., o>fn,... by the f i r s t , the next transfinite number after a l l of these oH-co = w2, by the second, and so on. Cantor stated that "Die beiden Erzeugungsprinzipe, mit deren Hilfe...die neuen bestimmt unendlichen Zahlen definiert werden, sind ^Briefwechsel Cantor-Dedekind, ed. E. Noether and J. Cavailles, Actualites Scientifiques et Industrielles, No. 518 (Paris, 1937), p. 55. solcher Art, dass durch ihre vereinigte Wirkung jede Schranke in der Begriffsbildung realer ganzer Zahlen durochbrochen werden kann" (GA, p. 166). If we wanted to formulate axioms of existence corresponding to Cantor's two principles of generation, we might do this as follows: (1) for each number there is a next greater, i.e. every number has a successor; and (2) for each set of increasing numbers having no great-est element, there is a next greater number. But now i t seems quite obvious that we must either have no set of a l l numbers or an absurdity. Thus, principle (2) needs some restriction in order to avoid the absurd-ity of supposing the existence of a number coming after the series of a l l numbers. Perhaps we could try to restrict (2) in the way suggested by Cantor's own principle of limitation, which "consists in the demand that the creation of a new whole number with the help of the two other factors be undertaken only when the totality of a l l preceding numbers 9 is countable in a known class already at hand in its whole extent." In view of this, let us formulate (2'): i f P_ is a property of numbers such that the numbers having P_ increase endlessly and there is a number a having P_ such that for any number u having P_ the set of a l l numbers less than u has at most the power of the numbers less than a, then there exists a set of all.ordinals having P_ and an ordinal next greater than a l l ordinals in that set. It does not follow from the principle (2') that there is either _ Briefwechsel, p. 56; cp. GA, p. 199. 95 a set of a l l finite numbers or a least transfinite number co. Furthermore, even if we assume these additionally, i t does not follow that to exists. It also does not follow that there is a set of a l l numbers and a number greater than a l l numbers. If we consider the property of being a number (in the extended sense) we shall not be able to find a number $ such that there are as many numbers less than 0 as there are numbers a l l told. We can't say: let $ be the ordinal of the set of a l l ordinals ordered according to magnitude, for we have not even introduced the concept of the ordinal number of a well-ordered set; moreover, we should have to prove the existence of the set of a l l ordinals, which we shall hardly be able to do with (2") in place of the naive comprehension axiom. Cantor said that at first i t seemed to him that the greatest significance of the introduction of the "new whole numbers" was "fiir die Entwicklung und Verscharfung des...Machtigkeitsbegriffes" (GA, p. 167). But he came to think that a greater profit from the new numbers consisted "in einem neuen, bisher noch nicht vorgekommen Begriffe, in dem Begriffe der Anzahl der Elemente einer wohigeordneten unendlichen Mannigfaltigkeit" (GA, p. 168). A well-ordered set, according to Cantor's original definition, is any well-defined set in which the elements are connected with each other in a definite succession, with respect to which there is a first element as well as an element coming next after each element which is not the last and, further, for any subset there is an element following next after a l l its elements, unless nothing succeeds a l l its elements (GA, p. 168). Thus, a set together with a particular succession of its 96 elements constitutes a well-ordered set i f f each of its subsets which has successors in that succession has an immediate successor. The fi r s t element of a non-empty well-ordered set is the immediate successor of its empty subset. The concept of the well-ordered sets, Cantor said, is fundamental for the entire theory of sets. He characterized the proposition that i t is always possible to bring any well-defined set into the form of a well-ordered set as a "grundlegende und folgenreiche, durch seine Allgemeingultigkeit besonders merkwiirdige Denkgesetz" (GA, p. 169). Two well-ordered sets E_ and F_ are of the same Anzahl i f f a one-one correspondence f. between them is possible such that for any two elements e_ and e_" of E_, j2 precedes e' in E i f f f (e) precedes f (e') in F. For any well-ordered set A there is always one and only one number (Zahl) <x of the extended system such that the set of a l l numbers preceeding a is a well-ordered set of the same Anzahl as A; this unique ordinal cx is the  Anzahl of the well-ordered set A (GA, p. 168). Thus, in his first exposition of the extended system of whole num-D bers which includes transfinite as well as finite numbers, Cantor 'defined' the numbers in terms of principles of generation. He then defined the well-ordered sets and defined the Anzahl of a well-ordered set to be one of the numbers of the extended system. In his 1883 work, the "Grund-lagen,"^ the terms "ordinal number," "simply ordered set," "similar," •^The fi f t h of Cantor's series of papers on infinite linear point manifolds was separately published with the t i t l e : Grundlagen einer  allgemelnen Mannichfaltigkeitslehre: Ein mathematisch-philosophischer Versuch in der Lehre des Unendlichen (Leipiz, 1883), and "order-type" are not introduced. Furthermore, the concept of a segment of an ordered set and a definition of the relation less than between the new numbers are lacking. For a l l this we must turn to subsequent works. (See below pp. lOOff). In 1885 Cantor reviewed Frege's Die Grundlagen der Arithmetic. That review contains the first publication of Cantor's later way of conceiving the ordinal numbers: ...verstehe ich unter der 'Anzahl oder Ordnungszahl einer wohlgeordneten Menge1 denjenigen Allgemeinbegriff, unter welchen alle wohlgeordneten Mengen, welche der gegebenen ahnlich sind, und nur diese fallen. 'Ahnlich' nenne ich zwei wohlgeordnete Mengen, wenn sie sich gegenseitig eindeu-tig, Element fur Element, unter Wahrung der gegebenen Ele-mentenfolge auf beiden Seiten, aufeinander abbilden lassen (GA, p. 441). Cantor first introduced the concept of a simply ordered set in his paper "Principien einer Theorie der Ordnungstypen, erste Mittheilung" which Cantor completed in early 1885, but never published."'""'' Cantor de-fined a simply ordered set as one among whose elements there is a relationship of rank according to which, for any two elements of the set, the one is the higher in rank and the other is lower, and further for any three elements e, e', e", i f e_ is of lower rank than e' and e' is lower than e", then e_ is also of lower rank than e" (PTO, p. 86). That is, the ordering relation must be connected, asymmetrical, and transitive. Two simply ordered sets are similar or isomorphic i f f i t is possible to completely coordinate their elements one to one so that the rank of any two elements of one of the sets is always the same as ^Ivor Grattan-Guinness, "An Unpublished Paper by Georg Cantor: 'Principien einer Theorie der Ordnungstypen, erste Mittheilung, 1" Acta Mathematica, 124 (1970), 65-107; hereafter PTO. 98 the rank of their correlates in the other set. Cantor 'defined' or explained the order type of a simply ordered set as the general concept under which f a l l that simply ordered set as well as a l l similar simply ordered sets and nothing else (PTO, p. 87). Now he identifies the transfinite numbers with certain order types: Doch sind auch diejenigen Gedankendinge, welche ich trans- finite oder uberendliche Zahlen nenne, nur besondere Arten von Ordnungstypen; sie sind n'amlich die Typen wohlgeordneter  Mengen (PTO, p. 84). "Die allgemeine Typentheorie," he says, "bildet einen wichtigen und grossen Theil der reinen Mengenlehre...also auch der reinen Mathematik, denn letztere ist nach meiner Auffassung nichts Anderes als reine Mengenlehre" (PTO, p. 84). Cantor thought that i t followed from his 'definition' of the type of a simply ordered set that (1) every simply ordered set has a unique order type (GA, pp. 297, 300), and (2) similar simply ordered sets have the same type (GA, pp. 298, 300, 321). Thus Cantor says, for example, "das Gleichheitskriterium...mit absoluter Notwendigkeit aus dem Begriffe des Ordnungstypus folgt" (GA, p. 300). But we really cannot agree. Nothing "follows with absolute necessity" from the rough psychologistic definitions which Cantor formulated. At best (1) and (2) must be taken as axioms in Cantor's system; but i f order types are conceived according to his idealistic explanations, then one can hardly believe that they are true! Cantor defined an order type as a general concept under which f a l l a l l similarly ordered sets. In his Principles, Russell extended and modified the definition of a type. Instead of the type of a simply 99 ordered set, Russell considered, in general, the relation type of a relation. A relation type is not conceived as a general concept (a mental entity), but is a class of similar relations. Relation types are to be distinguished from what Russell called a type of relation, which is a class of relations having certain logical properties in common. "And by the type of a relation I mean its purely logical properties, such as are denoted by the words one-one, transitive, symmetrical and so on" (Principles, p. 430). Russell also explains a type of relation as a class of relations which can be defined by propositional functions containing, besides variables, only logical constants (Ibid., p. 8). Russell also goes on to say that a type of relation is a class of relations having some formal properties in common (Ibid., p. 11). This concept of types of relation is connected with his view of mathematical theories in general: In mathematics, two classes of entities which have internal relations of the same logical type are equivalent. Hence we are never dealing with one particular class of entities, but with a whole class of classes, namely with a l l classes having internal relations of some specified type. And by the type of a relation I mean its purely logical properties (Ibid., p. 430). Further, he says that "Whenever two sets of terms have mutual relations of the same type, the same form of deduction will apply to both" (Ibid., p. 7). There is a connection between types of relation and relation types. 12 Relations of the same relation type have the same logical properties. For example, "the property of being contained in diversity is invariant for likeness-transformations" (PM, II, 203; cp. pp. 510 and 521). — The Analysis of Matter (1927; rpt. New York, 1954), p. 251. 100 Let us now return to the well-ordered sets and the ordinal numbers, first considering the important concept of the i n i t i a l segments of well-ordered sets which, together with the concept of similarity, is the basis for the definition of less than between ordinals. The theorems on seg-ments lead to the result that any two ordinals are comparable, and, ultimately, to the theorem that the powers of any two well-ordered sets are comparable. Then i t only remains to prove that every set is well-ordered—which Cantor had believed ever since his introduction of the concept of a well-ordered set—in order to conclude that any two powers are comparable. This will finally show that the powers of transfinite sets deserve the title of "cardinal numbers." The i n i t i a l segment A determined by the element a. of the well-3. ordered set A is the set of a l l elements of A which precede a. The remainder determined by a. is the subset of A containing a and a l l its successors (GA, p. 314). The segments determined by corresponding elements of similar well-ordered sets are also similar well-ordered sets. Let A and ji be well-ordered sets, and A' and B"* any (respective) i n i t i a l segments of these; in his 1897 papers, Cantor established the following theorems on i n i t i a l segments (briefly, segments) of well-ordered sets. If X£A', then A is not similar to X; however, an infinite well-ordered set may be similar to some of its remainders as well as to other subsets of itself. Different segments of a well-ordered set are not similar. It follows from this that a well-ordered set can be similar to at most one segment of another well-ordered set. If every A' is similar to a B' and every B' is similar to an A', then A and JB are similar. If every 101 A' is similar to a B' but there is a B' not similar to any A', then A is similar to some B'. If some A' is not similar to any B', then every B' is similar to an A'. Finally we have the most important comparability theorem for well-ordered sets: Two well-ordered sets are either similar, or else the one is similar to a segment of the other. These three alternatives are joint-ly necessary and mutually exclusive. It follows from this that i f a sub-set of A is not similar to any segment of A i t is similar to A. The comparability theorem for well-ordered sets provides a basis for a definition of less than between ordinal numbers from which i t follows immediately that any two ordinals are comparable. An ordinal number a' is less than the ordinal o^, a'< a, i f f any well-ordered set of ordinal (X* is similar to some segment of a well-ordered set of ordinal a. The ordinals a and 3 are identical i f f any well-ordered set of type a is 13 similar to any set of type 3. It follows that for two different ordin-als the one must be less than the other; indeed the relation < is a well-ordering relation (GA, p. 321). Let us denote the set of a l l ordinals preceding the ordinal number a by "W(a)"; thus, the i n i t i a l segment of the ordinals W(a) = {3: 3 < a}. For each ordinal ci, W(a) ordered according to magnitude is a well-ordered set of type ot; that i s , i f following Cantor we denote the ordinal of the — It would seem that this is best regarded as an axiom of Cantor's system, though i t is possible to define the ordinals in such a way that this proposition is a consequence of the definition. well-ordered set A by " A " , 1 ^ W ( a ) = a . To see that this is so, we con-sider any well-ordered set A of type ou Let < be the relation well-ordering A. ^  The set of a l l segments determined by elements a of A ordered by the proper inclusion relation c is a well-ordered set similar to A, for A = A , -*-»• a = a', a a i.e. the relation between elements and segments is one-one, and a < a'•«-»• A c A .. a a' A set similar to a well-ordered set is also well-ordered. By definition, the ordinal of a segment of A is less than the ordinal of A: A = a' < a = A. a Hence, every ordinal a' of a segment of A belongs to W(a). On the other hand, every ordinal in W ( a ) is the ordinal number of one and only one segment of A. Thus, the set of ordinals of segments of A is the set W ( a ) and there is a one-one correspondence between these segments and their ordinals. It is easy to see that i f a" and a' are elements of A, then a" < a'•••+-*• A „ < A , -«-»- A ,, c A ,. a a a a Thus, A is similar to W ( a ) , so that they have the same ordinal ou 14 The single bar above the letter denoting the well-ordered set was to suggest the single abstraction from the nature of the elements of that set, while the double bar above the letter denoting a set was to represent the fact that the cardinal is conceived by abstracting both from any order among the elements as well as from their characteristics. ^ I f we wanted to be more precise, we could say that a well-ordered set is an ordered couple (A,</, where < is a well-ordering relation whose field is the set A . ^The method of proof is already in Cantor ( G A , p. 3 2 9 ) . 103 Every set of ordinals has a least element. Let M be any set of ordinals, a_ any element of M, and M' the subset of W(ct) containing a l l elements of M, i.e. M' = MnW(a). If M' is the empty set, then a_ is the first element of M. But if M^  ^ 0, then M' has a first element 3, because M' is a subset of the well-ordered set W(a). Since M' contains a l l ele-ments of M which are less than cx, the number 3 is the least ordinal belonging to M. ^ Every set of ordinals ordered according to magnitude is well-ordered. This follows immediately from the preceding theorem, for i f M is a set of ordinals, then every non-empty subset of M has a least element. Any set of ordinals which contains a l l ordinals less than any ordinal which belongs to i t has an ordinal number which is greater than any of its elements. That is, any set X of ordinals such that i f aeX, then W(a) S X has X > a, for any ordinal aeX. This is so because i f W(a) £ X and aeX, then W(a+1) S X, so that a+1 < X. Now i t follows at once that there can be no set W of a l l ordinal numbers. For, i f W existed i t would have to be a well-ordered set, and for every ordinal a, W(a) <: W. Thus, i f W were a set, it would be a well-ordered set having an ordinal W = 3 greater than a l l ordinals. But 3 would belong — The proof is Hessenberg's, in "Grundbegriff der Mengenlehre," Abhandlungen der Friesschen Schule, N.S. 1 (1906), 551. "Grundbegriff der Mengenlehre" was also published as a book (Go'ttingen, 1906), 220 pp. 104 to W, so 3 < $» which would be in contradiction to the theorem that for every ordinal number a., -i (a < a). This result was the first published paradox of set theory; i t is usually referred to as "the Burali-Forti paradox" or "the paradox of the greatest ordinal," for W should be greater than a l l other ordinals. In The Principles, Russell thought that this paradox might be avoided by denying that W is well-ordered: This does not follow from the fact that a l l its segments are well-ordered, and must, I think, be rejected, since so far as I know, i t is incapable of proof. In this way, i t would seem, the contradiction in question can be avoided (p. 323). But as a matter of fact, i t can, as we have seen, be easily proved that every set of ordinals is well-ordered. The non-existence of a set of a l l ordinals also follows from the theorem that for every set M of ordinal numbers, there is an ordinal u immediately following a l l elements of M , which therefore does not belong to M. Let M* be the set of a l l ordinals which belong to M or which precede elements of M, i.e. aeM* «-> (33) (3eM & a _< 3). Moreover, M* = UW(a)u{a}, where aeM. We will show that u = M* . Let u a note first that M* = M (Va)(aeM + W(a)sM). In that case, i f aeM, then M q = W(a) < u = M. In any case, for every Cantor was the first (1895) to discover this paradox (see sect. 22 below). Burali-Forti did not actually conclude, as Cantor did, that there is no set of a l l ordinals; rather, he concluded, in contradic-tion to Cantor's comparability theorem, that there are incomparable ordinals ("A Question on Transfinite Numbers," [l897], van Heijenoort, pp. 105-111). a_ in M, W(a) < M*. We have only to prove that i f 3 is greater than every element of M, then u S 3. If Y < V, then y belongs to M*. There-fore, any ordinal greater than every ordinal in M and so not in M* must be j> P. For, i f a is in M*, then a <_ some element of M. Thus, M* = W(u) . In his later formulation of the foundations of transfinite set theory, Cantor did not use the principles of generation to define the ordinals. There is however an important distinction between ordinals corresponding to the two principles of generation. Every ordinal num-ber has an immediate successor, but not every ordinal number is the immediate successor of some ordinal; those that are, Cantor called numbers of the first kind, but now they are usually called successor  ordinals. The rest of the ordinals were called numbers of the second  kind by Cantor (GA, p. 330). If M' is a proper subset of an ordered set M and M'does not have a last element but there is an element next after the elements of M', then this element is the upper limit of *T (cp. GA, p. 308). Since every set of ordinals has a first member with respect to <, sets of ordinals ordered according to magnitude can only have upper limits, so that the upper limits of such sets may simply be called "limits." A limit number is the type of a non-empty well-ordered set not having a last element. The order type and the limit of a non-ending set of ordinals are always both limit numbers. If the non-ending set of ord-inals is an i n i t i a l segment of the ordinals, then its limit and its order type are the same ordinal number. Furthermore, i f X is a limit number, then X = lim W(X). 19 Hessenberg, "Grundbegriff der Mengenlehre," pp. 551-552. 106 Cantor called two subseries of type u of a simply ordered set "zusammengehorig" i f for every term of the one there are terms of the other which succeed i t (GA, p. 308). This concept was generalized by Hausdorff: The ordered set A is co final with the ordered set IJ i f f B_ is a subset of A and nothing in A follows a l l members of B. The ordinal cx is cofinal with the ordinal 3 i f f there is a well-ordered set of type which is cofinal with a well-ordered set of type 3. If £_ is cofinal with 3, then a < 3. The limit ordinal a. is cofinal with the limit ordinal B_ i f f is the limit of an increasing sequence of type _3, and so, i f f there is a set B of type _3 such that B_ is a subset of W(a) and W(ct) is cofinal with A. Any successor ordinal is cofinal with 1. If X is a limit number and W(X) is cofinal with A, then X = lim A. 18) Cardinal and Ordinal Numbers The purpose of this section is to describe the transition from Cantor's views about cardinal and ordinal numbers to precise concepts of these numbers suitable for systems of set theory avoiding paradoxes like the one given in the previous section. I will begin by emphasizing what is sound in Cantor's treatment of the number concepts. Cantor clearly realized what conditions satisfactory definitions of the cardinal numbers and order types should satisfy. (1) The respec-tive criteria of identity should follow from the definitions. (2) Every set should have a cardinal number (power) and every ordered set an order type; in particular, every well-ordered set should have an ordinal number. That is, the definitions should be justified by existence and uniqueness theorems. Cantor thought that his definitions certainly satisfied 107 requisite (1), but he had to invoke a doctrine about numbers similar to Leibniz's concerning the "eternal truths" and the possibles to secure anything remotely corresponding to (2). Thus, although Cantor stated that "Jeder Menge M kommt eine bestimmte 'Machtigkeit' zu" (GA, p. 282), we shall see that, given his definition of the power (cardinal number) of a set, i t is not possible to justify this assertion on the basis of set theory and logic. According to Cantor's definition, the power of a set M is the Allgemeinbegriff produced by abstracting from the characteristics of the elements of M as well as from any order among these. But Cantor immediately puts this "definition" aside in favor of another formulation from which he attempts the demonstration necessary to satisfy condition (1). Since every single element [of M ] , i f we abstract from its nature, becomes a 'unit,' the cardinal number M is a definite aggregate composed of units, and this number has existence in our mind as an intellectual image or projection of the given aggregate M.20 Thus, Cantor's powers are not actually general concepts or universals, 21 but sets of pure units. Cantor apparently thought the result of the ^Contributions to the Founding of the Theory of Transfinite Numbers, trans, and intro. P.E.B. Jourdain (1915; rpt. New York), p. 86, and GA, p. 283. Similarly, in the case of an ordered set M, Cantor says that "the order type M is itself an ordered aggregate whose elements are units (lauter Einsen) which have the same order of precedence amongst one another as the corresponding elements of M, from which they are de-rived by abstraction" (Contributions, p. 112; GA, p. 287). 21 It may be noted that Cantor's cardinals are very similar to Plato's mathematical numbers; there are, however, three differences. (1) Plato's numbers are finite (limited) sets of abstract units. (2) It does not seem that Plato considered his units to be in the mind. (3) Plato seems to have allowed that there are different sets of abstract units of the same number of elements (i.e. different mathematical numbers f a l -ling under or participating in the same Ideal number), while Cantor defin-itely thought that i t followed from his "definition" of the cardinals that equivalent sets have the same cardinal number. 108 abstraction process applied to two equivalent sets would be the same, so that equivalent sets have the same cardinal number (GA, pp. 283-284). The cardinal of M is supposed to be a set of pure units equivalent to M; thus, Cantor writes "M <v M" (GA, p. 284). But a l l this hardly amounts to a logical demonstration of the criterion of numerical identity from the definition of the power of a set as a set of distinct, abstract nothings produced in the mind by a process of abstraction. An argument that (2) is satisfied, even of the sort given for (1), is clearly out of the question. It is, however, possible to define the cardinals so that (1) and (2) are satisfiable, entirely within set theory, without resorting to the powers of the mind and unintelligible pure units. The Frege-Russell definition of the cardinals, explained in section 9, is such a definition. But in systems of set theory which avoid the paradoxes by "limitation of size" or by distinctions of type, this defin-22 ition must be modified. According to the original definition, the cardinal number of the class A is the class of a l l classes equipollent to A; in the modified versions, the cardinals are less comprehensive classes of equipollent sets. In the system of Principia Mathematica, the cardinals are defined to be classes of equivalent subclasses of a type, i.e., i f T is a type and XST, then X = {Y^T: Y * X}. Thus Russell was able to incorporate a modification of his definition — If the paradoxes are avoided by means of what Russell called the "zig-zag theory," i t is not necessary to modify the definition. 109 23 into his revised system of logic, and, according to the modified defin-ition, cardinals are not nearly so comprehensive as they were in the original version. For, i f X is a subclass of the type T_, the cardinal of X is a subclass of the power set of X, but the original Russell-cardinals contained classes of a l l orders and were themselves as large as the universe of a l l entities. A version of the Frege-Russell definition appropriate for axiomatic set theories which include the axiom of foundation has also been formu-24 lated. The cardinal number of the set A can be defined as the set of a l l sets Y equipollent to A and of rank less than or equal to the rank of any set which is also equivalent to A, where the rank of a set A is the least ordinal a such A9V , where, for each ordinal a, - a v a = u p ( v e ) . Hence, according to the present definition of the cardinals, the cardinal of A is the set of a l l subsets of which are equivalent to A, where is the smallest ordinal number such that V contains a subset equivalent a to A. With this definition, of course, only sets, and not also classes which are not sets, can have cardinal numbers. The doctrine for avoiding the paradoxes advocated in PM has the pecularity that "A ^ B" may be true, and yet "A = B" be meaningless. 24 Dana Scott, "Definitions by Abstraction in Axiomatic Set Theory," Bulletin of the American Mathematical Society. 61 (1955), p. 442; and "The Notion of Rank in Set-Theory," in Summaries of the Talks Present- ed at the Summer Institute for Symbolic Logic, Cornell University 1957. 2nd ed. (Princeton, 1960), pp. 267-269. See also Dieter Klaua, "Ein Aufbau der Mengenlehre mit transfiniten Typen, formalisiert im PrMdi-katenkalkUl der ersten Stufe," Zeitschrift fUr mathematische Logik  und Grundlagen der Mathematik. 3 (1957), pp. 303-316. 110 25 Let us now turn to definitions of the ordinals. Russell's definition of the ordinals was very similar to his definition of the cardinals. An ordinal number can be attributed either to a well-ordered set, conceived as an ordered pair (A,R} where A is a set and R is a well-ordering relation whose field is A, or to a well-ordering relation R, for the field is determined by the relation, but not vice versa. Russell defined ordinal numbers for well-ordering relations: The ordinal number of the w.o. relation R is the class of a l l relations isomorphic to R. Russell remarked on this definition that It would be more usual to regard the ordinal number as a common property of a certain class of series, but no common property appears except the mutual relation of ordinal likeness. Thus the definition avoids the introduction of a new indefinable and wholly unnecessary entity.26 Like his original definition of the cardinals, Russell's definition of the ordinals needs a modification. But there is another way of defin-ing the ordinals, first published by von Neumann. The basic idea Was not new with von Neumann, and I will remark on some earlier work after 27 presenting his method of introducing the ordinals. The class J_ is an incomplete ordering of the class H i f f J_ is an irreflexive, transitive class of ordered pairs of elements of IL The class H' is a segment of the domain H incompletely ordered by i f f H' 28 is a subclass of H_ and every element of H/ is a J-predecessor of any 25 On the other definition of the cardinals to be considered, the cardinals are some of the ordinals. 26"On Weil-Ordered Series," Russell Archives MS (1901), p. 7. 27 I follow the exposition given in chapters IV and V of "Die Axiomatisierung der Mengelehre," Mathematischen Zeitschrift, 27 (1928), reprinted in Collected Works, ed. A.H. Taub (New York, 1961), I. 28 A J-predecessor of y_ is anything x such that <x,y)eJ. I l l element of H - H'. The segment Seg (x; H,J) of the class H" Incompletely ordered by J_ determined by the element x of H is the class containing a l l J-predecessors of x. The segment determined by an element of H,J (H as ordered by J.) need not be a segment of H,J, for i f zeSeg(x; H,J) 29 it is not necessary that for every y_ in H - Seg(x; H,J), zJy. Like-wise, i t need not be the case that a l l segments of H,J be segments determined by an element x of H; for example, H is a segment of H,J which is not determined by any element of H. Of two segments of an incompletely ordered domain, one is a subclass of the other. The subordering (H'/H)J imposed on the subclass H" of H by the incomplete ordering J_ of the domain H is the class of a l l ordered pairs of elements of H' which also belong to J_, i.e., (H'/H)J = H'xH'nJ. The incomplete ordering [f]j transferred by the function f from the incompletely ordered H,J to H' = f[H] exists i f f is one-one on the class H: <f(u), f (v)> e[fjj •*-»• <u,v>eJ. (VU.VEH). [f]j orders the f-image f [H] of H, provided f is one-one on H, as J_ orders H. The incompletely ordered class H,J is similar or isomorphic to the incompletely ordered class H',J' i f f there is a function f which is one-one on H, H' = f [H], and J' = [f]j. The domain H incompletely ordered by J_ is completely ordered by J_ i f f J_ is connected, i.e. i t is never the case that for U,VEH, <U,V^^J "zJy" means the same as "(z,y)ej". 112 and <v,u>£J, so that for any two different elements of H, one is a J-predecessor of the other. If J_ is a complete ordering of the class H, then for every x. in H., Seg(x; H,J) is a segment of H,J. Since J is a complete ordering, any y_ in H but not in Seg(x; H,J) is a J-successor of every z_ in Seg(x; H,J). The element v of the class H incompletely ordered by J is a first (last) element of H,J i f f every other element of H is a J-successor (J-predecessor) of v. An incompletely ordered class has at most one first (last) element. The class H incompletely ordered by J. is well-ordered by J_ i f f for every non-empty subclass R' of R, the class H_' ordered by (H'/H)J has a first element. If H,J is a well-ordered class, then H,J is also a completely ordered class. If J is a well-ordering of H and u,v£H_, then {u,v} must have a first element. Thus, a well-ordering relation must be connected, and every well-ordered class, a completely ordered class. A function f is a numbering of the class H well-ordered by J_ i f f is a function of argument domain H and for each xeH, f(x) = f[Seg (x; H,J)]. Thus, for x in H, f(x) is the class of values of f_ for arguments which are J-predecessors of x. If x. is the first element of H,J, then f(x^) = 0. If x^ i-s t n e second element of H,J, then f(x£) = {0}, i f x^ is the third element of H,J, then f(x 3) = {0,(0}}, and so on. An ordinal number of the well-ordered class H, J numbered by f_ is the class f [ H ] . In fact, i t can be proved that every well-ordered class has one and only one ordinal number, and similar well-ordered classes have the same ordinal number. Moreover, a well-ordered class 113 is similar to its ordinal number. Let us designate the domain of a l l elements (individuals and sets which belong to other sets) by "U", the domain of a l l sets which are elements or, simply the sets, by "U*", and finally the domain of a l l 30 von Neumann-ordinals by "U**". Then, U**c\J*c\J. I have already intro-duced the symbol "E" for the relation (in extension) corresponding to the fundamental relation of class-membership e: <x,y)eE xey & x,y are elements. The class E can be proved to be a class which is not an element, thus "EeA" is always false. Now let us introduce the symbol for the relation in extension corresponding to the relation c: of class inclusion: (x,y>e£ +-*• xcy & x,y are elements. The class Z_ is also a class which is not a set. The class P_ is an ordinal number i f f (1) P £ U*, (2) _P is well-31 ordered by E_, and (3) for every element u of P, u = Seg(u; P, (P/U*)£). It is also the case that P_ is an ordinal number i f f P £ U** and every element of P is a subclass of P , We shall need to use this result in the"next section. The less than relation < for von Neumann ordinals is simply the proper subclass relation. Indeed, we have for any ordinal numbers P_ and Q_: P < Q P c Q ^ p e (j, — We can define the class IJ as follows: xeU x=x & 3y(xey). 31 These three conditions could be taken as a definition of the ordin-als; though they do not, of course, constitute a definition independent of the concept of well-ordering. Von Neumann notes the possibility of a "direct definition" but does hot give one (Collected Works, I, 391, n.24). 114 Each von Neumann ordinal is the class of a l l smaller ordinals. Thus, while Russell defined the ordinal numbers as classes of similar well-ordering relations, von Neumann defined the ordinals as particular classes, and in such a way that there is a particular relation £ such that, if P is an ordinal i t is well-ordered by (P/U*)£ = (U**/U*)£. The relation (U**/U*)E is a well-ordering of U** and (U**/U*)I = (U**/U*)E, where E_ is the relation in extension corresponding to e. Now the cardinal numbers may be defined so that they are some among the ordinal numbers. An ordinal is a cardinal number i f f i t is not equiv-alent to any smaller ordinal, i.e. the ordinal P is a cardinal i f f no Q_ c P is equipollent to P_. The cardinal number of the class A is the unique cardinal number equivalent to A. The theorems justifying these definitions can, of course, be proved. Furthermore, i t can be demonstrated that U** is both a cardinal and an ordinal, and, indeed, the greatest cardinal and ordinal number. U** is not a set, and is the only ordinal (cardinal) which is not a set. For every cardinal number which is a set, there is a greater cardinal number which is a set. It can also be proved that the class H is not a set i f f H = U**, or, as we could also put i t : M is a set i f f M e U**. P.E.B. Jourdain was the first that I know of to have the idea of defining the ordinals and cardinals in a way similar to von Neumann's. Already in 1906 Jourdain was attempting (in correspondence with Russell) to formulate a definition of the cardinals and ordinals which would be suitable as a method of avoiding the paradoxes by limitation of size. Jourdain explained the ordinals as certain well-ordered series. It is important to know that Jourdain did not understand a "series" to be a 115 3: class; rather, following Russell, he understood a series to be a relation. Each ordinal is to be the series of the preceding ordinals. Thus Jourdain did not define the ordinals in such a way that each ordinal is the set of a l l preceding ordinals, so that each ordinal is a set well-ordered by the membership (and also by the inclusion) relation. Moreover, Jourdain did not really define the ordinals at a l l ; at most, he only expressed some vague ideas which could lead to a logically satisfactory definition. Jourdain also had a 'definition' of the cardinals. As he explained his idea in 1912, the cardinal of a class could be defined as "a special-ized one" of the classes equivalent to that class. The method which immediately suggests itself is to start with the null-class and define the cardinal number 1 as the class whose only member is the null-class. Then 2 may be defined as the class whose members are the null-class and 1; and so on. 33 The cardinal numbers which Jourdain indicates here are the (finite) von Neumann ordinals. When Jourdain communicated his ideas about ordinals and cardinals to Russell in 1906, Russell gave the following precise formulations of the definition of the cardinals based on Jourdain's ideas: ...take as the Nc associated with the No_ £ {a: a < K & (3: 6 < a} i> {Bi 3 < U K Then a l l ordinals of the second class have the class of finite ordinals as their cardinal; and so on. Or even you might take to itself as the associated cardinal, putting as associated Nc_ "On the Question of the Existence of Transfinite Numbers," Proceedings of the London Mathematical Society (April, 1906), p. 267. 33 "On Isold Relations and Theories of Irrational Numbers," Proceed- ings of the International Congress of Mathematicians (Cambridge, 1912), p. 493. 116 Min {a: {&: g < a} ^ {&: B < £}} ...I cannot, of course, adopt these Dfs. because they depend upon the view that every class can be well-ordered. But on that basis, I see nothing whatever to object to in your l e t t e r . 3 4 The cardinal number associated with an ordinal number 5 is the cardinal number of any set which is the field of a well-ordering of type £. According to Russell's first suggestion, the cardinals would not be defined so that they are some of Jourdain's ordinals; on this suggestion, the cardinals would be sets of ordinals. A cardinal number would be a segment of the ordinals which is not equivalent to any proper subsegment of itself. According to the second suggestion, a cardinal number would be an ordinal K such that the segment of the ordinals deter-mined by K is not equipollent to the segment determined by any a < £• If the ordinals were defined as von Neumann defined them, i.e. in such a way that each ordinal is the set well-ordered by inclusion of a l l smaller ordinals, then both of Russell's suggested definitions would have the same extension and would define the von Neumann cardinals. According to the definition of the cardinals under consideration, i t will only follow that every class has a cardinal i f i t is the case that every class is the field of some well-ordering relation. It was skepticism on this point that prevented Russell from adopting the definitions he suggested. In 1917 Mirimanoff gave a definition of the 'von Neumann ordinals' which is independent of the theory of well-ordered sets and of the concept _ Unpublished correspondence, Bertrand Russell Archives (June 14, 1906). I have changed some of Russell's notations. of an order type."'"' But Mirimanoff did not take these sets, which he called the "sets S" (I will call them S-sets) as the ordinal numbers, rather he understood the ordinals in Cantor's way as order types of well-ordered sets, and merely thought the S-sets to be special well-ordered sets corresponding to ordinal numbers. The "S" stands for "segment." Mirimanoff apparently thought of the S-sets in the following way. Con-sider, for example, the well-ordered set {1,2,3} = A. The set of seg-ments of A is the set {0,{1},{1,2}} = A'; the set of segments of A* is {0,{0},{0,{1}}} = A"; the set of segments of A" is {0,{ 0} ,{0, {0}}} = A"', the set of segments of segments of segments of A. The set A"' is an S-set. Mirimanoff did not, however, give any exact definition corres-ponding to this idea. Von Neumann's definition of the ordinals (S-sets) given on p.112 above must have been suggested by the indicated relation between well-ordered sets and the S-sets. The S-set corresponding to the well-ordered set A is the image of the numbering whose domain is A. Mirimanoff defined the S-sets by the following three conditions. (1) Every descending element chain generated by one of these sets ends in the empty set. (2) For any two different elements of one of these sets, one is an element of the other. (3) If M is an S-set, then every element of an element of M is an element of M, i.e. any such set M is transitive. Mirimanoff also proved a number of the important properties 36 of the S-sets from this definition. — "Les antinomies de Russell et de Burali-Forti et le probleme fondamental de la theorie des ensembles," L'Enseignement Mathematique, 19 (1917), 37-52. 36 "Remarques sur la theorie des ensembles et les antinomies cantor-iennes," Ibid., pp. 209-217. Zermelo was the first to give a definition of the 'von Neumann' ordinals independent of the theory of ordered sets, but he did not pub-lish his work. Zermelo defined the ordinals as follows: A set M is an ordinal i f f i t satisfies the three conditions: (1) M = 0 or OeM, (2) (VXeM)(X' = M or X'eM), where X' = Xu{X}, (3) (VNSM)ftjN » M or U N e M ) . 3 7 The set X' is the successor of the set X. 19) Proof and Definition by  Transfinite Induction A principle of proof by transfinite induction and a theorem justifying definitions by transfinite induction are both proved by Cantor, but, like most of Cantor's other theorems in the theory of the transfinite, these results are special cases. It was typical of Cantor to give proofs of special cases by methods which could obviously be used to prove the general theorems. Even his famous diagonal argument was only applied to the particular case of the set of denumerable sequences of two distinct objects. In his 1897 paper, Cantor proves several theorems for the second number class which hold generally; among these are the theorems on induction. Zermelo formulated this definition around 1915 but did not publish i t ; the definition is, however, given by Paul Bernays in "A System of Axiomatic Set Theory—Part II," The Journal of Symbolic Logic, 6 (1941), p. 10. Zermelo's definition did not actually include 0 as an ordinal, and thus omitted the first disjunct of condition (1). Further defini-tions of the von Neumann ordinals, as well as proofs of the equivalence of the various definitions, are given in Heinz Bachmann's Transfinite  Zahlen, 2nd ed..(Berlin, 1967), pp. 24-26. 119 The principle of transfinite induction is the following assertion: If whenever a l l ordinals less than a_ belong to the class A, a_ also belongs to A, then a l l ordinals belong to A. Suppose the antecedent and the negation of the consequent. If there is an ordinal not belong-ing to A, then there is a least such ordinal a_. But, by the antecedent 37 of the theorem, i f every ordinal less than belongs to A, so does ou In Principia Mathematica, a much more general induction theorem is established. Whitehead and Russell define a class of relations which includes the well-orderings as a part; these relations are now called 38 "well-founded" relations, but Whitehead and Russell called them "well-ordered" relations and used the term "well-ordered series" for the relations we have been calling well-orderings. A relation.]? is well- founded i f f every non-empty subclass of the field of P_ has at least one P-minimal element (cp. PM, *250.01), where a P-minimal element of a. class A is a member of the field of P_ which is not a P-successor of any-thing in A (cp. PM, *2 05.1). The connected well-founded relations are exactly the well-orderings (well-ordered series). Whitehead and Russell prove the following general principle of 37 This proof was given by Cantor for the case of the ordinals of the second number class (GA, p. 337), but Cantor did not use the term "transfinite induction" or state the principle as a separate theorem. For early formulations of principles of transfinite induction see Schoenflies, "Die Entwickelung der Lehre von den Punktamannigfaltig-keiten, I Teil," Jahresbericht der Deutschen Mathematiker-Vereinigung, 8 (1900), 45; Hausdorff, "Unterschungen iiber Ordnungstypen," Verhand- lungen der koniglich sachsischen Gesellschaft zu Leibzig, matnematisch- physische Klasse, 58 (1906), 127. 38 The term "wohlfundiert" was used in Zermelo's paper "Grundlagen einer allgemeinen Theorie der mathematischen Satzsysteme," Fundamenta  Mathematica, 25 (1935), pp. 136-146. 120 induction for well-founded relations: wf (P) & VB(B^F(P)nM seq M{B}«M) -> F(P)*M, i.e. i f the P-sequents of every subclass of the intersection of the field of P_ and the class M belong to M, then M contains the field of P (PM, *250.3), where the P-sequents of a class 15 are the minima of the class of P-successors of a l l members of BnF(P)—i.e. the minima of Op""BnF(P). The proof goes as follows. Suppose the hypothesis of the theorem and the negation of its consequent. Let A be the class of things in F(P) which are not in M. Since AsF(P) and A 4 0, A has a P-minimal element x.- The class of P-predecessors of x is a subclass of M, but x is a P-sequent of the class of its P-predecessors, so that, by the hypothesis of the theorem, xeM (PM, III, 12). We now turn to theorems justifying definitions by transfinite induction. In an earlier chapter, we saw that a theorem justifying definitions of functions by finite induction is rigorously demonstrated in Dedekind's great masterpiece Was sind und was solien die Zahlen, by means of the principle of complete induction which is also established there. In the 1897 installment of his "Beitrage zur Begriindung der transfiniten Mengenlehre," Cantor proved a similar theorem justifying definitions of certain sorts of functions (normal functions) whose argu-ment domain is the second number class. The generalization of Cantor's theorem on definition by transfin-ite induction asserts that there exists exactly one function whose argu-ment domain is the class of ordinals W such that (1) f(0) = 6 > 0, (2) a < a' -»• f(a) < f(a'), (3) f(a+l) = f(a)y where Y is a particular ordinal > 1, (4) fC*) = lim f(V) for a limit ordinal X . This itself is a special u < X 121 case of the general theorem that there is exactly one function f such that f(0) = 6 f(ct+l) = g(f(a)) f ( X ) = h(f | w ( X ) , X ) , where is a particular ordinal, and g and h_ are particular functions. In Cantor's special case 6 > 0, g(B) = B'Y, and h(f | w ( X ) , X ) = lim f(y). y < X The main steps of the proof by transfinite induction of the exis-tence of f are the following. Suppose the theorem holds for 3 < a. Case 1: £ is of the first kind. By the induction hypothesis, there is a function ^ a_^ satisfying the conditions of the theorem; the function f o such that f Q(y) = f a _ i ( ^ ) f o r u £ a _ 1 » a n d f a( a> = g ( f a ^ i ( a _ 1 ) ) also satisfies those conditions. Case 2: ex is of the second kind. By the induction hypothesis, for every B < a, there is a function f^ satisfying the conditions of the theorem, and i t can be shown that they are also satisfied by the function f Q such that f a(8) = fg(3) for & < a, and f Q(a) = h(f Q | w(a),a). Thus, for every £, there is a function f satis-fying the conditions of the theorem. Let f(a) = f a(a); this function _f is the function whose existence was asserted by the theorem. The unique-ness of f must, of course, also be demonstrated. It will be noted that whenever a definition (of the appropriate type) is formulated in the proof just outlined, i t is immediately assumed that a function satisfying the defining equations exists, i.e. an unres-tricted axiom of abstraction is assumed for functions. The paradoxes, of course, make it necessary to reformulate this proof on the basis of 122 more restricted principles concerning the existence of functions. Von Neumann was the first to give such a proof, as well as the first to give a proof of any sort of a general theorem justifying definitions by transfinite induction. Von Neumann's theorem on definition by transfinite induction is the following very general assertion: For every function <(> of_ two arguments, there is a. function f with domain U** such that f (P) = <j>(P, f |P), and only one such function f . 4 0 Von 41 Neumann proved this theorem as follows. An ordinal JP is normal i f f there is a function £ of domain P_ such that for every QeP, g(Q) = <j)(Q, g|Q). Von Neumann shows that for every normal ordinal there is exactly one such function £ which is called "the function element up to P" and is designated by "FE(P)M. It follows that <j)(P, g|P) is unique for a given <l> and normal set P_; von Neumann calls this "the function value of P_" and denotes i t by "FV(P)". The function element of P satisfies the condition Vx[xeP {FE(P)}(x) = <j)(x, FE(P)|x)]. A normal ordinal is an ordinal having a function element. Let IT be the class of a l l normal ordinals which are sets. If P is a normal ordinal, then (1) every element ^  of P is a normal ordinal, and (2) FV(Q) = {FE(P)}(Q). In order to prove (1), we show that FE(P)|Q is a function element of Let m = FE(P)|Q; i t must — A similar theorem for well-founded relations was stated by Richard Montague in "Well-founded Relations; Generalizations of Principles of Induction and Recursion," Bulletin of the American Mathematical Society, 61 (1955), 442. 41 I present the part of the proof concerning the existence of f_ which is given in chapter IX of "Die Axiomatisierung der Mengenlehre." 123 be proved that m(x) = <}>(x, m|x), for every X E Q . If xeQ, then m|x = [{FE(P)} |Q] |x = FE(P)|x. Therefore, m(x) = <J) (x, m|x). Since ^ i s a normal set with function element FE(P)|Q, i t has a function value: FV(Q) = <j>(Q, FE(Q)) = +(Q,FE(P)|Q) = {FE(P)}(Q). Thus, (2) also holds. We may note that i f P_ i s an ordinal every element of which is.normal, then FE(P) = U { g : 3 X(xeP & g = FE(x)u{<x,FV(x)>})}. Next we prove that P i s a normal ordinal i f a l l i t s elements are normal. This i s done by defining a FE of P_. By hypothesis a l l elements x of P have FE's and FV's. Let j be the function: j = (<x,y>: xeN & y = <}>(x, FE(x))>. Thus, PeN -> j(P) = FV(P). (N is the class of a l l normal ordinals which are sets.) Now let n_ = j|P. We shall show that n = FE(P), and this w i l l establish the normality of P_. Every element x of P i s a normal set and n(x) = FV(x). By the preceding result, every element _y_ of x is normal and {FE(x)}(y) = FV(y). Hence, n(y) = {FE(x)}(y). It follows from this that n|x = FE(x)|x = FE(x). By definition n(x) = FV(x), therefore, n(x) = <|)(x, FE(x)) = <Kx, n|x), for every xeP. But this means that n i s FE of P_, which was to be proved. The normality of U** is easily established from the foregoing results., N i s a class of ordinals which are sets, so N c u**. Since every element of a normal set is normal, i f PeN, then PtN. From the theorem that a class of ordinals a l l of whose elements are subclasses of i t is i t s e l f an ordinal, i t follows that N i s an ordinal. Every ele-ment of N i s normal, therefore N i s also normal. Thus, i f were a set, 124 it would belong to itself, but, since N is an ordinal, this is impossible. U** is the only ordinal which is not a set, hence N = U**; U** is normal, and FE(U**) is the function _f_ whose existence was to be established. It follows at once from the general theorem on definition by trans-finite induction, that for any ordinal P e U** and any function <|> of two arguments, there is one and only one function ty such that for every xeP, ij>(x) = <J> (x, If f_ is the function of the general theorem, then f|P is the desired ty. A brief description of the sort of definition by transfinite induc-tion used in Principia Mathematica completes this section. In the case of the usual method of definition by transfinite induction, a new function whose argument domain is the class of a l l ordinals is defined by means of a "given" function. The procedure of definition by transfinite induction in PM is a means of defining well-ordered series in terms of known rela-tions which are not serial. Several definitions are needed in order to explain the method further. The definitions of the ancestral (transitive closure) of a relation R and of the R-posterity of a thing x. have already been given in section 2. For present purposes, i t is necessary to extend the latter concept and define the "transfinite posterity" of x with respect to relations R and Oj the following definitions are preliminary for this. The P-maxima of the class cx are the elements of the intersection of cx and F(P) which are not P-predecessors of elements of a. The P-limits of cx are the P-sequents of cx which are not P-maxima of o^. The derivative ^(a) of the class a_ with respect to the relation P_ is the class of P-limits of non-125 empty subclasses of the intersection of a_ and the field of P_. A class a_ is transfinitely hereditary with respect to R and 0^  i f f a_ is R-hereditary and contains the Q-limits of a l l non-empty subclasses of ar*F(Q), i.e. i f f R"au6^(a)sa. The transfinite posterity of x. with respect to R and Q is the class of a l l elements of F(Q) belonging to a l l transfinitely hereditary classes containing JC: (R*Q)(x) = F(Q)nfl{a: xea & R"au6QSa}. Finally, for each thing x and relations R and C;, the relation is the relation with field restricted to the transfinite posterity of x with respect to R and 0j QRx = < M R * Q X * ) x ( R * Q>00-It is such relations that figure in Principia definitions by transfinite induction, for i f x belongs to the domain of R, is transitive and irre-flexive, R is a many-one subrelation of 0^, and each non-empty subclass of (R*Q)(x) having a Q-limit has a unique Q-limit, then Q_ is a well-ordered Kx series and is, furthermore, the only transitive relation P_ whose field is contained in (R*Q) (x) such that (1) for any in the domain of P_, R(z) is its immediate successor, and (2) i f cx is any non-empty subclass of F(P) having no maximum, the Q-limit of a_ is the P-limit of ex. Whitehead and Russell remark that "This-proposition is essential for what may be called 'transfinite inductive definitions,' i.e. definitions of a series by defin-ing the successor of every term, and the successor of every class having no maximum" (PM, III, 87). The Principia method of definition by transfinite induction can be used for some of the same purposes as the ordinary method. Thus, Whitehead and Russell employ their method in the proof of Zermelo's well-ordering theorem, where the object is to show that every non-empty class is the field of some well-ordering relation (well-ordered series). Their demonstration of the theorem follows Zermelo's second proof. Let u be any set and M be the class of non-empty subsets of p. The well-ordering theorem follows from the supposition that there is a 42 selective relation S for the class M, i.e. a one-many subrelation of E_ whose argument (converse) domain is M. Thus, i f S is a selective relation for M, then for every aeM, S(a)ea9u. Now, let = £ and R be the relation on P(u) such that $Ra 6eM & B - {S(B)} = a. R is then a many-one subrelation of Q whose field is P(y) and whose domain is M. We want to define a well-ordering relation whose field is p. In fact, the relation s|QR)j|s = StQR^ is such a relation, for is a well-ordered series (*258.201) whose field is a subclass of P(y), s|(R*Q)(u) is one-one (*258.3), and F(StQ_ ) = y (*258.31). There will, of course, be some ordinal number B and some function f_ whose argument domain is W(n+1) such that f(0) = y, f( C+l) = f (?) - { S ( f ( 0 ) } for ? < n, f(X) = — For discussion of this axiom, as well as a proof of the well-ordering theorem based on von Neumann's axiom system for set theory, see section 31 below. Only the main steps of the proof in PM are given here as an illustration of how Whitehead and Russell's method of defin-ition by transfinite induction is used in a demonstration. This may be compared with the application of von Neumann's theorem on definition which is involved in the proof given in section 31. C\ f (£) for limit numbers X <_ n> and f (n) = 0. The well-ordered set £<X <u> StQj^> ^ s isomorphic to <W(n), <)• 20) Number Classes and Alephs In 1882 and 1883 Cantor not only introduced the extended system of whole numbers (the finite and transfinite ordinal numbers), he gave the first explanations of his newly discovered method of defining an increasing sequence of transfinite powers. Apparently Cantor had been seeking a way to define such a series of powers for some time. In 1882 he finally perceived how to define a series of transfinite powers—similar to the series of a l l ordinals—in terms of certain classes of transfinite ordinals. Cantor spoke of the principle of limitation as a means of defin-ing these number classes (GA, p. 167). However, the first number class (I) is not defined in accordance with the principle of limitation, as is the second number class (II); the class (I) is defined as the class of a l l finite numbers, while the class (II) is defined as the set of a l l ordinals the power of whose predecessors is that of the first num-ber class (I). The existence of the second number class follows from the principle 43 To take a simple example, let p be the set N of finite cardinal numbers and, for each non-empty subset a_ of N, S(a) be the least number in o. Then, f(0) = N, f(l) = N - {0}, f(n+1) = f(n) - {S(f(n))> = N - {0,1,..., n}, and f(u) = O f(n) = 0, for naff (n+1). n<co 128 (2'), and its type co ^  is the smallest ordinal greater than a l l ordinals with denumerably many predecessors. Cantor proved that the power of the ordinals of the second class is greater than that of the first class and that, in fact, i t is a next greater, i.e. there is no power greater than the power of (I) but less than the power of (II). Only the first and second number classes were explicitly considered by Cantor in his 1883 paper, and he gave no general definition of a num-ber class there. But i t is clear that an unending sequence of number classes can be defined, and that Cantor's method of proof, when supple-mented in an obvious way, can be used to show that the powers of these number classes also form an increasing sequence of cardinal numbers. In a paper written in 1884 but not published in his lifetime, Cantor says that: In the "Grundlagen" I have proved (or rather indicated complete-ly the way and the means for the proof) that the various powers of infinite sets constitute an absolutely infinite increasing series formed according to the same type as the real whole finite and transfinite numbers themselves; the more one seeks to comprehend the f u l l sense of these theorems, the more one must wonder at nature in its boundless size (PTO, p. 86). Thus, even before discovering the diagonal method, Cantor had already showed how to prove that there is an unending sequence of ever greater In general, regarding the existence of ordinals, (2') plus the axiom of infinity seems to have the same strength as the first edition of Principia Mathematica, the simple theory of types, and Zermelo's original system (1908) . The existence of co for every finite ordinal ii follows from these systems, but that is as far as we go. However, in a theory of Cantorian extent, the series of i n i t i a l ordinals and corresponding powers, must be isomorphic to the whole series of ordin-als. For this, the axiom of replacement is essential, as well as sets of "mixed" type. powers, the powers of certain classes of ordinals. By characterizing the series of ordinals and their corresponding cardinals as "absolutely infinite," Cantor apparently meant that these cannot be augmented in any way, whereas each transfinite number is surpassed by others. In his later formulation of the foundations of transfinite set theory, Cantor introduced the number classes and alephs without any reference to the principles of generation and restriction. The number  class Z(a) of the power ji is the class of a l l ordinals of well-ordered sets of power a. (GA, p. 325). The number class corresponding to any finite cardinal n is a unit class: Z(n) = {n}. This is so because any two finite well-ordered sets of a particular cardinal number ri are similar, and therefore have the same ordinal number. It is even the case that any two simply ordered sets of the same finite cardinal num-ber have the same order type, and this type is an ordinal number. For, any finite ordered set is well-ordered by any simply ordering relation whose field i t is (GA, pp. 298, 325). Cantor's first number class (I) is not = Z(m) for any power m, finite or transfinite. The first number class is the union of a l l the number classes corresponding to finite cardinals: UZ(n) = I . n To each infinite cardinal-number there corresponds an infinite class of ordinal numbers of well-ordered sets of that cardinal number of elements. Indeed, we shall see that the number class corresponding to a transfinite power is a well-ordered set having the next greater power, and its ordinal 130 is the least ordinal number of a well-ordered set of the next greater cardinal number of elements. Similar well-ordered sets are equipollent; therefore, i f two well-ordered sets have the same ordinal number, they have the same power. For each ordinal number a_, there is exactly one power corresponding to a_, which is the power of a l l well-ordered sets of ordinal number «_; Cantor — 45 used the symbol "a" to denote the unique power corresponding to a_. The alephs are the powers correxponding to transfinite ordinal numbers (Ibid.). Thus, the alephs are the powers of infinite well-ordered sets. Cantor proved that there is a least aleph KQ = co which is the smal-lest transfinite power. *<Q is the power of the set N of finite cardinal numbers, and the set N_ ordered according to magnitude is a well-ordered set of type OJ (GA, pp. 293, 325). It is evident that K Q is greater than any finite number, for N is not equivalent to any of its i n i t i a l segments. That is _< any transfinite power follows from the theorem that every infinite set contains a denumerable subset. Cantor proved this latter proposition as follows. Let _T be an infinite set. For every finite cardinal ri, T_ contains a subset of cardinal n_. If f is a function such that for every non-empty subset X of T, f(X) eX, let T f(l) = T, Tf(n+1) = T,(n) - {f(T-(n)) }, and M = T - T,(n+1); then UM is a denumerable r £ n t n n 46 subset of T_. But to complete the proof we have to infer by means of the axiom of choice that there is such a function jf. It must be noted that i f a were the power of a set which was not 4 5Van Heijenoort, p. 115; cp. GA, p. 298. ^6GA, p. 293; cp. The Principles of Mathematics, p. 122, par. -2. 131 the field of any well-ordering relation, then Z(a) would be the empty set, for no set of power a_ would be well-ordered, for i f a set is well-ordered, so is any equivalent set. If a_ is an aleph, i.e. a power corresponding to a transfinite ordinal number, then Z(a) is an infinite well-ordered set, which, thus, has a least element. Such a number is called an i n i t i a l ordinal, and to is the i n i t i a l ordinal of the number a class zC^ a) • The least transfinite number co is the smal- ^ lest i n i t i a l ordinal = COQ and is the first number of the class Z(XQ). The number cx is the type of the set of a l l i n i t i a l ordinals smaller than co^ . There is a one-one correspondence between i n i t i a l ordinals and alephs, and the greater i n i t i a l number has the greater corresponding aleph. It follows that every set of alephs ordered according to magni-tude is well-ordered. X A is the power corresponding to the i n i t i a l num-ber co , therefore: a X = co = W(w ) a a cr For every set of alephs, there is an aleph not belonging to i t which is the least aleph greater than every element of that set. If M is a set of alephs, let M* be the set containing a l l ordinals corres-ponding to alephs in M as well as those corresponding to any aleph smaller than some member of M. Let ti^ be the power of the set M*. It will be shown that fi is the smallest aleph following a l l alephs in M and that M* = co . M* immediately follows a l l ordinals in M*, therefore • u 3 ' tf^ is greater than a l l alephs in M. If ^ < ti , then any ordinal in Z ( i V a ) must belong to M* and consequently ti must belong to M. M* belongs to Z ( t f ) and therefore co < M*. In fact, M* = co , for otherwise fi would have to belong to M, contrary to what has been established 132 47 The number class Z(?<a) corresponding to the power X a is the set of ordinals: Z ( V - W(wa+1> " W<w«>» that is, the set of a l l ordinals u such that to < u < to , , . a — a+1 Cantor showed how to prove that 48 z ( * ) = * j . , . a a+1 and i t is also the case that <Z(tf ),<> = to . . . a a+1 Let a and 8 be ordinals greater than 0. a is singular i f i t is . cofinal with a smaller ordinal 8, but a is regular i f i t is not cofinal 49 with such a 8. The cofinality character, cf(a), of the ordinal a_ is the least 8 such that a is cofinal with 8; thus cf(a) <_ a, and cx is regular i f f cf(a) = a. The regular numbers can also be defined as those ordinals a_ which are not sums of a smaller number of smaller, ordinals. Let us take some examples. The only finite regular numbers are 1 and 2; every finite number except 1 is cofinal with 2. The least transfinite number to is regular, but the number to is singular, for to = lim to . ° to to ^ n n<to Let us say that the aleph is regular or singular accordingly as to^ is ^This proof is given in Hessenberg's "Grundbegriff der Mengenlehre," pp. 553-554. 48 GA, 333; see also pp. 198-200. 49 These concepts are due to Hausdorff, "Grundzlige einer Theorie der geordneten Mengen," Mathematische Annalen, 65 (1908), 442. 133 regular or singular. Every infinite regular number is an i n i t i a l number cu , but not every in i t i a l number is regular. If a. is not a limit number, then o>a is regular, i.e. every number C 0 g + ^ is regular. If a is a limit number, then co is cofinal with a and therefore co < a. The number co is singu-a a — a lar, i f a is a limit number and co > a. The i n i t i a l number co is an exorbitant number i f f i t is regular and a is a limit number."^ If the in i t i a l number co is exorbitant, then co = a, i.e. i t is a critical num-ct , a ber. That exorbitant numbers must, i f they exist, be very large is indicated by the fact that the least critical number, the limit of the sequence f_ such that f(l) = co and f(n+l) = ^ £ ^ n y * s singular. (The index of the i n i t i a l number which is the limit of the sequence f_ must itself be the limit of f.) We have seen that there are two ways of defining ascending sequences of powers which are similar to the series of ordinals. Let us describe the first way, based on Cantor's theorem, in more detail. The function V from ordinals to sets is defined by transfinite induction as follows: a v 0 - . Vi - P(V> V = U v , for limit ordinals X . 5 1 X or a<X The V for finite a are finite, but V = ; the sets V , for a > co, have a — co 0 or — ever greater transfinite powers. We may define this sequence f_ of powers ~^See Hausdorff, Ibid., p. 444, and Grundziige der Mengenlehre, p. 131, "^Cp. the definition of V on p.109; the two definitions are equival-ent. 134 as follows: « f < o > - V v f(a +l) - 2 f ( a ) - V ^ + 1 . f(A) = Ef(a) = V^^, for limit numbers A. a<A The increasing sequence of powers f is similar to the series of a l l ordinal numbers. It is a consequence of the well-ordering theorem that every trans-finite power is an aleph. The question arises: Where do the powers f(a) come in the sequence K of powers ordered according to magnitude? According to the generalized continuum hypothesis (GCH): *a a+1 hence, i f GCH is true, f (a) = = . aj+a a Thus we should have: V 2 V 2 • V i - 2 " - - v S i - 2 " -CHAPTER V THE THEORY OF LIMITATION OF SIZE 21) Introduction There have been many different views on the "source" of the para-doxes, as well as on their significance for set theory and for mathematics in general. In this chapter I present the analysis which originates with Cantor himself and which i s the analysis underlying the main current systems of axiomatic set theory. The general plan for avoiding the para-doxes based on this analysis was called by Russell "the theory of limita-tion of size." Since there is more than one analysis of the paradoxes which leads to a way of avoiding them, how could i t be claimed that any one of these reveals "the precise fallacy" to which the paradoxes are due?''" It seems to me that an analysis i s more plausible in this respect accordingly as i t isolates the paradoxical cases and does not affect those hot directly associated with a paradox. The vicious c i r c l e analysis as well as analyses 2 which attribute the paradoxes to "false abstractions," the supposition of the existence of i n f i n i t e sets, or to the supposition of "impure" sets are not precise analyses in this sense. A l l these succeed in avoiding the paradoxes only by going too far. '''The quoted phrase i s from Principia Mathematica, I, 1. 2 See Russell's "On the Substitutional Theory of Classes and Relations" in Essays in Analysis, ed. Douglas Lackey (London, 1973), pp. 165-166. - 135 -136 Despite their substantial difference, there is an important point of agreement between the analyses of Russell and Cantor: there is no reason to believe that the paradoxes are due to the supposition that there 3 are (actual) infinite sets. Thus, the paradoxes of set theory do not deserve to be called "paradoxes of the infinite." Generalizing a state-ment made already in Russell's Principles, i t may be said that none of the paradoxes of set theory "concern the infinite as such, but only very large infinite classes" (p. 362). The paradoxes are in a l l cases connect-ed with what Cantor called absolutely infinite multiplicities, i.e. with "certain very large infinite classes." The paradoxes might thus be refer-red to as paradoxes of absolute infinity, or, following G. Hessenberg,. they might be designated as "paradoxes of the ultrafinite;" the latter expression was adopted by Zermelo (van Heijenoort, p. 202). Poincare alleged that the "Cantorians" became involved in contra-4 dictions because they forgot that "there is no actual infinity." In his reply to Poincare's paper, Russell reaffirmed the point of view of The Principles: "the contradictions have no essential reference to infinity.""' Russell pointed out that semantic paradoxes such as the l i a r cannot be attributed to the supposition of an actual infinite and that Poincare himself had said that the paradoxes were the results of vicious -When I speak of "paradoxes" in this chapter, I mean only the set-theoretical or mathematical paradoxes, unless I explicitly mention the semantic paradoxes. 4 "The Latest Efforts of the Logisticians," trans. George Bruce Halsted, The Monist, 22 (1912), p. 538. 5"0n 'Insolubilia' and their Solution by Symbolic Logic" (1906) in Essays in Analysis, p. 197. 137 circles, and this theory "makes no reference to infinity, and by no means excludes infinite collections" (Ibid.). According to Whitehead and Russell, "An analysis of the paradoxes to be avoided shows that they a l l result from a certain kind of vicious circle." In Russell's earlier paper "Mathematical Logic as Based on the Theory of Types," he said of the paradoxes discussed there: we found that a l l of them arise from the fact that an expres-sion referring to a l l of some collection may itself appear to denote one of the collection.... We decided that, where this appears to occur, we are dealing with a false totality, and that in fact nothing whatever can significantly be said about a l l of the supposed collection (Logic and Knowledge, p. 101). In fact, Russell did not show that the paradoxes "arise" or "result" from such so-called vicious circles; he only showed that by eliminating these the paradoxes can be avoided. The paradoxes, however, can also be avoided in other ways. Moreover, i t was not even the case that every paradox was blamed on a vicious circle of the kind mentioned, i.e. not a l l paradoxes were said to be due to the use of an expression denoting one of the values of a bound variable which i t contains. Let us consider, e.g., the analysis of the definability paradoxes. This is explained in PM with reference to Berry's paradox, which runs as follows. Only a finite number of integers can be named by means of English phrases having fewer than some particular finite number of sylla-bles, say 19. Therefore, there must be a least number not definable by a °PM, p. 37. Whitehead and Russell considered a l l the paradoxes, both set-theoretical and semantic paradoxes, to be "vicious circle" fallacies. Many years later Ramsey classified the paradoxes in terms of Russell's own concept of logic as "logical" and "epistemological" or "linguistic" ( = semantic) paradoxes. In My Philosophical Develop- ment, Russell says that Ramsey is "definitely right" about the classi-fication of the paradoxes (p. 126). 138 phrase containing fewer than 19 syllables. But the phrase "the least integer not nameable in fewer than nineteen syllables" names the least integer not nameable in fewer than 19 syllables and i t has fewer than 19 syllables. Whitehead and Russell say that this paradox obviously embodies a vicious circle fallacy. Now i t is true that we have an obvious violation of the definability form of the vicious circle principle here, for an integer is defined by means of an expression containing a bound variable whose range of values is the set of a l l integers. In general the vicious circle principle rules out a l l definitions of the form "The least integer 3C such that ty (x)," because i t is short for "The x. such that [ty (x) & (Vy) (<j>(y) x <_ y)] ," where "x" and "y" range over a l l integers. According to the form of the vicious circle principle which rules out impredicative definitions, an expression containing a bound variable cannot denote one of the values of that variable: "Vicious circles arise where a phrase containing such words as a l l or some (i.e. containing an apparent variable) appears itself to stand for one of the objects to which the words a l l or some are applied." 7 It is no surprise, however, that Whitehead and Russell do not blame Berry's paradox on the 'vicious circle' involved in the use of the expres-sion "the least integer x . . . T h e y blame the Berry paradox on the follow-ing vicious circle fallacy: the word 'nameable' refers to a totality of names, and yet is allowed to occur in what professes to be one among names. Hence 7"0n 'Insolubilia' and their Solution by Symbolic Logic," Essays in Analysis, p. 213. 139 there can be no such thing as a totality of names,1 in the sense in which the paradox speaks of 'names' (I, 63). But a name referring to a totality of names is not a case of an expression referring to something which is a value of a bound variable i t contains, for i t is the expression, not what i t denotes, which is a value of a bound variable i t contains. As a final criticism of the vicious-circle analysis of the paradoxes, i t may be said that just as Russell himself did not accept the proposal that the paradoxes are due to the supposition of infinite sets, we cannot accept the vicious circle analysis of the paradoxes. Only very large sets having impredicative definitions are paradoxical! While Russell may have found a common characteristic of the paradoxes, there is no reason to believe that the characteristic is peculiar to the paradoxes. As far as I can see, the vicious circle analysis fixes upon an innocent characteris-tic which is present in the paradoxes as well as many non-paradoxical cases, and the paradoxes are not due to the so-called vicious circles in-dicated by Whitehead and Russell. These authors say that "whenever we have an illegitimate totality, a l i t t l e ingenuity will enable us to con-struct a vicious-circle fallacy leading to a contradiction..." (I, p. 64). They give no evidence for this assertion, and I certainly do not know any reason to believe i t . I should like to see anyone derive a contradiction from, e.g., the supposition that there is a set of a l l finite sets of individuals, i.e. the intersection of a l l sets of individuals which con-tain the empty set of individuals and, when they contain the set of indiv-iduals A and _b is an individual, contain Au{b}. Al l in a l l , we have to agree with Hilbert that the exclusion of impredicative definitions 140 8 (definability form of the vicious circle principle) is heresy. According to P. Bernays, the "essential importance" of the antinomies is that they show the impossibility of supposing a totality of a l l sets and 9 relations as well as that every set is an element of further sets. Indeed, according to the analysis of the paradoxes due to von Neumann, both of these suppositions are to be denied, and i t is maintained that because some sets are not elements of further sets, there can be no set of a l l sets, but only a set of a l l sets which are elements. The sets which are not ele-ments are those which are as large as the universe of a l l elements.^ It is clear that Cantor did not perceive from the beginning that the absolute totalities and classes as large are illegitimate. He did not any-where make use of a set of a l l things (as Dedekind did) or of a set of a l l sets. But, at the end of the paper in which his famous theorem is first proved, he says that he has shown that "the aggregate (Inbegriff) of a l l powers, i f we think of the latter as ordered according to magnitude, forms 'a well-ordered set,' so that...for every set of powers which increases without end there is a greater" (GA, p. 280). These assertions quite obviously cannot both be true, but Cantor does not seem to have noticed this at the time. "Probleme der Grundlegung der Mathematik," Mathematische Annalen, 102 (1929), p. 2. 9 "On Platonism in Mathematics" in Benacerraf and Putnam, p. 277. ^ I t is convenient to use the word "class" as I have used "set" and to use "set" for those classes which are elements of other classes. This usage is purely technical and is not to be understood as implying that classes and sets (in the special sense) are somehow objects of differ-ent "natures." 22) The Cantor-von Neumann  Analysis of the  Paradoxes Cantor himself was the first to become aware of the paradoxes. According to Felix Bernstein, one of Cantor's students, Cantor found the paradox of the ordinal number of the set of a l l ordinals "in 1895 and communicated i t by letter to Hilbert in 1896 and to Dedekind in 1899."''"''' There is even some reason to think that Cantor discovered the paradox of the set of a l l alephs by the time of the publication of his first article on the transfinite numbers in 1895, for he says there that the transfinite cardinals ordered according to magnitude "jedoch in einem erweiterten Sinne eine 'wohlgeordnete Menge' bilden" (GA, p. 295). The phrase "in an extended sense" suggests that Cantor may have already discovered that the multiplicity of a l l alephs is inconsistent. Further-more, Cantor speaks as i f he were in possession of an argument for the comparability of powers, and this would have to be the argument, given below, which makes use of the so-called "inconsistent multiplicities." He says that the truth of the comparability theorem will be established after "wir einen Uberblick uber die aufsteigende Folge der transfiniten Kardinalzahlen und eine Einsicht in ihren Zusammenhang gewonnen haben werden" (GA, p. 295). It seems most likely that Cantor discovered the paradoxes, or, as he says, the necessity of distinguishing two kinds of multiplicities, ___ "Uber die Reihe der transfiniten Ordnungszahlen," Mathematische  Annalen, 60 (1905), p. 187. Cantor also communicated the paradox.of the set of a l l alephs to Hilbert, who mentioned i t in his paper "Uber den Zahlbegriff," Jahresbericht der Deutschen Mathematiker-Vereinigung, 8 (1900), p. 184. 142 while trying to establish the comparability of powers. According to Bernstein, Cantor found the paradox of the set of a l l ordinals while 12 attempting to prove the well-ordering theorem. But the problem of well-ordering is closely connected with that of comparability, and I suspect that, above a l l , Cantor wanted to prove that any two powers are comparable and, thus, fully deserve to be called "numbers." The comparability of powers follows at once from the proposition that every power is an aleph and Cantor's theorem on the comparability of well-ordered sets. As Cantor put i t in a letter to Dedekind: "The big question was whether besides the alephs, there were s t i l l other cardin-13 alities of sets" (van Heijenoort, p. 114). In a later letter to Jourdain, Cantor remarked that "Den unzweifelhaft richtigen Satz, dass es auser den Alephs keine andern transfiniten Cardinalzahlen giebt, habe ich vor liber 20 Jahren (bei der Entdeckung der Alephs selbst) intuitive erkannt."14 After realizing that the supposition that certain sets exist leads to contradiction, Cantor began to distinguish between inconsistent and consistent multiplicities. I shall briefly discuss his explanations of 12 Bernstein is quoted by the editors of Dedekind's Werke, Vol. 3, p. 449. 13 Cantor says in this letter, written in 1899, that he had been in possession of a proof that a l l powers are alephs for two years, thus casting doubt on my speculations on the previous page. But in a letter to Jourdain written in 1903, Cantor said that he had communicated his proof to Hilbert about seven years ago. Moreover, i t seems most like-ly that Cantor discovered the paradoxes and the argument for compara-bility at the same time, which, according to my speculations, was in early 1895. 14 Ivor Grattan-Guinness, "The Correspondence between Georg Cantor and Philip Jourdain," Jaresbericht der Deutschen Mathematiker- Vereinigung, 73 (1970), p. 116 143 the distinction, but, first, the most important point: which multi-plicities are inconsistent? Cantor's opinion is clear from his descrip-tion of the inconsistent multiplicities as "absolutely infinite." Though what he meant is not in doubt, Cantor did not define his meaning in characterizing a multiplicity as "absolutely infinite." It is quite easy to supply an appropriate definition. A multiplicity is absolutely  infinite i f f i t is at least as big as some absolute totality of mathema-tical objects such as the totality of a l l ordinals, the totality of a l l  sets, or the totality of a l l things!""* It seems evident from the fact that Cantor identified inconsistent multiplicities with those which are absolutely infinite that i t was his view that a l l and only those multiplicities which are absolutely infinite are paradoxical.^^ Or, as we should prefer to say, a formula (or property) does not determine a set i f f i t is satisfied by (or belongs to) an abso-lute infinity of things. This is in fact provable on the basis of the axioms formulated by Cantor; alternatively, the statement "a class is a; set i f f . i t is of smaller power than the class of a l l elements" can be taken as an axiom of set theory, replacing the ones given by Cantor. Cantor says that, "If we start from the notion of a definite multi-plicity (a system, a totality) of things, i t is necessary, as I discovered, ^We might prefer to say that a formula is satisfied by (or a property belongs to) an absolute infinity of entities i f f there are at least as many things satisfying i t (or having it) as there are ordinal numbers or sets, or entities a l l told. ^One might say that the thesis that only absolutely infinite multi-plicities are inconsistent expresses the conviction that there is no reason to expect paradoxes of set theory of another kind than the known ones—the paradoxes of absolute infinity. 144 to distinguish two kinds of multiplicities..." (van Heijenoort, p. 114). This suggests that the general concept of a multiplicity is to be taken as a primitive idea. But of the two kinds of multiplicities, an incon-sistent multiplicity is explained as "such that the assumption that a l l of its elements 'are together' leads to a contradiction, so that i t is impossible to conceive of the multiplicity as a unity, as 'one finished thing'" (Ibid.), which makes an inconsistent multiplicity a sort of non-entity. Indeed there is a passage in one of Cantor's letters to Jourdain in which he says quite clearly that there really are no such things as inconsistent multiplicities: Only complete (fertig) things can be taken for elements, only sets, but not inconsistent multiplicities, in whose essence it lies that they can never be thought as complete and actually  existing.17 Inconsistent multiplicities are not elements because they do not "actually exist." We may say that, according to Cantor's analysis, the paradoxes result from the supposition that absolutes or totalities as large exist, 18 i.e. totalities which are "too big" or "too comprehensive." Later von Neumann recognized that the paradoxes can be avoided by rejecting the assumption that every set is an element of further sets. Now i f we maintain that not every set is an element, then the totalities which are 'too comprehensive to exist' are those which would have to contain "^7"The Correspondence between Georg Cantor and Philip Jourdain," p. 119. 18 Cantor, of course, did not put i t exactly this way himself; he spoke of the assumption that the elements of a totality are together, rather than of the assumption that the totality exists. 145 a non-element. i f a formula is satisfied by classes which are not ele-ments, then that formula does not determine a class which has as elements exactly those things satisfying i t . Von Neumann's analysis agrees with Cantor's in making size a signifi-cant factor: the classes which are not elements are those which are as large as the totality of a l l elements. The classes which are not elements are not, however, inconsistent multiplicities in Cantor's sense; they are not totalities the supposition of whose existence (or the "being together" of whose elements) leads to contradiction. It is the supposition that a class (of elements) as large as the class of a l l elements is an element that leads to contradiction, not the supposition that such a class exists. The classes which cannot be supposed to exist are not the classes of ele-ments which are not themselves elements, but the classes which would have to contain such classes (non-elements) as well as elements. Using von Neumann's ideas, we really can distinguish two kinds of multiplicities (both of which are "consistent"), classes which are elements 19 (or sets) and classes which are not elements; and Cantor's axioms can be reformulated as axioms of the existence of sets, i.e. axioms stating — The concepts of class and membership are primitive, and the property of being a set is defined: a class is a set i f f there is a class to which i t belongs. I should like to emphasize that classes which are elements and those which are not elements are not things of different "type" or "category" in the sense of being things of alto-gether different natures; moreover, no doctrine that certain state-ments are meaningless is involved. It would, in some respects, be better to use the term "set" instead of the term "class" and speak of sets which are and sets which are not elements, but this would not be convenient or in line with customary usage. 146 20 that certain classes are elements. (1) If the class M is a set, so is UM. (2) If M is a set and M' is a subclass of M, then M' is also a set. (3) If the class M is a set and A is equipollent to M, then A is a set. Cantor also accepted as an axiom the proposition that i f a multi-plicity does not have an aleph as its power, then i t contains a sub-21 multiplicity equivalent to the multiplicity W of a l l ordinals. Now a multiplicity has an aleph as its power i f f i t is equivalent to W(a) for some ex. Therefore, we may formulate Cantor's well-ordering axiom thus: (4) If the class A is not equipollent to any i n i t i a l segment of the ordinals W(cx), then some subclass A' of A is equipollent to the class W of a l l ordinals. Axiom (4) has as a consequence the theorem that the power of any set is an aleph and, hence, the well-ordering theorem for sets. Let us consider the grounds in its favor which may have made i t seem evident to Cantor, keeping in mind that Cantor's principal aim was to establish the comparability of powers. It follows from the comparability theorem for well-ordered sets that any two alephs are comparable. Cantor also proved -»3b(*0 < b < ^ ) , and his argument can be used to prove a much more general theorem. But Cantor did not prove that Va(tfQ < a -»• ^  < a). 20 Cantor stated these axioms in one of his letters to Dedekind, and based his derivations of various paradoxes as well as the com-parability theorem on them. They were, of course, formulated in terms of consistent and inconsistent multiplicities (see van Heijenoort, p. 114). 21 In a letter to Jourdain, Cantor said that he considered this as "einleuchtend," which amounts to taking i t as an axiom. 147 This leaves open the possibility that there is a power a_ greater than r\Q but incomparable with It can be proved that any power smaller than an aleph is an aleph or a finite power, but a power greater than the aleph (V^  might be incomparable with every aleph greater than &a. This is where Cantor's axiom (4) comes in. Cantor no doubt thought i t evident that every power must be comparable with any aleph and, conse-quently, that, i f there be such, incomparable powers must be greater  than every aleph. Cantor would, then, have tried to argue that there cannot be a power greater than every aleph. It is quite possible that in this attempt he discovered the paradoxes. Perhaps he began to wonder what the power and the order type of the "well-ordered set" of a l l ordin-al numbers would be; its power should be an aleph, and yet i t must be greater than every aleph. Moreover, a multiplicity having a power great-er than every aleph would have to have a submultiplicity equivalent to the multiplicity of a l l ordinal numbers. There is another point of interest concerning comparability which I would like to discuss briefly here. In proving that is the smallest infinite power, Cantor proved that every infinite power is comparable with tiQ. As indicated in section 20, his argument needs to be supplement-ed with the axiom of choice. Cantor might have used essentially the same argument with the implicit use of the axiom of choice to prove that an infinite power which is not an aleph must be greater than any aleph. Indeed, this is precisely what G.H. Hardy did in his paper "A Theorem 22 Concerning Infinite Cardinal Numbers." — The Quarterly Journal of Pure and Applied Mathematics, 35 (1904), pp. 87-94. 148 On the basis of his axioms (l)-(4) and the result that W is not a set, Cantor proved that every power of a set is an aleph. His proof is as follows. If A is a class which does not have an aleph as its power, then A is not similar to any i n i t i a l segment of the ordinals W(a). By axiom (4), some subclass A' of A is equipollent to W. W is not a set, so neither is A. This is so because i t is a consequence of axiom (2) that a class containing a subclass which is not a set is not a set. Hence, i f a class does not have an aleph as its power, i t is not a set; therefore, i f a class is a set, i t has an aleph as its power. Later Fritz Hartogs.proved from Zermelo's axiom system, without using the axiom of choice, that no set has a power greater than every 23 aleph. Let M be any non-empty set, _N the set of a l l well-ordered sub-sets of M, and L the set of a l l equivalence classes of members of N with respect to the relation of similarity (isomorphism); also let -< be the relation on L^  such that for any £ and h in L, g < h i f f any element of £ is similar to some segment of any member of h.. The relation •< well-orders L. Every element of N is similar to the segment of L determined by its equivalence class. The supposition that a subset MQ of M is equipollent to L implies a contradiction. For, i f M^  ^  L, then would belong to N , and therefore MQ would be similar to a segment of L. Hence, i t is not the case either that L^ M or that L < M. This means that L is c — either greater in cardinality than M or else incomparable with M. If M is well-ordered, then M < L, while if M is not well-ordered, M and L — c — — — are incomparable. Suppose the power of L, is tf^; i f i t is assumed that 23 •• "Uber das Problem der Wohlordnung," Matematische Annalen, 76 (1915), pp. 438-443. 149 every power is comparable with any aleph, then M < ^  , from which i t follows that M is an aleph. Cantor also stated that i t is evident that the totality of a l l conceivable entities is not a set. Russell discovered this paradox 24 independently in January of 1901. Cantor seemed to have proved that there is no greatest cardinal number, yet i t also seemed to Russell that there must be such a number. In fact the greatest number should be the cardinal number of the class of a l l entities, for there cannot possibly be a larger class. Every other class must be a subclass of the class RU which contains absolutely everything. Thus, the contradiction: P(RU)cU < P(RU) . c When Russell became aware of this situation in early 1901, he wrote that if Cantor's proof that there is no greatest number were valid, "the contra-dictions of infinity would reappear in a sublimated form. But in this one point, the master has been guilty of a very subtle fallacy, which I hope 25 to explain in some future work." Thus Russell minutely examined the proof of Cantor's theorem expecting to find some error. Russell thought that there must be some limitation on Cantor's argument which is not satis-fied in cases like the class of a l l entities, the class of a l l classes, or 26 the class of a l l propositions. Therefore he examined the results of 24 So far as I am aware, Russell was the first to publish this paradox, in his article "Mathematics and the Metaphysicians." Hilbert no doubt knew of i t , but did not mention i t in print, as he did the paradox of the set of a l l alephs. 25 Mysticism and Logic, p. 84. 26 "Thus i t would seem as though Cantor's proof must contain some assumption which is not verified in the case of such classes" (Principles, p. 362 ) . 150 applying the method of Cantor's proof to such classes. The result was his discovery of the contradiction which has come to be called "Russell's 27 paradox." We shall see how this comes about for the class RU. Let J_ be any subclass of RU, and let B_ be any subclass of P (RU) equipollent to T. Further, let f be a one-one relation whose value domain is B_ and whose argument domain is T_. It was shown above how to define a subclass K of RU which is not a member of B_. In fact, K = {t: t e RU & t i f (t)}. But now let us consider a particular f_; the corresponding K will be denoted by "r". Our particular _T will be Cls = P(RU) c RU, and _f_ will be the identity relation restricted to Cls, so that f(c) = c, for every c_ in Cls. Of course the value domain of £_ is also Cls = 15. It looks as i f there should not be a K. Yet, as explained above, the class jr defined by the following condition ought to be what we want: c e r i f f c t f(c) (for every c_ in Cls). Taking account of the definition of f_: c e r i f f c t c. r is the class of a l l classes which are not members of themselves. Mysticism and Logic, p. 24. In his Introduction to Mathematical Phil- osophy, Russell says that when he first came upon the contradiction of the greatest number in 1901, "I attempted to discover some flaw in Cantor's proof that there is no greatest cardinal.... Applying this proof to the supposed class of a l l imaginable objects, I was led to a new and simpler contradiction...." It seems, however, that Zermelo was actually the first to discover the paradox of the class of a l l classes which do not belong to themselves. The following information concerning this is provided by Christian Theil: "eine unverdffentlichte Postkartenmitteilung Hilberts an Frege vom 7.November 1903, deren erster Teil wie folgt lautet: 'Sehr geehrter Herr College. Besten Dank fur 2ten Band Ihrer "Grundgesetze, " der mich sehr interessiert. Ihr Beispiel am Schlusse des Buches (S. 253) ist uns hier bekannt' (Am Rand oben steht die Anmerkung dazu: 'Ich glaube vor 3-4 Jahren fand es Dr. Zermelo auf die Mitteilung meiner Beispiel hin'); 'andere noch iiberzeugendere Widerspruche fand ich bereits vor 4-5 Jahren ...'" (Grundlagenkrlse und Grundlagenstreit [Meisenhein am Glan, 1972], p. 94, note 63). 151 Since r is a class, i t ought to be the case that f (r) = r. But _r is so defined that i t is not the f_-correlate of any class. Furthermore, r e r i f f r i r, hence, the contradiction: r e r & r i r. In a letter of July 9, 1904 to P.E.B. Jourdain, Cantor commented on roughly this derivation of Russell's paradox. Corresponding to the axiom of set theory that every set has a power set, Cantor implicitly 28 had the axiom that every set M has a Belegungsmenge G_. Cantor said: Were we now, as Mr. Russell thinks, to let an inconsistent multiplicity (perhaps the totality of a l l transfinite ordinal numbers...) take the place of M, then a totality G_ correspond-ing to i t would not be formed at a l l . The impossibility is based on the fact that an inconsistent multiplicity, because i t cannot be conceived as a whole, therefore not as one thing, is not to be used as_ an element of a multiplicity. 29 It is most interesting that in response to a paper (now lost) on the paradoxes which he received from Jourdain in April of 1905, Russell considered the "assumption that a class can always be an element of an-30 other class." But Russell did not recognize that the contradictions can be avoided by rejecting this assumption. He thought the idea would not avoid his contradiction because the assertion that the paradoxical class is not a member of itself is equivalent to the assertion that i t is a member of itself. But using the comprehension principle that goes along with this idea for avoiding the paradoxes we do not get Russell's contra-— See section 28 below. 29 "The Correspondence between Georg Cantor and Philip Jourdain," p. 119. 30 My guess is that the letter from Cantor quoted above is the indirect cause of Russell's consideration of this assumption. 152 diction. The principle is: For every propositional function of one free variable, there is a class containing exactly those elements which satisfy that function. Thus, in the case of the condition "x^x", we have 3AVx(xeA <-+• 3B(xeB) & x£x) . Hence, AeA 3B(AeB) & A^ A (*), which is not a contradiction, and is indeed true i f and only i f A—the class of a l l elements which are not members of themselves—is not an element of any class, for then the left and right sides of the above formula (*) are both false, and the biconditional is true. 23) Russell on the Theory of  Limitation of Size In The Principles, Russell remarks in discussing the problems con-nected with the greatest number that: The difficulty arises whenever we try to deal with the class of a l l entities absolutely, or with any equally numerous class; but for the difficulty of such a view, one would be tempted to say that the conception of the totality of things, or of the whole universe of entities and existents, is in some way i l l e -gitimate and inherently contrary to logic. But i t is undesir-able to adopt so desperate a measure as long as hope remains of some less heroic solution (Principles, p. 362). Here we have an expression of the idea of the theory of limitation of size: the paradoxes are associated with the totality of a l l entities and "any equally numerous class." Although Russell would eventually adopt a no-class theory, at the time he was writing The Principles holding the absolute totalities "in some way illegitimate" seemed to Russell to be a "desperate" measure f u l l of "difficulty." Why should this have been the case? I do not know for sure why Russell said these things, but I suggest that the following may have been the main reason. Russell must have been very reluctant to accept the theory of limitation of size in 1902 on account of his views about the range of variables and the nature of "formal truth:" It seems to be the very essence of what may be called a formal truth, and of formal reasoning generally, that some assertion is affirmed to hold of every term; and unless the notion of every term is admitted, formal truths are impossible (Ibid., p. 40). Regarding the range of variables, Russell said that "in every proposition of pure mathematics, when fully stated, the variables have an absolutely unrestricted field: any conceivable entity may be sub-stituted for any one of our variables without impairing the truth of our proposition" (Principles, p. 7). "The variable is, from the formal standpoint, the characteristic notion of Mathematics. Moreover i t is the method of stating general theorems..." (Ibid., p, 90). "By making our 3C always an unrestricted variable, we can speak of the variable, which is conceptually identical in Logic, Arithmetic, Geometry, and a l l other formal subjects. The terms dealt with are always a l l terms; only the complex concepts that occur distinguish the various branches of Mathematics" (Ibid., p. 91). In a rejected portion of the May 1901 draft of The Principles, Russell states that "logical constants are classes or relations whose 31 extension includes everything." In this manuscript, he gives the following characterization of pure mathematics: Thus pure mathematics must contain no indefinables except logical constants, and consequently no premisses, or indemon-31 " : Russell Archives MS. 230.03400, p. 1. 154 strable propositions, but such as are concerned exclusively with logical constants and variables whose possible values form a class which is a logical constant. It is precisely this which distinguishes pure from applied mathematics. In applied mathematics, results which have been proved by pure mathematics for a whole class of entities are applied to one or more particular members of a class (Ibid., p. 5). Thus Russell was unwilling to conclude that there are no "true propositions concerning a l l objects or a l l propositions" (Principles, p. 368) because, on the view he was inclined to hold, this seems to amount to saying that there are no formal truths, no propositions of pure mathe-32 matics. Russell did however consider one way of dealing with this prob-lem by using the notion of any in place of a l l : A natural suggestion for escaping from the contradictions would be to demur to the notion of a l l terms or of a l l classes. It might be urged that no such sum-total is con-ceivable; and i f a l l indicates a whole, our escape from the contradictions requires us to admit this.... Thus the cor-rect statement of formal truths requires the notion of any term or every term, but not the collective notion of a l l terms (Principles, p. 105). The term "theory of limitation of size" first occurs in Russell's article "On Some Difficulties in the Theory of Transfinite Numbers and Order Types," which was published in 1906, but was finished by November of 1905. In this paper Russell said that the theory is the one "naturally suggested" by the paradoxes, especially the Burali-Forti paradox and the paradoxes connected with the class of a l l entities. He even said that, 33 "This theory has, at first sight, a great simplicity and plausibility." In his work on the paradoxes, Russell gave a special significance Cp. Essays in Analysis, p. 205, and Logic and Knowledge, pp. 71-73. 'proceedings of the London Mathematical Society, 4 (1906), p. 43. 155 to the paradox of the class of a l l classes which do not belong to them-selves. Russell says that not until 1905 did he "work seriously on Burali-34 Forti's contradiction." He had felt " i t quite impossible to know what one ought to say about Burali-Forti 1s contradiction until the other is 35 settled." Thus, the ways of avoiding the paradoxes Russell tried were such as were most directly suggested by a particular paradox. When Russell finally turned his attention to the Burali-Forti para-dox and the paradoxes in general, he succeeded in showing that Burali-Forti' s paradox is a purely logical one, in the sense of not being essenti-ally connected with numbers. For this paradox is only a special case of a certain general form, of which a l l known "logical" paradoxes are instances. We have seen that for every set of ordinals there is an ordinal not belong-ing to that set. Now let us consider any propositional function j>_ and descriptive function f_ such that *) Vu3z[Vx(xeu •+ (fix) & z = fu <j>z & zeu]. For any such and f_, the supposition that {x:<j>x} exists and belongs to the domain of f_ leads to a contradiction. Russell drew the conclusion that for and f_ satisfying (*), either (1) does not determine a class, or (2) j>_ determines a class, but this class does not belong to the argu-ment domain of f_. But for Russell's paradox, which results when "x^x" and "u" are put respectively for "cjix" and "fu" in (*) together with the supposition that (x: x^ x} exists, only (1) is a possible conclusion (since fu = u). "Bertrand Russell on his Paradox...," p. 107. Unpublished letter to Jourdain (June 4, 1904). 156 Russell thought that the anslysis of the paradoxes he gave in his paper "On Some Difficulties in the Theory of Transfinite Numbers and Order Types" pointed to the conclusion that the contradictions result from the fact that, according to current logical assumptions, there are what we may call self-reproductive processes and classes. That is, there are some properties such that, given any class of terms a l l having such a property, we can always define a new term also having the property in question. Hence we can never collect a l l the terms having the said property into a whole; because, whenever we hope we have them a l l , the col-lection which we have immediately proceeds to generate a new term also having the said property (p. 36). Thus, there are properties such that no class of things having that property contains a l l things of that property. On the basis of von Neumann's analysis of the paradoxes, we may say that the reason that such a property does not determine a class is that the class of a l l classes which are elements and have that property will be a class of that property but not an element of classes. Russell showed that there is a series ordinally similar to the series of ordinals each of whose terms is a set which is not a member of itself. This means that there are at least as many sets not members of themselves as there are ordinals. Let x be a set of sets which do not belong to themselves. The set x. does not belong to itself, nor does xu{x} belong to itself. Likewise, the union of a set of sets whose members do not belong to themselves is a set of sets not belonging to themselves, and i t does not belong to itself. A series with the required properties is 157 defined thus: G 0 G a+1 G u{G }, a a ' OG a<X for limit numbers X. This series is well-ordered by the proper subset relation c, so that we have: G c GQ i f f a < 8 . Every term G of the series G is a set of sets a $ J a — 36 not members of themselves, and no term of G_ belongs to itself. 37 Using Russell's own method of definition, we can derive a contra-diction. Let _T be the intersection of a l l classes K such that (1) xeK, (2) XeK + Xu{X}eK, (3) i f X^ K and X is an element, then UXeK. If T is an element, then UT will belong to T, and UT I OT. If UT belongs to T, then so does UTu{UT}. But now we have a contradiction, for i t follows from this that UT e UT. The main point in this is that Russell shows that the contradiction concerning the class of a l l classes not members of themselves is a paradox of absolute infinity, for an absolute infinity of classes satisfy the propositional function "x^x". But he did much more. For example, he showed how to generalize the method of obtaining a contradiction given in section 22 above. In fact, a contradiction can be derived from the supposition of a class u which contains a subclass equipollent to the universe RU of a l l entities. Let us suppose that RU is equipollent to a subclass u" of u, i.e. Russell does not give exactly this argument. See Ibid.; first footnote to p. 36. Russell does not give exactly this argument either. that _u is a class at least as big as the universe of a l l entities. The class P(u) of a l l subclasses of u is a subclass of RU. Therefore, P(u) is equipollent to some subclass u" of u1 , and hence of _u. Since 11 is also equipollent to a subclass of P(u), by the Schroder-Bernstein theorem, P(u) is equipollent to _u. Furthermore, i t follows that there is a one-one relation f_ whose domain is ?(u) and whose converse domain is u. The class K = {x: xeu & x^f(x)} is a subclass of u, and so K = f(b), for some b_ in u. In the proof of Cantor's theorem, this is only a supposition which is proved false. Here, i t follows from previous suppositions. Since b_ is an element of 11 , b_ is either a member of K or i t is not. If b_ is in K, then b^f (b) = K. If b is not in K, then b is in u but not in f(b) = K, so beK. Thus we have a contradiction: beK i f f bj^ K. Some examples of absolutely infinite totalities pointed out by Russell are, classes containing a l l entities not having some property belonging only to individuals, e.g. the totality of a l l non-men, and, for any non-empty class A, the class of a l l classes equivalent to A. It is particularly evident that the cardinal number 1 (as Frege and Russell define it) is equipollent to the class of a l l entities. More-over, for any non-empty A, U{X: X is equipollent to A} = RU. Indeed, we can derive a contradiction from the supposition that the Russell cardinal of any non-empty set exists. If there were a class of a l l classes equipollent to the class M, then there nevertheless should be a class N of greater power. There would also be for each X in N a class P of a l l pairs <X,Y) with Y in M, as well as a set P_ of a l l the sets P with X in N. For every X in 159 N, is equipollent to M; therefore, P is a subclass of the class of a l l classes which are equivalent to M, which means that P_ has at most the cardinality of this class. Yet, P_ is equipollent to N. Thus we have 39 a contradiction. On the basis of his analysis of the paradoxes in "On Some Difficul-ties..." (especially the Burali-Forti paradox), Russell formulated the following hypothesis: If (j) is a property which does not determine a class, there is a (definable) series of terms having <J> which is similar to the series of a l l ordinals. He also says that the theory of limitation of size naturally becomes particularized into the theory that a proper class must always be capable of being arranged in a well-order-ed series ordinally similar to a segment of the series of ordin-als in order of magnitude.40 Thus, we get the principle: The property ty determines a class i f f there  are not as many things having ty as_ there are ordinal numbers. In particu-lar, there would be no class of a l l things, or any class which would have to be at least as big as the class of a l l things, for there are at least as many things as there are ordinals. The only reason given by Russell for not accepting the theory of limitation of size is that he was not able to find satisfactory axioms concerning what sets and numbers there are. Russell thought that i f a class is equipollent to a segment of the ordinals, then i t is a consistent class. "But this proposition is of l i t t l e use, until we know how far the Heinz Bachmann, Transfinite Zahlen, p. 113. 40 "On Some Difficulties in the Theory of Transfinite Numbers and Order Types," p. 43. 160 series of ordinals goes." A great difficulty of this theory is that i t does not t e l l us how far up the series of ordinals i t is legitimate to go. It might happen that co was already illegitimate: in that case a l l proper classes would be finite. For, in that case, a series ordinally similar to a segment of the series of ordinals would necessarily be a finite series. ... We need further axioms before we can t e l l where the series begins to be illegitimate (Ibid., p. 44). Russell did, however, clearly recognize which ordinals would have to be obtained by satisfactory axioms: It is no doubt intended by those who advocate this theory that a l l ordinals should be admitted which can be defined... without introducing the notion of the whole series of ordin-als. Thus they would admit a l l Cantor's ordinals, and they would only avoid admitting the maximum ordinal (Ibid.). So Russell's whole objection to the theory of limitation of size comes to just this: But i t is not easy to see how to state such a limitation precisely: at least, I have not succeeded in doing so (Ibid.). 24) Paradoxes of the Ultrafinite: Hessenberg and Zermelo Unlike Russell, Hessenberg and Zermelo did not consider the paradoxes to be a l l of one kind. They distinguished between paradoxes of finite definability and paradoxes of the ultrafinite ( = paradoxes of absolute infinity). In "Grundbegriff der Mengenlehre," Hessenberg explained his use of the term "ultrafinite" as follows: Von den paradoxen Mengen, die wir hier untersuchen, lasst sich leicht zeigen, dass sie von grosserer Machtigkeit als jedes Alef sind. ... Wir wollen diese Art Mengen and die ihnen eigenen Widerspruche als ultrafinit bezeichnen, hier-"On Some Difficulties in the Theory of Transfinite Numbers and Order Types," p. 36. 161 mit das Wort transfinlt auf die die Lehre vom Transfiniten anwendbar ist (pp. 627-628). He also remarked that: Die Menge aller Ordnungszahlen ist nur ein typisches Beispiel paradoxer Mengen. Aus jeder zu ihr 'ahnlichen Menge muss ein analoger Widerspruch entspringen. Und nicht nur der Ordnung-stypus dieser Mengen, sondern auch ihre Machtigkeit besitzt die widerspruchsvolle Eigenschaft, letze ihrer Art zu sein: Die Machtigkeit von W ware grosser als jedes Alef, obwohl sie selbst ein Alef sein m'usste. Diese Tatsache mag die Bezeich-nung 'ultrafinit' rechtfertigen (Ibid., p. 635). Regarding the "paradoxes of finite definability," Zermelo referred to Peano (1906) as well as to the chapter in "Grundbegriff der Mengenlehre." The paradoxes of the ultrafinite are eliminated by Zermelo's axiomatiza-tion, in particular the axiom of separation, while the paradoxes of finite definability are eliminated by the restriction to "definite" prop-ositional functions.^3 The aim of Zermelo's investigation into the foundation of set theory was, as he explained i t , "to seek out the principles required for estab-lishing the foundations of this mathematical d i s c i p l i n e . - T h e problem was that of avoiding the two sorts of antinomies which had emerged, while retaining the theorems proved by Cantor and Dedekind. The choice of the axioms was made with the characterization of the paradoxes of set theory as paradoxes of the ultrafinite in mind. Thus, Zermelo eliminated the "A New Proof of the Possibility of a Well-Ordering," van Heijenoort, p. 192, note 11. 43 See Zermelo's "Investigations in the Foundations of Set Theory," in van Heijenoort, and also Hessenberg's "Willkurliche Schbpfungen des Verstandes?", Jaresbericht der Deutschen Mathematiker-Vereinigung, 17 (1908), 145-162, and "Kettentheorie und Wohlordnung," Journal fur Mathematik, 135 (1910), 81-133. 44 "Investigations in the Foundations of Set Theory," p. 200. I 162 only defect which Russell found in the theory of limitation of size. He stated axioms which say what sets there are in such a way that i t does not follow that the absolutely infinite (or ultrafinite) 'sets' exist (Ibid., p. 202). The paradoxes show that there are no sets corresponding to certain perfectly legitimate concepts; thus, for example, there is no set deter-mined by the propositional function "x is an ordinal number" (Ibid., p. 195). The problem then, is to say what sets there are. Zermelo's solu-tion of this problem consisted in the formulation of axioms of two kinds. (1) There is an (actual) infinite set of sets. The paradoxes are not associated with the concept of infinity in general, but only with ultra-finite 'sets' like the set of a l l ordinals and the set of a l l sets. No paradox other than those of the "linguistic" category concerns, say, denumerable sets. (2) Zermelo further postulated that there are certain relations R between classes such that i f A is a set which is not 'too big' and R(A,B), then B is a set which is not too big. Of course he didn't put i t quite this way. What he said was that " i f in set theory we confine ourselves to a number of established principles...—principles that enable us to form i n i t i a l sets and to derive new sets from given ones—then a l l such contradictions can be avoided" (Ibid.). Zermelo's axioms may be regarded as specifying (at least part of) the domain of sets which are not big enough to be associated with para-doxes. The possibility of formulating further axioms is left open. In fact, Zermelo later accepted Fraenkel's axiom of replacement and formu-lated the axiom of foundation and even an axiom asserting the existence of inaccessible ordinals (Grenzzahlen). On the basis of his system of axioms Zermelo proved that no_ set  contains a l l sets. This is a consequence of his theorem that every set M has a_ subset which is not an element of M, which is proved as follows. Let MQ be the subset of the set M consisting of a l l elements of M which are not members of themselves. MQ is not an element of itself, for none of its members are members of themselves; but i f MQ belonged to itself i t would contain a set which is a member of itself, which i t does not. Since MQ is not a member of itself, i t cannot belong to M either. If MQ were in M, i t would have to be in itself, because i t is the subset of M containing those sets which do not belong to themselves. It is not the case, as has often been alleged, that Zermelo's sys-tem is not based on an analysis of the paradoxes. For example, Kurt Grelling said that: Die Methode Zermelos, mit den Widerspr'uchen der Mengenlehre fertig zu werden, hat bei aller Eleganz und Einfachheit doch etwas Unbefriedigends an sich. Sie setzt uns zwar in den Stand, die Mengenlehre und die 'ubrige Mathematik aufzubauen, ohne in Widerspruche verwickelt zu werden; aber sie deckt die eigentliche Quelle der Widerspruche nicht auf. In dieser inehr philosophischen Hinsicht sind die beiden anderen Versuche befriedigneder, allerdings auch schwerer verst'andlich. Der eine von ihnen stammt von Bertrand Russell...der andere von den Mathematikern Weyl und Brouwer.45 This criticism is totally unjustified. Zermelo's system is based on a doctrine about the "source" of the contradictions every bit as much as Russell's system is. Indeed Zermelo's view on the "source" of the para-doxes is a possibility considered also by Russell, who thought i t quite Mengenlehre, Math.-Ph. Bibliothek, No 58 (Leibzig, 1924), p. 46. 164 plausible, but rejected i t for the very defect which Zermelo's system removed! 25) Mirimanoff's Solution of  "The Fundamental Problem  of Set Theory" By the fundamental problem of the theory of sets, Mirimanoff meant the problem of finding necessary and sufficient conditions for the exis-tence of sets.^ Reformulated in terms of von Neumann's analysis of the paradoxes, this is the problem of giving necessary and sufficient conditions for a class to be a set. In preparation for his solution of the fundament-al problem for the case of what he called the "ordinary" sets, Mirimanoff presented a most interesting discussion of the paradoxes, including several new ones. Reflection on Russell's paradox seems to have led Mirimanoff to formulate two somewhat similar paradoxes which were not previously known. In order to give the first of these, some preliminary definitions are necessary. For each set A, let A + be the intersection of a l l classes K con-taining A as an element such that whenever X E K , then also U X E K . We may think of the members of the class A + as being the classes of things in-volved to various degrees in A. The elements of A are involved in A to the first degree, the elements of U A to the second degree, the elements of UL)A to the third degree, and so on. Now we may define the class of a l l things involved in A as the class A* = L ) A + . ^ 7 Thus, A * contains a l l — "Les antinomies de Russell et de Burali-Forti et le probleme fondamental de la theorie des ensembles," p. 38. ^1 use the word "involved" here to call to mind a form of Russell's vicious circle principle. 165 members of A, a l l members of members of A, and, in general, a l l members of members of...A. A* contains a l l terms of any element chain descending 48 from A, or, as we shall say for brevity, any A-descents. The sets A and 15 are M-isomorphic, in symbols A==B, i f f there exists a one-one correspondence f_ between A* and B* such that for any elements 49 a_ and a* of A*: aea' -«-»• f(a)ef(a'). Thus, for M-isomorphic sets A and B_, there is a one-one correspondence between A and B_ such that correspond-ing elements are either both indecomposable (i.e. memberless) or M-isomorphic sets. Instead of sets which differ from each of their elements, Mirimanoff considers sets which are not M-isomorphic to any of their elements; and, instead of sets which belong to themselves, he considers sets which are M-isomorphic to at least one of their elements. We may note that i f a set belonged to itself, i t would contain a set M-isomorphic to itself. But i t does not follow from the hypothesis that a set contains an element M-isomorphic to itself that i t belongs to itself. The class R* of sets which are not M-isomorphic to any of their elements is a subclass of the class R of a l l sets which do not belong to themselves. A set of sets which do not contain elements M-isomorphic to themselves does not contain an element M-isomorphic to itself. This is so for the following reason. Suppose E'eE & E'=E. E 1 must contain an element E" such that f(E") = E'eE, 48 An A-descent is any element chain ...x E . . . E X . E A descending from — n 1 A, and its links x may or may not belong to A. Descending element n — chains are either finite or of type *to. 49 Mirimanoff does not define the relation - as I do here, but only gives a rough indication of the relation he has in mind. 166 and E"=f(E") = E'. Thus, i f for every set X in E, X is not M-isomorphic to any element of X, then E is not M-isomorphic to any element of E_. By this argument, R' is not M-isomorphic to any of its elements, but then i t must belong to itself, and thus contain an element M-isomorphic to itself. Mirimanoff also formulates another paradox similar to Russell's. In this paradox, sets which do not generate infinite descending element chains take the place of sets which do not belong to themselves. If a set belonged to itself, i t would also generate an infinite descent, and, likewise, i f a set belonged to a member of a member of itself, or, more generally, to a member of a member of a member of...itself, i t would generate an infinite descent. But i t does not follow from the hypothesis that a set generates an infinite descent that i t belongs to itself, and so on. Let V be the class of a l l sets which do not generate infinite des-cending element chains. A class of sets which do not generate infinite descents also does not generate an infinite descent; a class generates an infinite descending element chain i f f i t contains an element which does. Therefore V does not generate an infinite descent. But, then V must belong to itself; hence, V generates the infinite descent: ...VeV eV. Mirimanoff called the sets which do not generate infinite descents ordinary sets. Extraordinary sets are, then, those sets which generate at least one infinite descent. Every descending element chain generated by an ordinary set ends after a finite number of links either with the 167 empty set or an individual (an "indecomposable element"). If A* contains no indecomposable elements, a l l descending element chains generated by A are infinite. A kernel of the ordinary set A is any end link of a descending element chain generated by A. A kernel of an ordinary set may or may not belong to i t . Every set which contains an element M-isomorphic to i t is extra-ordinary. For suppose A is M-isomorphic to its element A', then A' will contain an element A"=A', and so on; hence, there will be an infinite descending element chain of M-isomorphic links: ...eA"eA'eA. Thus, the class .V of a l l ordinary sets is a subclass of R'. It does not follow, however, that every extraordinary set contains an element M-isomorphic to i t . Now for a couple of paradoxes. Let V' be the class of a l l ordin-ary sets whose kernels belong to the set _N of indecomposable elements (including the " f i c t i f " empty set e_, which could of course, be taken as a particular individual). V' is a subclass of V and is identical to V, if N is the set of a l l indecomposable elements. V (if i t were an element) would be an ordinary set whose kernels a l l belong to N.» but then V would belong to itself, and, hence, be an extraordinary set. The equivalence relation - partitions V into pairwise disjoint equivalence classes. Each of these classes is a set,"*1 therefore there is a class of them a l l . If a choice class C from these sets were a set, then C_ would be an ordinary ^ 1 will actually present these "paradoxes" as proofs that certain classes are not elements, i.e. not sets. This is of course a departure from Mirimanoff's presentation. ^This follows from the fact that i f A and B_ are ordinary sets- and A=B, then A and B_ have the same rank; see below for the definition of the rank of an ordinary set. 168 set belonging to V'; consequently, C_ would be M-isomorphic to one of its elements and hence extraordinary. As has already been stated in an earlier chapter, Mirimanoff con-ceived the ordinals as order-types of well-ordered sets and thought of the "von Neumann ordinals" (S-sets) as sets in one-one correspondence with the order-types of well-ordered sets (ordinals). Thus, the class W of a l l ordinals is equipollent to the class W' of a l l S-sets. Since that W', _V', _V, R' and R are not sets follows directly from the Burali-Forti paradox and the axioms: (1) a subclass of a set is a set (axiom of subsets), and (2) i f a class is not an element, neither is any class 52 equivalent to i t . From these principles i t follows that a class con-taining a subclass equivalent to a class which is not a set is itself not a set. W' is equivalent to W and W'SV'^ VfeR'tR. From the Burali-Forti paradox we conclude that W is not a set, and, therefore, neither are W' , V, V, R', and R. All the paradoxes considered by Mirimanoff are paradoxes of absol-ute infinity. Mirimanoff concludes from his analysis of the paradoxes that in many cases there are larger and larger sets of things of a certain kind, and no set of a l l things of that kind. With von Neumann's distinc-tion between classes which are and classes which are not elements, we would say that in such cases there is a class which is not an element of a l l sets Mirimanoff, of course, formulates the results and principles in terms of the "non-existence of sets." For example, he would say that W, W', etc. "do not exist" and formulate axiom (2) thus: i f a set does not exist, neither does any set equivalent to i t . Of course, the principles (1) and (2) in this form (which can be made unobjectional) are already in Cantor. 169 the kind in question. For every set A of things of such a kind, there are things of that kind which do not belong to A: "on est bien en presence 53 d'une extension indefinie qui ne comporte pas d'arret ou borne." Mirimanoff attempts to state a precise solution of the fundamental problem of set theory for the case of ordinary sets based on his analysis of the paradoxes and three postulates concerning the existence of ordinary sets: (1) the power set of an ordinary set exists; (2) the union of a set of ordinary sets exists; (3) i f a set A exists, then so does any set of ordinary sets which is equivalent to A. He also assumes the axiom of extensionality for ordinary sets, and that the class N of kernels of the ordinary sets under consideration is a set. It follows from postualte (3) that for every ordinal a_, the corres-ponding S-set = a exists. Each a is the set of a l l S-sets of lower s s rank than _a, where the rank of an S-set is its order type—cx is the rank of a . Hence, for any ordinal cx, the set a is equivalent to the set s s W(a). The sets S_ having a certain property have a Cantorian bound i f f there exists an ordinal number greater than the ranks of a l l those sets. Mirimanoff's solution of the fundamental problem for the special case of S-sets is then the following: (Cl) The class A of S-sets is a set 54 i f f A has a Cantorian bound. If the class A of S-sets does not have a Cantorian bound (C-bound), 53 "Les antinomies de Russell et de Burali-Forti et le probleme fondemental de la theorie des ensembles," p. 48. 54 I have altered this and the following results in accordance with von Neumann's analysis of the paradoxes. 170 the supposition that A is a set implies that W' is a set, for the union of such a class A is W' . But i t has already been shown that the class W' is not a set. If the class A does have a C-bound cx, then A is a set because A£cxs. This completes the demonstration of (CI). It follows at once that (C2) a class A of ordinals is a set i f f A has a C-bound. If A is a class equipollent to a class M of ordinal numbers, which does not have a C-bound, then A is not a set. Moreover, any class containing a subclass A 1 equivalent to a class of ordinals not having a C-bound is not a set. These results are, however, not quite what is needed in Mirimanoffs argument that I give in the next para-graph. What he needed is the following: If f is a function whose values are ordinals, then, i f f[A] does not have a C-bound, A is not a set. For the solution of the fundamental problem for a l l ordinary sets, the concept of rank is defined generally for these sets. The rank of a kernel is 0; the rank of an ordinary set is the least ordinal number greater than the rank of any of its elements. I shall denote the rank of the ordinary set A by "r ( A ) " . Mirimanoff proves that every ordinary set has a rank. If every member of the ordinary set A has a rank, A also has a rank. For, i f A is a set, T [A] has a C-bound, and therefore a least C-bound. The least C-bound of is the rank of A . There-fore, i f an ordinary set does not have a rank, at least one of its ele-ments does not have a rank. Moreover, a rankless ordinary set would have to generate a descent of rankless links ending in a rankless kernel, but every kernel has rank 0. Consequently, every ordinary set has a rank. 171 The reasoning used here could be used to establish the general principle of induction for ordinary sets: If a l l indecomposable elements have the property cp and i f whenever every element of the ordinary set A has <}> A also has <j>, then every element of the class V* of a l l ordinary sets and indecomposable elements has cp. Of course, a f u l l justification of Mirimanoffs definition of rank would require the proof of the existence and uniqueness of _r. As a matter of fact, i t is possible to establish a general principle justifying inductive definitions of functions whose domain is the class V*. Mirimanoffs solution of the fundamental problem for ordinary sets is the following criterion: (C3) The class A of ordinary sets is a set i f f r[AJ has a Cantorian bound. In order to establish this criterion, Mirimanoff proves that for every ordinal number cx, there is a set 0^  of a l l ordinary sets of rank a. The proof is by transfinite induction. Suppose that for every cx < 8, the set 0„ exists. 0o is the set of a l l sets containing at least a set cx p of rank a for a l l cx < 8. The class of a l l the 0 for cx < 8 is a set — — a (by postulate 3), and U 0 , the set of a l l ordinary sets of rank < 8, ct<B " is a set (by postulate 2). Thus, sets of rank 8 must be subsets of U 0a, and consequently Og is a set because i t is a subset of the power cx<8 set of the set U 0 . Indeed, i f for some ordinal cx, there were no set cx<B a 0 , then the same would hold for some a' < a, and so on. It would follow cx that there is not a set 0 n = N, contrary to the postulate that N is a set. If A is a class of ordinary sets whose ranks do not have a C-bound, r[k] is not a set; therefore, A is not a set. On the other hand, if the ranks of the elements of the class A have a C-bound 6 , then A c. U 0 , hence, A is a set. This establishes the criterion ct<8 a (C3). When the ordinals are understood to be the von Neumann ordinals, i t is the case that A is a set i f f sup(r[AJ) < U** i f f Ur[A] 4 U**. 26) Von Neumann's Axiom of  Limitation of Size When the assumption that every class is an element of further classes is rejected, the question of an axiom stating necessary and sufficient conditions for a class to be an element arises. According to the limitation of size theory, only absolutely infinite classes are not elements (i.e. not sets, in the technical sense of "set"). We have just seen that Mirimanoff proved a criterion for being a set on the basis of certain axioms. But this criterion could hardly be taken as an axiom of set theory, using the ordinal numbers as i t does. If an axiom of limitation of size is to be satisfactorily formulated, then the class of a l l elements, not the class of a l l ordinals (which are elements), must be taken as the typical absolutely infinite class. Thus, von Neumann formulated the following axiom of limitation of size, i.e. axiomatic criterion of elementhood based on the limitation of size analy-sis of the paradoxes: 3A(xeA) ~«3f (f [x] = U), 173 i.e. the class x. is an element (a set) i f f there is not a function f map-ping x onto the class of a l l elements IJ = {y: 3A(yeA)}. We shall see that this axiom has the consequence that a class is not an element i f and only i f i t is equipollent to U. Von Neumann gave a simple argument showing that the class R of a l l sets which are not members of themselves is at least as big as U, i.e. that i t is absolutely infinite. This is established by showing that there exists a one-one relation f_ whose value domain is R', a sub-class of R, and whose argument domain is U. If x 4 y, then the set {{x,y}} is not an element of itself. Furthermore, for any element x, {x} 4 0. Now, the functional relation f defined by the equation f(x) = {{0,{x}}}, has the desired properties, f is one-one, i.e. f(x) = f(y) -> x=y, because UJf (x) = {x}, and the value of f for each set JC as argument is a set not a member of itself. Another example is provided by the Russell cardinals. Russell himself noted already in his Principles that, as he defines the cardinals, every cardinal of a non-empty class is as large as the class of a l l enti-ties. Let us consider the simplest case of the Russell cardinals of infinite sets A. For each infinite set A, there exists a one-one function f whose value for each element x is a set equivalent to A. We may define such an f_ as follows: (1) If btA, then f(b) = Au{b}. (2) If aeA, then f(a) = A - {a}. For every element x, f(x)eA, and f is obviously one-one. The function f maps the universe of elements IJ one-one onto a proper subclass of A: 174 f[u] = {z: 3y[(yeA & z=A - {y} v (y&V & z=Au{y})]} c A. It is of course an immediate consequence of the axiom of limitation of size that the class of a l l elements U_ is not a set, since a class is not a set i f some function maps i t onto U. ^  Furthermore, U* = P(U), the class of a l l sets, is not itself a set, for, if U* were a set, then UU* = U would also be a set, but i t is not. The principle of limitation of size is logically a very strong axiom; among its consequences are the axiom of subsets, the axiom of re-placement, the well-ordering theorem, and, hence, the axiom of choice. Von Neumann proved, without the axiom of choice, that there is a class W which well-orders U. Moreover, U,W is similar to U**,(U**/U*)£. Proof. U** is a class which is not a set. Hence, there is a function f_ such that every element x. is a value of f_ for at least one yeU**. Let H be the class of a l l ordinals y_ such that x = f (y). H x x is a non-empty set of ordinals; therefore, H^  has a first element u. Thus, for every element x there is one and only one ordinal u, the least ordinal of the class H« . Let 0 be the class of these u. x ~ The function f_ with domain restricted to 0 is one-one. Let W be the class of a l l pairs <f(a),f(6)> such that ct,6e0. and <ct, g> e(0/U*)E, then U,W is similar to 0,(0/U*)E. Since U,W is well ordered i t has an ordinal number P. In fact P_ = U**, for U is not a set and U,W is similar to P,(P/U*)Z, so P_ is not-a set. U** is the only ordinal which is not a set. ^.It may also be noted that i f U were a set, then i t would follow from the axiom of subsets (a subclass of a set is a set) that {x:x^x} is also a set, for {x:xj£x & xeU} G U. 175 Von Neumann can also prove that a l l classes which are not sets have the same power = the greatest cardinal number. The domain of a l l cardinals which are sets is not a set, and its ordinal number = its cardinal number is U** = the greatest cardinal = the greatest ordinal. A class H is not a set i f f H is equivalent to U**. Since U is not a set, U is equivalent to U**. Hence, a domain E which is not a set is equivalent to the universe of elements U. Thus, i t has been established that a_ class H JLS_ not a_ set i f f H equivalent to IJ. Further, i t is the case that a class H ±s_ a set i f f E jls of smaller power than U, i.e. H is a set i f f H < U**. There is a largest class TJ_ and a greatest number, but the paradox of the greatest number does not arise, because IJ contains only those of its subclasses which are sets. If we apply Cantor's argument to TJ_, we do not conclude that the diagonal class belongs to U* = P(U), because it may not be an element. Indeed, i f the diagonal class is equipollent to U, then i t is not an element and, therefore, does not belong to U* in particular. It should be noted, however, that i f the class of a l l sub-classes of the class A exists, then i t is of greater power than A, but i f A is not an element, there is only the class of a l l subsets of A, and this is equipollent to A. CHAPTER VI THE AXIOMS OF SET THEORY 27) The Extensional Concept of  Sets In this section I give some explanations of the pure extensional concept of sets and contrast i t with the view of classes underlying Russell's skepticism concerning the axiom of choice. 1 The latter view had also been vigorously advocated by Frege. I w i l l explain and c r i t i c -ize some of their reasons for preferring this concept. The most s i g n i f i -cant difference between the two concepts, so far as the theory of the transfinite i s concerned, i s that the axiom of choice seems evident on 2 the pure extensional view, but not on the other. The explanations of both views concern the naive concepts, i.e. the concepts as they might i have been explained without regard to the paradoxes. According to the extensional view of classes, a class (set) i s a 3 plurality, t o t a l i t y , or multitude of things, and i s determined by and consists of those things. Thus, Cantor spoke of "einer Menge M (die aus wohlunterschiedenen, be g r i f f l i c h getrennten Elementen . . . besteht und insofern bestimmt und abgegrenzt ist) . . . " (GA, p. 397). As von Neumann and Morgenstern put i t : "'"See section 31 below for Russell's skepticism. 2 Actually i t i s the equivalent multiplicative axiom which seems particularly evident. 3 See Kurt Godel, "Russell's Mathematical Logic," in The Philosophy  of Bertrand Russell, ed. P.A. Schilpp, pp. 137, 140. -176-177 A set is an arbitrary collection of objects, absolutely no restriction being placed on the nature and number of these objects, the elements of the set in question. The elements constitute and determine the set as such, without any order-ing or relationship of any kind between them.^ By saying that a set is determined by its elements i t is meant that for certain things there is one and only one set which contains exactly those things as elements, i.e. to which exactly those things have the membership relation. Whenever there are certain things^ whenever many different things exist, so does the set or plurality of them a l l ; a set exists when its elements exist. Let us note the circularity of this explanation. Phrases like "certain things" or "any things" (when used as they were above) really mean about the same as "a set of things" or "a plurality or multitude of things." When i t is said that a set is constituted by its elements, one in-tends to emphasize that a set is a "mere" aggregate, so that its existence does not depend upon anything more than the existence of its elements; in particular, the existence of a set does not depend in any way upon proper-ties common and peculiar to its elements, ~* formulas satisfied exactly by the elements, or acts or operations of the mind—perhaps, thinking them together or regarding them as one. Furthermore, no connection between the elements is relevant to their constituting a set—no sort of glue, rope, rake, force, or "synthesis" (physical, mental, or conceptual) is necessary for the existence of a mere set or plurality of things. 4 John von Neumann and Oskar Morgenstern, Theory of Games and  Economic Behavior (Princeton, 1944), p. 61. "*Cp. Georg Kreisel, "Mathematical Logic" in Lectures on Modern  Mathematics, ed. T.L. Saaty (New York, 1965), III, 100. 178 The explanations just given have of course been formulated from the native point of view, from which i t would seem that when certain things exist, they always do determine a set constituted by those things.^ Thus, according to the naive pure extensional view of sets, i t would seem that the existence of the elements i s sufficient for the existence of the. set: Whenever many different things exist so does the set of them a l l . The antinomies show that this i s not the case. This really does seem remarkable, for given the existence of certain things, the existence of the "mere aggregate" of them might seem almost t r i v i a l . Thus we have what Mirimanoff called "the fundamental problem of set theory": One believed, and i t seemed evident, that the existence of individ-uals must necessarily entail that of their set; but Burali-Forti and Russell have shown . . that a set of individuals may not exist, although the individuals exist. Since we cannot reject this new fact, we are obliged to conclude that the proposition which seemed evident to us . . i s true only under certain conditions. Then the problem of the theory of sets: What are the necessary and sufficient conditions for a set of individuals to exist?^ The part of the problem concerning necessity is s t i l l easy—what i s nec-essary is the existence of the "individuals". But, according to von Neumann's analysis of the antinomies, we have to say that the existence of a class depends not only on the existence of i t s elements, but also on their not being classes which are 'too big': Whenever many different things exist, provided none of them are equipollent to the class of a l l elements, then so does the class of them a l l . As Bolzano put i t : "that every arbitrary object A whatever can be united with any other arbitrary objects B_, C_, D_ . . . whatever to form an aggregate, or to speak s t i l l more rigorously, that these objects already form an aggregate, without our intervention (an sich selbst schon) . . ." (Paradoxes, Section 3). 7"Les antinomies de Russell et de Burali-Forti . . . ," p. 38. 179 I would like next to contrast the "pure" extensional concept of set just explained with the concept preferred by Frege and Russell, which, for convenience, I will refer to as the "Fregean doctrine" of classes. According to the Fregean doctrine, neither the general concept of class nor the membership relation is a primitive idea. In a manuscript written in 1905, Russell laid down the "fundamentals" of the Fregean doctrine. There is one primitive idea: "u Kl <j> . = . u is a class determined by <f>," and three primitive propositions: (1) V<j>3u(u Kl i.e. every proposi-tional function (property) determines a class; (2) u Kl <j> & v Kl <j> -»• u=v, i.e. a propositional function determines only one class; (3) 3u(u Kl $ & u Kl ^) «-»- Vx(<j>x +-y ipx), i.e. two functions determine the same class i f and only i f they are formally equivalent. The membership relation and the property of being a class are introduced by the definitions: xeu . = . 3<J)(u Kl (J) & <j>x) Df. Class(u) . = . 3<j>(u Kl •) Df. That is, " J C belongs to u" means "x satisfies some function which deter-mines the class u_," and vu is a class" means "u is a class determined by some propositional function." The motivation for formulating the theory of classes in this way was the conception of a class as something determined by a property, g concept, or propositional function. As Frege himself expressed i t , "classes are determined by the properties that individuals in them are to have." Frege regarded concepts (properties) as more 'primitive' than 8 The Principia Mathematica no-class theory is, so to speak, a no-class theory corresponding to this theory of classes "as entities." 9 Philosophical Writings, p. 104. 180 classes; Russell was also much inclined to this view. Frege said, "The class...is something derived, whereas in the concept—as I understand the word—we have something primitive." 1^ Thus, according to Frege, the class is "derived" from the concept. He said that "the concept is logi-cally prior to its extension."'''1 The following passage seems to give one • of his reasons for holding this: We can only determine a class by giving the properties which an object must have in order to belong to the class. But these properties are the attributes (Merkmale) of a concept.12 But this is not a reason for asserting that a concept is "logically prior" to its extension or that a class is "derived" from a concept. For this certainly does not follow from the way in which we must "determine" (specify) a class. One reason why Russell favored the view that classes are extensions of properties (functions, intensions) over the purely extensional view of classes concerned the definition of infinite classes: "The need of in-cluding infinite classes was one of my reasons for emphasizing functions 13 as opposed to classes in PM." But here again, while we can only specify an infinite set by means of a formula representing a property common and peculiar to the members of the set, the existence of the set depends only upon the existence of the elements. Frege's notes to Jourdain's "The Development of the Theories of Mathematical Logic and the Principles of Mathematics," Quarterly Journal  of Pure and Applied Mathematics, 43 (1912), 251. ^Philosophical Writings, p. 106. 1 2Frege's notes (1912), p. 252. 13 The Autobiography of Bertrand Russell, 1914-1944 (Boston, 1968), p. 324. 181 Thus, in contrast to the view of classes favored by Frege and Russell, a set is not thought of as being determined by a concept, property, or de-fining propositional function. Sets are conceived independently of any considerations of definability as "arbitrary multitudes."^ As Quine puts i t , "there is in the notion of class no presumption that each class is specifiable;"'''"' rather, "Things are viewed as going together into sets in any and every combination."'''^ It is from this concept of set (class) that the axioms formulated in the subsequent sections of this chapter most clearly follow; and, as Gbdel says, that sets be definable in some way or other, or in a particular way "is neither formulated explicitly nor contained implicitly in the accepted axioms of set theory" ("What is Cantor's Continuum Problem," p. 266). Another factor which apparently influenced both Frege and Russell to favor the view that classes are determined by or are dependent upon concepts or properties is the problem of the unity of a class. Thus Frege said: "I characterize as a concept that which has number, and in so doing indicate that the totality...is held together by characteristics."^"7 In My Philosophical Development, Russell said that 14 Godel, "What is Cantor's Continuum Problem?" in Philosophy of  Mathematics: Selected Readings, ed. P. Benacerraf and H. Putnam (Englewood Cliffs, N.J., 1964), p. 265. ^Set Theory and its Logic, p. 2. Ontological Relativity and Other Essays (New York, 1969), p. 117. "*"7Qn the Foundations of Geometry and Formal Theories of Arithmetic, p. 144; cp. Foundations of Arithmetic, p. 61. To avoid possible mis-understanding, I should perhaps say that I am not discussing a l l reasons motivating Frege and Russell's preference for their view of classes as dependent upon concepts, properties, or propositional functions. 182 It seemed to me—as, indeed, i t s t i l l seems—that, although from the point of view of a formal calculus one can regard a relation as a set of ordered couples, i t is the intension alone which gives unity to the set. The same thing applies, of course, also to classes. What gives unity to a class is solely the intension which is common and peculiar to its members (p. 87). In Principia Mathematica, i t is even stated that arguments of some degree . of cogency against the existence of classes "can be elicited from the ancient problem of the One and the Many" (I, 72). The essential point is the following: If there is such an object as a class, i t must be in some sense one object. Yet i t is only of classes that many can be predicated. Hence, i f we admit classes as objects, we must suppose that the same object can be both one and many, which is impossible (Ibid., n.). This argument is invalid, i f for no other reason, because "many" cannot properly be "predicated" of classes or of anything else; we may say "the class A has many elements," but "the class A is many" is either a shorter formulation of this or else nonsense. There is no contradiction whatever in saying that the class A is a single entity having many elements, just as there is no contradiction in saying that the Eiffel Tower is one thing having many parts. Whatever may be cogent arguments against the existence of classes, the argument from the supposed problem of the one and the many is not among them. To the extent that the problem of unity is genuine, i t is solved very easily for classes, but not for properties. We have a criterion of identity and difference for classes (in the extensional sense), but not for properties. But when classes are understood in Frege's manner as being entities determined by concepts, then, by Frege's own admission, 183 the criterion of identity for such classes i s not evident! Thus Frege tried to avoid Russell's contradiction by modifying his view on this . 1 8 point. The "principle of unity" for classes in the extensional view i s the axiom of extensionality. In general, there i s nothing at a l l to the question of unity besides the c r i t e r i a for identity and difference. In the f i r s t appendix to The Principles of Mathematics, Russell said: Although a class i s many and not one, yet there i s identity and diversity among classes, and thus classes can be counted as though each were a genuine unity; and in this sense we can speak of one class and of the classes which are members of a class of classes (p. 516). But nothing more is ever necessary for attributing "genuine unity" to anything. Nothing more is necessary for saying that a class i s a single entity than that we be in possession of a criterion of "identity and diversity." It i s not at a l l necessary that in addition we be able to describe some special sort of "glue" which makes the many into the one. Supposing this necessary leads straight to the Leibnizian doctrine that the only true unities ( = the real beings) are simple (monads). For no satisfactory glue w i l l be found for any type of complex entity, including material bodies and persons. This concludes my discussion of the extensional concept of set. Be-fore going on to the axioms, I would like to point out that axiomatiza-tion designed to avoid the antinomies need only involve a change in our view on what sets there are. It was Ramsey's impression that Zermelo and See Grundgesetze der Arithmetik (1893; rpt. Hildesheim, 1966), I, • v i i ; II, 253. 184 others who were engaged in formulating set theory axiomatically "propose to cease using the word class or aggregate in its ordinary sense, what-19 ever that may be, but to mean by it anything satisfying certain axioms." But this is by no means necessary, even if it was the attitude taken by some; that it was not Zermelo's attitude seems clear from his statement that "we are dealing with a productive science resting ultimately on in-tuition" (van Heijenoort, p. 190). 28) Axioms Implicit in Dedekind  and Cantor E. Noether has said that some of Zermelo's axioms could have been taken over from the first chapter of Dedekind's "Was sind und was sollen die Zahlen." But we ought also to note that a number of axioms are immediately suggested in Cantor's writings. As a matter of fact, most of Zermelo's axioms can be obtained by formulating existence axioms corresponding to special notations introduced by Cantor. In what follows I point out what corresponds to Zermelo's axioms in the writings of Dedekind and Cantor. The axiom of extensionality is stated by Dedekind: "The system S_ is . . . the same as the system T_ . . . when every element of is also 20 an element of T_, and every element of T_ is also an element of S_." 19 "Mathematics, I. Mathematical Logic," The Encyclopaedia Britannica, 13th ed., Supp. Vol. 2 (London, 1926), p. 83. 20 Essays on the Theory of Numbers, p. 45. The axiom of extensional-ity was already stated by Boole: Classes are identical "if all the mem-bers of the one are members of the other" (The Laws of Thought (1854; rpt. New York, n.d.). 185 Cantor introduces the sign "0" expressing the absence (Abwesenheit) of points. "P=0" really only expresses the fact that there is no such set as P_ is supposed to be: "P=0 bedeutet also, dass die Menge P_ keinen einzigen Punkt enthalt, also streng genommen als solche gar nicht vor-handen ist" (GA, p. 146). This corresponds to Zermelo's axiom of the empty set: "There exists a (fictitious)set...0, that contains no ele-ment at a l l . " Corresponding to the axiom of unit sets, we have Dedekind's statement, "For uniformity of expression i t is advantageous to include also the special case where a system S_ consists of a single. . .element a." 2 1 What corresponds to Zermelo's power set axiom in Cantor would be a Belegungsmenge axiom. Cantor explained the Belegungsmenge of the set K[ with the set M as the set of a l l laws correlating with each element of N a unique element of M. Every element of N. is "covered" by one and only one element of M, but different elements of N may be covered by the same element of M. The concept of a Belegung is a primitive idea for Cantor; likewise, the concept of an Abbildung is a primitive idea for Dedekind. Both Cantor and Dedekind introduce the concept of the union of a set of sets. The least common multiple or Vereinigungsmenge (union) of any number of sets ?^Y^t?y •• is the smallest set which contains every 22 element of these sets. 21 Essays on the Theory of Number, p. 45. 22 Peano had formulated an axiom asserting that the intersection of two classes is a class. Russell commented on this axiom as follows: "If class is taken as indefinable, i t is a genuine axiom, which is very necessary to reasoning. But i t might perhaps be somewhat generalized by an axiom concerning the terms satisfying propositions of a given form: e.g. 'the terms having one or more given relations to one or more given terms form a class'"(Principles, p. 30). 186 In effect, Cantor assumed an axiom of infinity. He did not attempt a proof that there exists an infinite set, as Dedekind and others did. Rather, he assumed that the finite cardinals form an infinite totality (GA, p. 293). Moreover, in his first writing introducing the ordinal numbers, Cantor assumed the existence of the least transfinite ordinal number to. Zermelo says in explanation of Cantor's later procedure which had been incorrectly represented by SchSenflies: ...Cantor 1897 defines the numbers of the second number class as the order types that can be associated with well-ordered denumerable sets, and he proves everything else from this definition. Only the existence of denumerable sets, or the type u>_ by which they are defined, is assumed; there is no need of a further axiom (van Heijenoort, p. 196). Cantor remarked on the problem of supposing the "consistency" in his sense of multiplicities of transfinite power in a letter to Dedekind written oh August 28, 1899: We must raise the question of how I know that the well-ordered multiplicities or sequences to which I ascribe the cardinal numbers o K * M ^ 0 ' 1* O ) Q CO^' are also real 'sets' in the sense of the word explained, i.e. 'consistent multiplicities.' Would i t not be conceiv-able that these multiplicities were already 'inconsistent,1 and that the contradiction of the assumption of a 'gather-ing of a l l their elements' had been produced only not not- iceably? My answer to this is that this question is to be extended to the finite multiplicities also and that a close consideration leads to the result: even for finite multi-plicities, a 'proof of their 'consistency' is not to be managed. In other words: The fact of the 'consistency' of finite multiplicities is a simple unprovable truth; i t is 'the axiom of arithmetic' (in the old sense of the word). And likewise the 'consistency' of the multiplicities to which I ascribe the alephs as cardinal numbers is 'the axiom of the extended transfinite arithmetic' (GA, p. 447). 187 Cantor's axiom of transfinite arithmetic is just an axiom of infinity. It is of course only necessary, as Zermelo points out in the quotation on the preceding page, to assert the existence of a denumerably infinite set. The existence of sets of higher power then follows by means of the other axioms. The axiom system we are considering does not make i t pos- • sible to prove the existence of sets of nondenumerable inaccessible cardinal number. Zermelo's axiom of infinity states that there is a set which con-tains infinitely many sets: this set contains the empty set, and, when-ever i t contains the set a_, i t also contains the set {a}. This axiom of infinity replaces the proofs of the existence of infinite sets given by Bolzano, Dedekind, Frege, and Russell, which were based on the false com-23 prehension principle. Yet the formulation of the axiom is in accordance with the idea of Dedekind's proof. Zermelo even went so far as to say that his axiom of infinity "is essentially due to Dedekind" (van Heijenoort, p. 204). By contrast to Zermelo's axiom and other axioms of infinity formu-lated in set theory which assert the existence of an infinite set of sets, Russell's axiom of infinity asserts that there are infinitely many C.J. Keyser had maintained that i t is necessary to set up an axiom of infinity, but his reason was wrong. Furthermore, the axiom he formu-lated was not of the type formulated by either Zermelo or Russell; rather, i t was an axiom of potential infinity. See Russell's criticisms, which will apply with suitable modifications to many another idealistic doct-rine, in "The Axiom of Infinity," pp. 811-812. (Keyser's papers were cited in footnote 37 to chapter II.) 188 24 individuals in the world of concrete things; the set-theoretical axioms, unlike Russell's, do not suppose that there are infinitely many concrete individuals. Whitehead and Russell say of their axiom of infinity that, "It seems plain that there is nothing in logic to necessitate its truth or falsehood, and that i t can only be legitimately believed or 25 disbelieved on empirical grounds" (PM, II, 183). This is certainly not the case with Zermelo's axiom of infinity, or with other similar axioms. Their truth or falsehood must be recognized, i f at a l l , a priori. In the system of Principia Mathematica, i t is possible to prove the existence of a set of any inductive (i.e. finite) cardinal number: For i f the total number of individuals be n, the numbers of n 2 n objects in succeeding types are 2,2 , etc., and these num-bers grow beyond any assigned inductive cardinal. Owing, how-ever, to the fact that we cannot add together an infinite number of classes whose types increase without cardinal limit, we cannot hence show that there is a type in which every induc-tive cardinal exists, though we can show of every inductive cardinal that there is a type in which i t exists (II, 183). It is also the case that the existence of such a set cannot be proved (independently of the axiom of infinity) in Zermelo's system. Yet there is nothing to prevent the supposition of such a set in Zermelo's set theory, as there is in Russell's type theory. In type theory, the sup-position would be declared meaningless. According to one of the forms of the axiom of infinity given in PM, i f A is any inductive (finite) class, there are objects not belong-ing to A (II, 203). This means that the lowest type is a class such that no finite subclass contains a l l its elements. 25 Though Russell held that i t could not be known whether his axiom of infinity is true, he was actually quite inclined to believe that i t is true. For example, he said in his book Introduction to Mathematical  Philosophy that although i t is possible that "the total number of things in the world be finite," this "seems unlikely" (p. 13). 189 Though Russell thought that "there is no conclusive logical reason" for believing his axiom of infinity to be true, he also held that "there is certainly no logical reason against infinite collections, and we are therefore justified, in logic, in investigating the hypothesis that there are such collections" (Intro. Math. Phil., p. 77; cp. p. 141). Thus, in Principia Mathematica, the "axiom" of infinity is treated as an hypothesis (II, 203). 29) The Axioms of Separation  and Replacement Cantor's 1895 paper on transfinite set theory begins with his famous 'definition:' "Unter einer 'Menge' verstehen wir jede Zusammenfassung M von bestimmten wbhlunterschiedenen Objekten m unsrer Anschauung oder unseres Denkens (welche die 'Elemente' von M genannt werden) zu einem Ganzen. In Zeichen driicken wir dies so aus: M = {m}" (GA, p. 282). I would like to lay particular stress on the fact that no reference to laws, formulas, or common properties is made in Cantor's statement: only the elements are mentioned. If the slight idealistic cast of the formulation is removed, we have an exposition of the extensional concept of sets. It would seem that we could say, in accordance with Cantor's explanation, that whenever certain objects exist so does the set of them a l l . It seems natural to say further that in order to specify a set, i t is only necessary to construct a formula satisfied by exactly the members of the set, and, indeed, any formula ought to specify a set having a l l those objects which satisfy the formula as its elements. The paradoxes show that this is not the case. As Zermelo put i t , the paradoxes show 190 that "the customary definition of set i s too wide" (van Heijenoort, p. 195). In virtue of von Neumann's analysis of the paradoxes, only a very slight modification of the principle of the existence of classes is needed. Instead of the naive principle that every propositional function determines a class, we have the principle: For each propositional func-tion <{>, not containing the variables "A" and "B", the universal closure of 3AVx(xeA «->• 3B(xeB) & cf>) is an axiom. The propositional functions to which no classes correspond as their extensions are those which are satisfied by at least one class which i s not an element; but for each such propositional function <j>, con-taining only the free variable "x" and provided i t does not contain "B", the propositional function "3B(xeB) & <f>" determines a class. Zermelo thought of his axiom of separation as "in a sense...a sub-stitute for the general definition of set" given by Cantor (van Heijenoort, p. 202). This axiom states that for every "definite" propositional func-tion whose free variable ranges over the set M, there is a corresponding subset of M which contains as elements a l l those members of M which satisfy the function. The replacement of the naive comprehension principle considered in the previous paragraph concerned classes in general, i.e. those which are and those which are not elements of other classes. By contrast, Zermelo's axiom of separation, as well as most of the other axioms, i s to be understood as an axiom concerning the existence of sets, or as asserting that certain classes are sets. Zermelo defined a definite assertion as one for which "the funda-191 mental relations of the domain, by means of the axioms and the universally valid laws of logic, determine without arbitrariness whether i t holds or not." A propositional function containing a variable whose values are the elements of a class K is definite i f a definite assertion results for 2 6 each member of K. Propositional functions involved in the axiom of separation are restricted to those which are definite, in order to avoid the paradoxes of finite definability. The concept of definiteness was thought unsatisfactory by many. In particular, Russell took a critical view of Zermelo's axiom on its account; he writes in a letter to Jourdain: I have only read Zermelo's article once as yet, and not care-fully, except his new proof of the Schroder-Bernstein, which delighted me. I agree with your criticisms of him entirely. I thought his axiom for avoiding illegitimate classes so vague as to be useless; also since he does not recognize the theory of types, I suggest that his axioms will really not avoid contradictions, i.e. I suspect new contradictions.27 Russell himself could have easily removed the defect from Zermelo's axiom if he had cared to. He needed only to use his concept of an extensional function: Extensional functions may be defined as follows. An elemen- tary proposition is one which contains no propositions as constituents and contains no apparent variables. An elemen- tary function is one whose values are elementary propositions. An extensional function is one whose values result from apply-ing negation, disjunction, and variation any finite number of times in any order to elementary propositions and functions.28 ^"Investigations in the Foundations of Set Theory" in van Heijenoort, p. 201. 27Unpublished correspondence (March 15, 1908). This is the only comment I know of by Russell on Zermelo's way of avoiding the paradoxes. 28"Fundamentals," Russell Archives MS. (1907). 192 The elementary functions of Zermelo's system are those of the form "x£y" and "x=y". The definite propositional functions are the extensional functions resulting from these in the way Russell indicates. Russell's other objection seems to show him becoming a bit dogmatic by 1908. He had no more reason to expect further contradictions in Zermelo's system than he had in his own; both systems avoided a l l known kinds of paradoxes. If Russell thought there might be paradoxes of some unknown type, they might just as well have been of a kind which his own system would not exclude. Perhaps after about six years of hard work on developing a sat-isfactory solution of the paradoxes, he was not so willing to consider alternatives any more. The vagueness introduced into the axiom of separation by the con-cept of definiteness is quite easily eliminated; the concept was not really so vague that what was intended could not be recognized. A precise characterization of a definite propositional function seems to have been first published by Herman Weyl (1910): Unter einer definiten Beziehung soil eine solche verstanden werden, welche auf Grund der beiden Beziehungen = und E durch endlichmalige Anwendung unserer in geeigneter Weise modifizierten Definitions-prinzipien erklart ist. Ist den M irgend eine Menge, a irgend ein Ding, 11 eine definite Zweidingbeziehung, so bilden diejenigen Elemente x von M, welche zu a_ in der Beziehung y stehen, stets eine Menge. An Stelle einer definiten Aussage im Zermelo'schen Sinne t r i t t also hier die definite Beziehung zu einem festen Element a_ (oder auch zu mehreren festen Elementen).29 Zermelo himself did not approve of such explications of the concept of definiteness. In particular, he objected to Fraenkel's presentation — Gesammelte Abhandlungen, Vol. I, 304. He used five "principles of definition." 193 for the following reasons: ...bei der Charakterisierung der zugelassenen Funktionen verfahrt er konstruktiv, was dem Zweck und Wesen der axio- matischen Methode im Grunde widerspricht und ausserdem vom Begriffe der endlichen Anzahl abh'angt, dessen Klarung doch gerade eine Hauptaufgaben der Mengenlehre sein sollte.30 Thus Zermelo preferred an axiomatic explication of the concept of definiteness. This, he said was what he always had in mind, and what is realized in von Neumann's system. Zermelo, however, found von Neumann's system too complicated in its foundations, so he presented his own axioma-tic clarification of the concept of definiteness in general, without special reference to set theory. First Zermelo explained what he meant by a "definite" property or relation. However, his explanation is unsatisfactory because it is not clear whether he is talking about linguistic entities such as sentences and propositional functions (forms) or non-linguistic entities such as propositions, properties, and relations. On the whole i t seems best (at least for the present exposition) to take the "S'atze," "Aussagen," and "Functoren" to which definiteness is attributed as linguistic enti-ties, whatever Zermelo's actual intentions may have been. Let B_ be any class of entities and R a set of relation symbols de-noting basic relations among the elements of B_ as well as variables whose values are relations on B_. Zermelo's "axiomatic" characterization of the definite statements and functions amounts to the following definition, which has the same effect as the "genetic" or "constructive" method but does not employ the concept of finite number. The class P_ of statements 30 .. "Uber den Begriff der Definitheit in der Axiomatik," Fundamenta  Mathematica, 14 (1929), p. 340. 194 and propositional functions definite with respect to R is the smallest class K satisfying the following conditions. (0) K contains a l l atomic formulas formed with relation symbols from R and the appropriate number of variables whose values belong to B, i.e. atomic propositional functions are definite. (1) The negation of an element of K belongs to K. (2) The conjunction or disjunction of any pair of elements of K belongs to K. (3) The universal or existential quantification of any formula in K belongs to K—the quantification may be either with respect to first or second order variables. In his 1930 paper, Zermelo formulated the axiom of separation as follows: "Durch jede Satzfunktion f(x) wird aus jeder Menge m eine Untermenge m^  ausgesondert, welche alle Elemente x umfasst, fur die f(x) wahr ist. Oder: jedem Teil einer Menge entspricht selbst eine Menge, 31 welche alle Elemente dieses Teiles enthSlt." This last statement seems to be the best formulation of what is really meant—every part of a set is a set, and the subsets determined by a propositional function construct-ed out of a finite number of symbols are, in the case of an infinite set, only some among the subsets of the set. In von Neumann's system, the statement: A subclass of a set is a set corresponds to Zermelo's axiom schema of separation. _ "Uber Grenzzahlen und Mengenbereiche," Fundamenta Mathematica, 16 (1930), p. 30. 195 The fact is, however, that the axiom of separation is not strong 32 enough for a f u l l Cantorian set theory. Indeed, the necessary axiom is already suggested in Cantor's correspondence with Dedekind. Though Cantor did not actually formulate exactly this proposition, I shall re-fer to the following statement as "Cantor's axiom:" If f is a one-one function and A is a set, then f[A] is a set. In particular, a class equipollent to a set is a set. By the axiom of replacement, I shall mean the statement that i f A is a set and £ is a function, then g[AJ is a set. The axiom of replacement is a consequence of Cantor's axiom and the multi plicative axiom, for we can prove, using the multiplicative axiom, that for every function f_ there is a one-one function f' with the same values and whose domain is a subclass of the domain of f_. According to Cantor's axiom, i f A is a set, so is f ' [ A J = f[A]. Hence, i f f is a function and A is a set, f[A] is a set. The axiom of separation is an immediate con-sequence of this proposition. Let A' be a subclass of the set A and f_ be the identity function with argument domain restricted to A, then f[A] A' is a set. The axiom of separation is a main point of difference between the theory of limitation of size and what Russell called the "zig-zag theory. — See A. Fraenkel, "Zu den Grundlagen der Cantor-Zermeloschen Mengen-lehre," Mathematische Annalen, 86 (1922); there the axiom of replacement is formulated as follows: "Ist M eine Menge und wird jedes Element von M durch ein 'Ding des Bereiches JJ'... ersetzt, so geht M wiederum in eine Menge uber" (p. 231). 33 Perhaps the most convenient axiom yielding a set theory of Cantorian extent was formulated by von Neumann in his "Uber eine Widerspruchfrei-heitsfrage in der axiomatischen Mengenlehre:" If A is a set and j* is a function and B is a subclass of g [ A J , then B is a set. Axiom schemata of replacement were formulated by Fraenkel and Skolem. 196 Following Russell's usage of 1905, let us say that a propositional func-tion is "irreducible" if i t does not determine a class. According to the zig-zag theory, in contrast to the theory of limitation of size, i t is not the being satisfied by too many things that makes a function i r -reducible, and the negation of a reducible function is also reducible. Thus, the zig-zag theory postulates that every class has a complement and that there is a universal class which contains everything including itself. By the definition of an irreducible function, for any class u and irreducible function f_, either f is not satisfied by some elements of u, or f_ is satisfied by things not belonging to u. The zig-zag theory which Russell had favored at one time held that " i t is just as much the terms not satisfying <{>x as the terms satisfying i t , that make i t irreducible." According to the theory of limitation of size, a function does not determine a class when too many things satisfy i t . Thus, this theory says that a propositional function does not determine a class because of what does satisfy i t . The zig-zag theory, on the other hand-, holds that there are functions <f> such that there are classes u containing a l l things satisfying <(>, but $ is always unsatisfied by some members of such a u. We could say of such a <\> that i t did not determine a class because of what does not satisfy i t , because every class containing everything satis-fying 4> also contains elements not satisfied by <{>. But the existence of such functions <j> is just a point of contention between the theory of limit-ation of size and the zig-zag theory. The theory of limitation of size denies that there are such functions, for i t holds in general that a class which is no bigger than a set is again a set, or any propositional function 197 satisfied by no more things than belong to some set determines a set. According to the theory of limitation of size, an irreducible function is satisfied by some non-u's, for every class _u. 30) Zermelo's Strong Axiom of  Infinity In his paper "Uber Grenzzahlen und Mengenbereiche: Neue Unter-suchungen uber die Grundlagen der Mengenlehre," Zermelo postulated a new axiom of infinity asserting the existence of an absolutely infinite in-creasing sequence of very large ordinals. This section deals with the nature of these ordinals and their relation to sets of sets satisfying the axioms of "Zermelo-Fraenkel" set theory, which Zermelo called "normal domains." A normal domain is a set of sets satisfying the axioms of ex-tensionality, power set, union, replacement and foundation with respect 34 to the membership relation restricted to that domain. A Grenzzahl is the least ordinal greater than a l l ordinals belong-ing to a particular normal domain P_. In fact, i f Va,(Va/U)E is a normal domain, then cx is its Grenzzahl. No Grenzzahl is cofinal with an ordinal smaller than itself, i.e. every Grenzzahl is a regular i n i t i a l number. This is proved by showing that i f the Grenzzahl or characteristic y of the normal domain P_ were cofinal with a lesser ordinal number 3 , then y would have to be an ordinal belonging to P_, which, by definition, — Zermelo actually allowed normal domains to contain individuals (Urelemente), as well as sets. But, for the sake of simplicity, i t is assumed in the rest of this chapter that the variables only have classes as values. On this stipulation, U = U*. Furthermore, the ordinals are always to be understood as von Neumann ordinals, and the notation "W(a)" is short for "Seg(a; U**,(U**/U*)E)." 198 i t is not. If the Grenzzahl p were cofinal with the lesser ordinal 8 , then the i n i t i a l segment W(u) of the ordinals would contain a subsequence of type 8 consisting of numbers a < p, which would not a l l belong to any proper section W(a) of W(u). This would be the case because, i f u is cofinal with 8 < P, then, by definition, p is the limit of an increasing 8-sequence containing terms larger than any ordinal <p. Every ordinal less than p belongs to P_. Hence, 8 would belong to P_, and, for each n < 8 , the number a would also belong to P, since a < p. From the axiom of replacement, i t follows that there is in the normal domain P_ a set containing a l l the cc^ . The set UT also belongs to P_ and is an ordinal a = lim a = p. n<8 n Characteristics of normal domains are not, therefore, cofinal with lesser ordinals, i.e., they are a l l regular i n i t i a l numbers. In fact, characteristics of normal domains are Hausdorff exorbitant numbers, i.e. regular i n i t i a l numbers having limit number indices. An exorbitant num-ber is not the limit of an increasing sequence of smaller type, and when it is greater than co^ , i t is also greater than ^ ^ j * If the normal domain P were of characteristic p = co , ,, then P would contain the ordinal co and — cx+1 — a also the power set of co , whose cardinal number would be > co and there-of a fore >_ p. The ordinal of P0»J ) with respect to any relation well-ordering i t would be ^ p, which contradicts the fact that p is the characteristic of P. Every characteristic of a normal domain is a critical number of the function ty which is defined by transfinite induction as follows: (1) ty(Q) = 0, (2) = the i n i t i a l number of the number class Z ( 2 * * 5 * ) , ( 3 ) for limit numbers a , ty(a) = lim i | > ( 0 . £<ct Proof. If <5 is the characteristic of the normal domain p_ and a < 6, then i K a ) < 6. This assertion is demonstrated by transfinite induction. The induction hypothesis is that for a l l £ < a , ty(K) < 5. If £ is a successor ordinal, ct = £+1 for some il < a. belongs to p_, because normal domains satisfy the power set axiom. Hence, ^ ( a ) < 6. On the other hand, i f ot is a limit ordinal, then the theorem follows because normal domains satisfy the axioms of replacement and union. To each element K_ of the ordinal cx, there corresponds a unique ordinal ty(K) < 5. Since a is a set in p_ and each <K£) is also a set in p_, there is a set P_ in p_ which contains a l l the ty(Z), by the axiom of replacement. The union of P_ is also a set in D, and, since the union of a set of ordinals is itself an ordinal, P is an ordinal = lim < K £ ) = <!>(«)• The ordinal ^ (ct) is therefore < 6. Now, i f 6 were < ty(&) = H m ^ ( o c ) » then there would already be an ct<6 a < 6 such that 6 < i p ( a ) (since ij>(6) is the smallest number greater than a l l ^ ( o t ) ) . But this contradicts the lemma just proved in the last para-graph, so 6 is a critical number of the normal function ty, that i s , ty(6) = 6. (6 cannot be > i|>(6) since ty is an increasing function.) > The smallest transfinite number co is the characteristic of the smallest normal domain. Let M be any normal domain and its subdomain containing everything in M except the links of any descending element chain one of whose links is an infinite set, i.e. 200 M' = M - {XeM: X* contains an i n f i n i t e set}. (See section 25 above for the definition of X*.) is an i n f i n i t e set of f i n i t e sets and is the smallest normal domain. In fact, M' = V . to Zermelo called the smallest (standard) normal domain containing an in f i n i t e set the Cantorian normal domain; the characteristic of this domain i s the smallest nondenumerable Grenzzahl. The existence of Grenzzahlen cannot, of course, be proved from the Zermelo-Fraenkel axioms. Therefore, Zermelo formulated a new axiom or "hypothesis" which, altered in accordance with von Neumann's analysis of the paradoxes, asserts that a sequence of ever more comprehensive models of set theory which are sets exists, which i s isomorphic to the absolutely i n f i n i t e sequence of elements of U**, and, consequently, for every Grenzzahl there i s a greater Grenzzahl. We might say that the universe of elements IJ, which i s not a set, contains sets in which the axioms are already satisfied, and, indeed, an absolutely i n f i n i t e sequence of ever more comprehensive such subuniverses. 31) The Axiom of Choice The f i r s t proof of the well-ordering theorem, other than Cantor's argument given in section 22, was given by Zermelo in 1904. His proof was based on the following axiom of choice: For every set M of non-empty sets, there i s a function f such that for M' belonging to M, f(M'')eM'. Zermelo characterized this axiom as a "logical principle" which "cannot, to be sure, be reduced to a s t i l l simpler one, but 201 which ... i s applied without h e s i t a t i o n everywhere i n mathematical 35 deduction." i-Zermelo's f i r s t proof of the well-ordering theorem (1904) uses only d e f i n i t i o n s and theorems from Cantor's general theory of well-ordered sets and makes no mention of o r d i n a l numbers. Zermelo's second proof (1908) of the well-ordering theorem does not so much as use an ordinary d e f i n i t i o n of well-ordering, but proceeds by the methods of Dedekind's theory of chains. It i s e x p l i c i t l y based on the axiom system which he expounded more f u l l y i n another paper of the same year. The proof which I w i l l expound here was given by von Neumann i n 1929. It features an a p p l i c a t i o n of the general theorem on d e f i n i t i o n by t r a n s f i n i t e induction and i s based on the following axiom of choice: for every binary r e l a t i o n R there i s a_ function g such that for every x, i f there i s a y such that <tx,y>eR, then < x,g(x) )^eR. That i s , i f a thing JC has the r e l a t i o n R_ to at l e a s t one thing, then x. has K. to g(x). Let H be any non-empty set, i . e . a c l a s s which i s an element, and K any proper subset of H. Then, there i s at l e a s t one x. such that xeH and x|:K. It follows from the axiom of choice that there i s a g_ such that f o r every proper subset K of the c l a s s _H, g(K) belongs to _H, but not to K. Now l e t h be the function such that h(K) = Ku[g(K)}. Then, K<= h(K) <== H. 35 "Proof That Every Set Can be Well-Ordered" (1904) i n van Heijenoort, p. 141. 36 In von Neumann's system the axiom i s : (Vf)(3g)(Vx) [ ( 3 y)([f,<x,y>] i A) -+ [f,<x,[g,x]>] + A] 202 Von Neumann proved the following well-ordering theorem: If H_ is a set, then there exists a well-ordering relation whose domain is H. Let ty be the function such that for each xeU** and function m which is a set (i.e. a class of ordered pairs which is a set), <Kx,m) = g(m[x]), where £ is a function such that for every proper subset K of H_, g(K)eH - K. Then, for every function ty: 4>(x,*|x) = g(ip|x[x]) - g(*[x]). It follows from the theorem on definition by transfinite induction that there is a function ty such that for xeU**: *(x) = +(x,*|x) = g(*[x]). Von Neumann showed that there is an ordinal P_ which ty maps one-one onto H. Thus, [ty]CB/ll*)Z is a relation well-ordering H. An ordinal number x_ is normal (for this proof) i f f ty [x] c H. (If x is a normal ordinal, then ^ (x)eH - <J»[x].) Let P_ be the class of a l l normal ordinals. It is evident that an element of a normal ordinal is again a normal ordinal, hence, i f xeP, then x£P. Consequently, P_ is an ordinal. It remains to be shown that ty is one-one on P and that ^[P] = H. Let x and y_ be any two different elements of P_. From the fact that they are von Neumann-ordinals, i t follows that one, say x.» is an element of the other. Now ty(x) but not ij;(y) belongs to ty [yj , for ty(y) = g(^[y]) and g (ty[y~\)lty[y]. Therefore, for any normal ordinals x. and y_, x^y ty(x) t *(y). It is obvious that ty[P] is a subset of H. It follows from this and the axiom of replacement that P is a subset of H. If ^ [P] H, i t 203 would be a proper subset of H; in that case, P_ would be normal and thus a member of itself. But this is impossible, for P_ is an ordinal, and no ordinal belongs to itself. This completes the proof of the well-ordering theorem. This method of proof will not work for classes H which are not sets, for the means of proving that ^[P] = H would be lost. It does, however, follow that if H is not a set, then H has a subclass equivalent to U * * , i.e. <JJ[U**] - H. To see why, let us compare the cases in which H is a set with that in which H is not a set. (1) H is a set. It follows from the fact that I|J [ P ] S H that ^[P] is a set. Since <JJ[P] ^ P , P_ is also a set. A contradiction follows from the supposition that i|>[p] 4 H, therefore, I|>[P] = H. (2) H is not a set. In (1) i t was proved that P is a set from the hypothesis that H is a set; now we assume that P_ is a set. It follows that <J>[P] is a set.- If I|>[P] is a set and H is not, then IJ> [ P ] f H. But as before, <J;[p] 4 H implies a contradiction. There-fore, our supposition that P is a set is false, and we have proved that P_ is not a set, i f H is not. Since U * * is the only ordinal number which is not a set, U * * = P . The class H' = ^ [u**] is a subclass of H which is equipollent to U * * . Thus, every class which is not a set contains a subclass equipollent to U * * , the class of a l l ordinals which are sets. In the next section, we shall see that i t is possible to prove (by means of the axiom of foundation) that every class which is not a set is the field of some well-ordering relation. The most evident of the propositions which are equivalent to the axiom of choice is the multiplicative axiom, which states that for every 204 set of non-empty, pairwise disjoint sets there is a set containing exact-ly one element from each member of this set of sets. Russell seems to have recognized independently of Zermelo's work that this proposition is either not demonstrable or very hard to prove; he says in a letter written to Jourdain in 1906 that: As for the multiplicative axiom, I came on i t so to speak by chance. Whitehead and I make alternative recensions of the various parts of our book, each correcting the last re-cension made by the other. In going over one of his recen-sions, which contained a proof of the axiom, I found that the previous proposition used in the proof had surreptitiously assumed the axiom. This happened in the summer of 1904. At first I thought probably a proof could easily be found; but gradually I saw that, i f there is a proof, i t must be very recondite.37 As in the case of the axiom of infinity, Russell did not accept 38 the multiplicative axiom except as a hypothesis. In The Principles of Mathematics, Russell did not regard the axioms of the various mathe-matical theories as propositions asserted by those theories. He main-tained that what is asserted in a particular theory is that the axioms imply the theorems. In Principia Mathematica, Whitehead and Russell apply this doctrine to the axiom of infinity and the multiplicative axiom. They do not assert either of these statements in PM; rather, "Bertrand Russell on his Paradox and the Multiplicative Axiom, An Unpublished Letter to Philip Jourdain," Journal of Philosophical Logic, 1 (1972), p. 107. 38 He once stated, however, that "the multiplicative axiom...appears highly self evident" ("Mathematical Logic as Based on the Theory of Types," p. 99), and, for at least a short period, he was convinced by arguments of Scheffer and Ramsey that the axiom of choice is true (The  Analysis of Matter, p. 299). But, for the most part, he apparently thought that "although, at first sight," the multiplicative axiom seems "obviously true, yet reflection brings gradually increasing doubt, until at last we become content to register the assumption and its consequences, as we register the axiom of parallels, without assuming that we can know whether it is true or false" (Intro. Math. Phil., p. 122). 205 they prove a large number of theorems having the form of conditionals whose antecedent is one of these "axioms." So far as I can t e l l , Russell had only one objection to the multi-plicative axiom. In his paper "On Some Difficulties in the Theory of Transfinite Numbers and Order Types," he says that "the doubt as to the truth of Zermelo's axiom arises from the impossibility of discovering [a propositional function] by which to select one term out of each set of classes" (p. 30). At -that time Russell maintained that a proposition-al function is "a necessary but not a sufficient condition of the exis-tence of an aggregate" (Ibid.). Likewise, in The Principles of Mathematics, he said that "every class can certainly be defined by a propositional function" (p. 103). Thus, Russell's doctrine that classes are determined by properties (or propositional functions) is the source of his skepticism concerning the multiplicative axiom. One of Ramsey's main criticisms of the system of Principia Mathematica was that, on the theory of classes expounded there, the axiom of choice is not evident: If by 'class' we mean, as I do, any set of things homogeneous in type not necessarily definable by a function which is not merely a function in extension, the Multiplicative Axiom seems to me the most evident tautology. I cannot see how this can be the subject of reasonable doubt, and I think i t never would have been doubted unless i t had been misinterpreted. For with the meaning i t has in Principia, where the class whose existence it asserts must be one definable by a propositional function of the sort which occurs in Principia, it becomes really doubtful ... (Foundations of Mathematics, p. 58). The multiplicative axiom may be doubtful on the Fregean doctrine of classes, but on the extensional view, which was explained in section 27, there is no axiom which is more evident: "the axiom of choice is just as evident 206 139 as the other set-theoretical axioms for the 'pure' concept of set...." Indeed, as Godel has said, the axiom of subsets and the axiom of-choice (or rather the multiplicative axiom) are the axioms which best express 40 what is meant by a set in the extensional view of sets. 32) The Axiom of Foundation It seemed to Hausdorff that Russell's paradox turns out to be "das philosophisch wohlbekannte Paradoxon der schlect definierten Allheit-s k a t e g o r i e . I n fact, Hausdorff took i t as an "axiom" that no class can be an element of itself. On this ground, wir erkennen, dass das Wort 'alle' nicht immer eine erfiillbare Forderung bezeichnet, dass wir in manchen Fallen zwar distri-butiv j edes Objekt einer bestimmten Definition gemass denken, nicht aber kollective alle Objekte 'uno intellectus actu' zusammenfassen konnen (Ibid.). If i t were thought that Hausdorff's axiom is certainly true, that would be a reason for thinking the naive comprehension principle false. Thus, to use Hausdorff's own examples, there are no such classes as the class of a l l things, the class of a l l classes, the class of a l l judgments or propositions; and there are no negative classes such as the class of non-men or the class of a l l non-green things. Perhaps it is not going too far to say that the naive comprehension principle's implication of the existence of classes which would have to belong to themselves makes this 39 "What is Cantor's Continuum Problem?" p. 259, n. 2. 40 "Russell's Mathematical Logic," p. 151. 41 Rev. of The Principles of Mathematics by Russell, Vierteljahrsschrift  fur wissenschaftliche Philosophie und Sozialogie, 29 (1905), p. 123. It is not of course true that the supposed class of a l l things is "ill-defined," for i t would seem to be defined by means of the condition "x=x". 207 principle seem somewhat doubtful on reflection. For my own part, I am inclined to think this consideration has some force. Yet this conviction i s not, in my opinion, strengthened by i d e a l i s t i c considerations. In his Grundzlige der Mengenlehre, Hausdorff explained that "A set is a collection (Zusammenfassung) of things into a whole (Ganze), i.e. into a new thing" (p. 1). Here an i d e a l i s t i c view of sets i s used to support the assertion that a set is never identical with one of i t s own elements: xeM ->- x?*M. It seems to me, however, that the i d e a l i s t i c explanation of a set as something "resulting from" or determined by arbitrary choices or random selections of things is depen-dent upon the non-idealistic concept of set. For i t might as well be said that for any set of acts of selection there i s a set containing exactly the selected objects. From a non-idealistic point of view, arbi-trary or random selections of things merely specify a set containing just the "elected" things as elements; they do not somehow "create" the sets. In his 1908 system, Zermelo did not exclude the existence of sets which are elements of themselves by an axiom. He did, however, adopt such an axiom in his 1930 paper—the axiom of foundation. A similar axiom was formulated by von Neumann. Though this i s not exactly Zermelo's formu-lation, the axiom of foundation states that the membership relation i s well-founded, i.e. every non-empty class K contains an element (individual or set) which has no members in common with K. This means that every non-empty class K contains at least one e-minimal element, i.e. an element to which no element of K has e. 208 An axiom of foundation may also be formulated as follows: It is not the case for any class A that a l l elements of A are non-empty subsets of A. If there were an A of that sort, then, i f x^ ^ 0 were in A, any element x^ of x^ would be an element of A, and, therefore, a non-empty subset of A containing an element x-j, which would again be an element of • A, and so on. In fact, A would contain an infinite descending element chain: • * • c x ^ • This axiom implies Hausdorff's axiom as a special case. The axiom of foundation excludes not only classes which would belong to themselves, but also classes which would be elements of elements of themselves, and so on; i t also excludes infinite descending element chains. For the most part, the only reason to believe in the existence of classes not satisfying the axiom of foundation was provided by the classes which are "too big," such as the class of a l l things, the class of a l l classes, and the class of a l l classes equipollent to a particular class (i.e. the original Russell cardinals). According to Russell's (original) definition of the cardinal number of a class, the greatest number, i.e. the cardinal of the class of a l l entities RU, is the class of a l l classes equivalent to RU. This number would belong to itself, but the Russell cardinals would not generally belong to themselves. They would, however, belong to members of members of themselves. For example, {2,A} would belong to 2 = the class of a l l couples, provided A ^ 2. Thus, the theory of limitation of size eliminates the large classes which would not be well-founded, such as RU. The axiom of foundation does not, however, quite follow from the theory of limitation of size, for the 209 classes which are "too big" were not the only classes which were thought to belong to themselves. Some logicians, including Russell, thought that on the pure extensional concept of classes i t ought to be the case that {x} = x, so that {x}e{x}. It came to be realized that when x is the empty set or a class having more than one member, x cannot be iden-tical with its unit class. However, that an individual may be identical with its own unit class is not ruled out. The axiom of foundation is in accordance with a certain Interpre-tation of a form of Russell's vicious circle principle. Russell gave several formulations of the principle, but in Principia Mathematica the tit l e "vicious circle principle" seems to be officially bestowed upon the following statement: 'Whatever involves a l l of a collection must not be one of the collection;'or conversely: 'If, provided a certain collection had a total, then i t would have members only definable in terms of that total, then the said collection has no total 1 (I, 37). We shall ignore the second form in which the phrase "only definable in terms of" occurs, and consider only the first form. The axiom of founda-tion agrees with this form i f i t is interpreted as stating that no totality (class) can contain as an element anything which involves or presupposes the existence of a l l its members. According to the extensional concept of a class (set), a class depends on the existence of a l l its elements. Therefore, according to our understanding of the vicious circle principle, a class cannot contain itself as an element and neither can i t belong to any element of an element of...an element of itself. Thus, we may say that a class A cannot belong to the class A* of things whose existence i t involves. 210 We shall now turn to some of the most interesting consequences of the axiom of foundation which were established by Zermelo and von Neumann. Some of these may be seen as to a certain extent a clarification of the paradoxes of absolute infinity. First we consider Zermelo's demonstration that every normal domain is well-stratified. Lemma on subdomains of a normal domain: A subdomain M of a normal domain P_ is identical to P_, i f (1) M contains a l l elements and subsets of any set i t contains, and i f (2) M contains any set m of the normal domain P_, a l l of whose elements belong to M. If the sets x and y_ belong to M, then, by (1) a l l their elements belong to M. That P_ satisfies the axiom of extensionality means that for sets x and y_ in P_, (Vz) [zeP -> (zex •«->• zey)] -»• x=y. If JC and y_ are in M, then they are also in P_ and, i f z^  is in x and y_, then (zex -«-+ zey). Hence, x=y. Therefore, the axiom of extensionality holds for M. If xeM, then P(z)eM and a l l subsets of x belong to M> hence, by (2), P(x)eM. Likewise, the union axiom holds for M because of (2), since xey & yez & zeM zeM, by (1). If the domain of a function whose values are a l l in M belongs to M, then the set of its values belongs to P_, and, by (2) also to M; hence the axiom of replacement holds for M. Since M is a subdomain of P_, the axiom of foundation holds for M. If P - M = R is non-empty, then any set in R, since i t does not belong to M» must contain at least one element of R. This contradicts the axiom of foundation, which states that every non-empty domain contains at least one element which has no element in common with i t . 211 Every normal domain P_ of characteristic if decomposes into a well-ordered TT-sequence {S } . of disjoint strata such that S contains a l l n a a<iT J a sets whose elements are in U Sg but which do not themselves belong to 8<a P any S„ for B < a. The strata S are defined in terms of the sets V p ct ot which were defined in section 20 above. The set S = V ,, - V . Thus, ct ot+1 a ' S consists of those subsets of V which do not belong to V = U S„. ot a a w B 8<a P The strata S Q are disjoint, and i t is also the case that they are non-empty, for each stratum contains the ordinal ot_ which is its index. This is shown by transfinite induction. If for a l l B < ot 8eS„, then p a l l elements of ja belong to strata preceding S a and, hence, to V . Therefore, a must belong to S = V , , - V . ' — & a a+1 a It follows in virtue of the lemma on subdomains that the subdomain of the normal domain P_ is identical to P_. Let r_ be an element of P_ which is a subset of V . The elements r 0 : belong to strata S„ . The in-P 6 dicies a of the strata to which the elements of r belong form a well-ordered p . — set of type u < TT, since its power is <_ r". This sequence has an upper limit a < ir, for ir, being a Grenzzahl, is not cofinal with any lesser ordinal. Therefore, rtV and reV ,,, hence reV . V is a subdomain of a a+1 TT IT P_ containing any set r_ belonging to P_, whenever rtV^; therefore, = P, by the lemma on subdomains. In his paper "Uber eine Widerspruchfreiheitsfrage in der axiomatis-chen Mengenlehre," von Neumann proved that there is no more danger of contradiction in his system S_, containing the axiom of limitation of size, than there is in the system of set theory S*+F, which is like S_ except 212 that i t contains the axiom of foundation F_ and, instead of the axiom of limitation of size, an axiom of choice and the axiom that i f A is a set, j> is a function, and B?g[AJ , then B is a set. As we have seen in section 26, von Neumann's system has the peculi-arity that the well-ordering theorem is derivable by means of the Burali- . Forti paradox and the axiom of limitation of size. It is concluded from the paradox that U** is as big as IJ, but no function mapping U** onto U 42 is specified. The axiom of replacement, as well as the axiom of union, is also provable. Thus, von Neumann's system is remarkably strong. Von Neumann thought that his system seems to go beyond "die eigentliche Mengenlehre." But he showed that S_ is actually equivalent to one of Cantorian extent in the sense that i t Is based essentially on the thesis that "es zu jeder (nicht 'zu grossen') Menge von Machtigkeiten (nicht 'zu grosser' Mengen) eine noch grossere Machtigkeit (die noch immer zu einer nicht 'zugrossen' Menge gehort) gibt" (p. 497). Thus, von Neumann showed that, i f set theory of Cantorian extent is consistent, then so is his system, which seems to go beyond Cantorian set theory. His proof has two parts. He showed that (a) i f S* is con-sistent, then so is S*+F, and that (S) S_ follows from S*+F, where F_ is the axiom of foundation. The function V such that for every ordinal ot, V = U P(V R), is A. Levy, "On von Neumann's Axiom System for Set Theory," American  Mathematical Monthly, 75 (1968), 762-763. 213 4 3 used in the following demonstrations. Let II =UV . a If H is a subset of II, then there is an ordinal a such that H 9 V . — — a Von Neumann proves this as follows. Every element u of H belongs to some V . It follows from the axiom of choice that there is a function k such a — that for every ueH, k(u) is one of the ordinals a_ for which ueV^. Since H is a set, so is k [ H J = H'. Further, H S U V [ H / ] . H' is a set of ordinals, soUH' is an ordinal 8 at least as great as any in H'. For every aeH', V • therefore, U V [ H ' 1 Q V q and so H & v o . It follows that HeV-,., a 8 1 J 8 8 8+1 and hence Hell. Thus, i t has been proved that every subset of II is an element of II, i.e. P (II) ^ II; this, of course, has the consequence that II is not a set. If a set u belongs to II, then _u belongs to V q for some ordinal cx; moreover, there must be a least ordinal a such that ueV . This ordinal — a ct = G(u) is the degree (Grad) of the set _u. Von Neumann remarked that Dieser Begriff des 'Grades' entspricht dem Russellschen Begriff der 'Stuffe,' in einer konsequent aufgebauten und den 'Cantorschen Umfang' erreichenden Mengenlehre kann man. nicht vermeiden, alle (transfiniten!) Ordnungszahlen als 'Grade' ( = 'Stufen') zuzulassen (p. 505). Russell had been inclined to attribute the paradoxes to the failure to exclude "impure" classes. Thus, in his Introduction to Mathematical Philosophy, he said: "The fallacy consists in the formation of what we may call 'impure' classes, which are not pure as to 'type.'"^ Russell 4 3 Cp. section 20 above. Von Neumann's system is, of course, actually an axiom system for a certain kind of function. I have reformulated his definitions, theorems and proofs in terms of set theory. These re-formulations are always entirely obvious. 44 P. 137. A class M containing an element which is also a subset of M would be an example of an "impure" class. 214 used an argument (which he also used against the supposition of the exis-tence of classes)to support this contention, namely the paradox of the greatest number: But i f we could...add together into one class the individuals, classes of individuals, classes of classes of individuals, etc., we should obtain a class of which its own subclasses would be members. The class consisting of a l l objects that can be count-ed, of whatever sort, must i f there be such a class, have a cardinal number which is the greatest possible. Since a l l its sub-classes will be members of i t , there cannot be more of them than there are members. Hence we arrive at a contradiction.^ These considerations do not, however, show that "the fallacy" con-sists in supposing the existence of impure classes. The inconclusiveness of Russell's recommendation of type theory is clearly indicated in his own formulation: I_f_ there be a class of a l l entities.... Indeed, we are by now quite familiar with a way of excluding a class of a l l entities other than ruling out "impure" classes (simple type theory): namely, the thesis that not a l l classes are elements of classes. This doctrine excludes a class of a l l entities as well as does the doctrine that a class does not contain any of its subclasses as its elements. Simple type theory excludes small as well as large "impure" classes. This exclusion of classes like {x,{x}} just because paradoxical classes would be impure Ibid, p. 136. In his introduction to the second edition of The  Principles of Mathematics, Russell says that the contradiction of the greatest cardinal "has the merit of making peculiarly evident the need for some doctrine of types" (p. x i i i ) . 2 1 5 seems to me definitely unjustified. Furthermore, i t does not seem that when the classes that Russell indicates are "added together," the resulting "impure" class is anything like the class of a l l entities. Let R ( 0 ) = the class of a l l individuals, and R(n+1) = P( R(n)). The class Russell indicates would be R(co) = (jR(n) n where "n" ranges over the finite ordinals. But this is not the class of a l l entities, for R(io+l) = P(R(co)) is a more comprehensive "impure" class, and, in general, for every ordinal cx, R(cx) is a more comprehensive class than any R(8) where 8 < ex. Even U = U R(C X) is not the class of a l l things, a but the class of a l l elements. Russell, of course, would not admit the existence of R(co) because i t is a class of mixed type. Thus, Principia Mathematica does not incorporate a theory of Cantorian extent. In the system of PM, the existence of only denumerably many transfinite cardin-als can be proved. Whitehead and Russell say that, "There is not...so far as we know, any proof of the existence of Alephs and Omegas with infinite suffixes, owing to the fact that the type increases with each suc-cessive existence-theorem, and that infinite types appear to be meaning-— Such sets are, of course, not ruled out by any form of the vicious circle principle. I cannot believe, moreover, that anyone could really suppose that a contradiction results from supposing the existence of small "impure" classes. This indicates (at least to me) that impurity is a harmless characteristic which would belong to paradoxical classes, but which is not responsible for the paradoxes. It is interesting to note that the exclusion of impure classes did not seem particularly plausible to Whitehead, as the following extract from a letter written to Russell in 1 9 0 7 shows: "Two men are in a room, they form a class of two things, they and the class form a class of three things. This lat-ter class, i t is true, is a class of these things in another sense to that in which the former class is a class of two things. But your theory of types refuses a l l sense to the latter class (Does i t not?). This is very paradoxical." Cp. Principles, p. 5 2 6 , par. 2 . 216 less" (PM, III, 173). Let us now proceed to some of the main properties of the function G. For x and v_ in II, i f xey, then G(x) £ G(y). The degree of any set is always a successor ordinal, for whenever a set _u belongs to V^ , where A is a limit ordinal, u_ already belongs to some V^ , with a < X. Hence, G(y) is a successor ordinal (say) B+l. Since, y e v"g +p V-Vg« Therefore, xeV., and consequently G(x) < B < B+l = G(y). P For every non-empty subclass A of II, i t is not the case that for every element x. of A there is an element y_ of A such that yex. If this theorem were false, there would be a subclass of II containing an infinite descending element chain. The class of ordinals which are the degrees of these sets would not have a fi r s t element, but this is impossible. It can be shown that each of the axioms of S* is satisfied in the domain of sets II with respect to the relation of class membership e' which holds between the elements of this domain and also between the mem-bers of II and the subclasses of II which are not sets. In this way the assertion (a) is established (cp. p. 210 above). Von Neumann proves (B) by deriving the principle of limitation of size from the axiom system S*+F. He first establishes the theorem that n = U. If this did not hold, then U - II = IT ± 0. IT would have to contain an element x having no e-predecessors in U'. This means that any elements of x would have to belong to II. Therefore, xtll, and since xeU, x is a set. It follows that xeP(II) and so xell. But this contradicts our assumption. 217 In section 31, i t was shown that the method of proof used to prove that every set is the field of some well-ordering relation could not also be used to prove that every class is the field of some well-ordering. On the basis of a set theory which includes an axiom of foundation, how-ever, the following theorem can be proved: There is a well-ordering W of U such that for each ueU, Seg(u;U,W) jjs a_ set. Proof. For each element x of U**, let H be the class of a l l ele-— x ments of U whose degree is x. Since H = V , H is a set, for V is the — — x x x x set of a l l elements whose degree is <^  x. Every element of U_ belongs to one and only one H . x It can be proved in S* that every set is well-ordered. Thus, the 47 class of well-ordering relations whose field is H X is a non-empty set. By the axiom of choice, we are justified in letting Wx be a particular well-ordering relation whose field is U.^. The relation W is defined in terms of the W as follows: — x uWv i f f [G(u) < G(v)] or [G(U) = G(v) = x & uWxv]. W is the class of a l l ordered pairs belonging to any of the sets H XH or x y for x,ycU** and x < y. The relation W is a well-ordering whose field is U. Furthermore, for each ueU, Seg(u; U,W) is a set. This may be seen as follows. If vWu, then G(v) s. G(u) and, hence, G(v)eG(u)u{G(u)} = G(u)+1. Therefore, every W-predecessor of u belongs to some for some xeG(u)+l, consequently, Seg(u;U,W) Q H[G(u)+l] . By means of the well-ordering W of U i t is possible to prove that U is equipollent to U**. Let P be the ordinal of U,W. Since, U,W is similar to P,(P/U*)i:, U^ P. That U is not a set follows (independently — If A is a set, then the class P(AxA) of a l l relations on A is also a set. 218 of the axiom of limitation of size) from the axiom of separation and either the Russell or the Burali-Forti paradox. Therefore, by the axiom of re-placement, P_ is not a set. Since U** is the only ordinal which is not a set, P = U**, and, consequently, U^ U**. Finally, there is the proof of the principle of limitation of size. Let H be a class which is not a set, and let ^ be a one-one correspondence between U and U**. Further let H' = g[H]. H' is the set of a l l ordinals which are g-correlates of members of H. Let P_ be the ordinal of the class H in the order (H/U*)E. Then, HM>. P is not a set, and therefore P = U**, because H is not a set. Hence, HMJ**, and, consequently, HMJ. If f is a mapping of the class H onto the universe of elements U and i f JH were a set, then (by the axiom of replacement) U would be a set, which i t is not. Hence, H is not a set. It follows from the axiom of foundation that the universe of elements decomposes into an absolutely infinite sequence of strata containing sets of ever greater complexity, and the segments of the universe of elements corresponding to these strata are ever more comprehensive. 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