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I’ ON THE PARADOXES OF SET THEORY A TIlES IS SUBMITTED TO THE FACULTY OF ATLANTA UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE BY PRINCE ILONA WINSTON DEPARTMENT OF MATHEMATICS: ATLANTA, GEORGIA AUGUST 1959 ) ACKNOWLEDGEMENT The writer wishes to express sincerest appreciation to Professor Lonnie Cross for suggesting this problem and for his assistance and advice in its development. TABLE OF CONTENTS Page AC~OWL~G~NT. • . ii LIST OF FREQUENTLY OCCURRING SYMBOLS................. iv Chapter I. INTRODUCTION, • • • • • • • , • • • • • • • . • • . , , . • • . 1 II. THEPARADOXESOFSETTHEORY......,.,,.,,,0.., 5 The Cantor Paradox.... .... .. .. 5 The Russell Paradox. • • . • . 8 The Burali— Forti Paradox......,....,.... 12 The Richard Paradox. ... ...... ...... .... .... 15 III. CAUSES OF AN) PROPOSES FOR SOLVING PARADOXES... 17 Axiomatic Set Theory....................... 17 Impredicative Definitions. .. .... ,...... 22 Proposals of the Various Schools of Mathematics Logic isni. • . • . 24 Intuitionisni. 25 Fo~alis~. • • • . • a • . • • • a a a a a a a . • 28 IV. SUMMARY AND CONCLUSION.........,.....,......... 31 BIBLIOGRAPHY. * . 33 iii LIST OF FREQUENTLY OCCURRING SYMBOLS AND NOTATIONS U(S) — If S is~a set, then U(S) denotes the set of subsets of S. S The cardinal number of the set S. - not equal to inequality signs G(S) the sum of the sets belonging to the set S whose members are sets. - is an element of — is not an element of U U - union fl fl — intersection - is contained in - is equivalent to 0 -. empty or null set iv CHAPTER I INTRODUCTION Modern theory of sets is usually considered to have be gun with Georg Cantor (1845—1918), who devised the first number for infinite sets during the latter part of the nine teenth century. It is revealing to read the “definItion” of set given by Cantor: By a “set” we shall understand any col lection into a whole, M, of definite, distinguishable ob jects in (which will be called “elements” of M) of our in tuition or thought.1 “Geometry and analysis, differential and integral cal culus deal continually, even though perhaps in disguished ex pression, with infinite sets.”2 Thus wrote Felix Hausdorff (1914) in his Fundamentals of the Theory of Sets. In order to acquire a genuine understanding and mastery of these various branches of mathematics it is necessary to obtain a knowledge of their common foundation, namely, the theory of sets. Georg Cantor had delayed publishing his work on the theory of sets for ten years. It was not until he recognized that his concepts were Indispensable to the further develop ‘Raymond Wilder, Introduction to the Foundations ~ Mathematics (New York, 1952), p. 54. 2 Joseph Breuer, Introduction to Theory of Sets (New Jersey, 1958), p. 1. 1 2 ment of mathematics that he decided to publish it. The first sentence of the first of his papers was the definition of “set.” This was written by a first class mathematician who did not know, as no other mathematician, at the time, that the word “set” was “loaded.” However, enlightenment was not long in forthcoming. As a matter of fact, it was virtually ready not long after Cantor published his ideas, as a result of the announcement by an ItalIan logician, Burall-Forti, of a fundamental difficulty with one of Cantor’s basic defInitions. Unfortunately, Buralj-Fortj misinterpreted the definItion, so that his in sight did not first win recognition. Soon after, Bertrand Russell announced his famous “antinomy” and this time there was no attempt to avoid the difficulty by subterfuge or other wise, and the problem of “what to do?” had to be met. This was a development of great importance which occur red at the end of the nineteenth and at the beginning of the twentieth century. That this new theory, so beautiful and fruitful, should lead to such logical consequences came as a profound shock to mathematicians. Ferge, for example, con sidered that all his work based on the theory of sets, was jeopardized. Many mathematicians, as a result of the an tinomies, have ceased to work on aspects of mathematics which , depend upon an unqualified acceptance of set theory. Poincare characterized the theory of sets as “a disease from which we 3 will someday recover.”’ Others, more courageous perhaps, or more convinced of the ultimate validity of the theory, set out to correct errors into which mathematics had drifted. Perhaps, the greatest paradox of all is that there are paradoxes In mathematics. It is not surprising to discover inconsistencies in the experimental sciences, which periodi cally undergo revolutionary changes. Yet, because mathematics builds on the old but does not discard it, because it is the most conservative of the sciences, because its theorems are deduced from postulates by the methods of logic, in spite of its having undergone revolutionary changes it is not suspected of being capable of engendering paradoxes. Nevertheless, there are paradoxes which do arise in mathematics. There are contradictory and absurd propositions which arise from fallacious reasoning. There are theorems which seem strange and incredible, but which because they are logically unassailable, must be accepted even though they transcend intuition and imagination. The third and most im portant class consists of those logical paradoxes which arise in connectIon with the theory of sets, and which have re— suited in a reexamination of the foundations of mathematics. These logical paradoxes have created confusion and consterna tion among logicians and mathematicians. ‘Raymond Wilder, op. cit., p. 200. 4 It is with the latter type of paradox with which this paper is concerned. It is because of the concern that this crisis in the foundations of mathematics created and because of the profound effect that it had not only on the theory itself but with the other subjects in mathematics, that these paradoxes are presented. SinDe the discovery of paradoxes in set theory, a great deal of literature has appeared offering solutions. in con nection with the paradoxes, some of the solutions that have been propos ed will be given. It is by no means the purpose of this paper to draw any conclusions as to which proposal is the more acceptable, but to merely give the proposals and a critical analysis of each. CHAPTER II THE PARADOXES OF SET THEORY The Cantor Paradox One of the most important paradoxes which arises in the theory of sets is the Cantor paradox. This paradox oc curs In connection with the theory of transfinite candinals. It was discovered idependently by Cantor in 1899. It can be derived by considering the following theorem: Theorem,--To any set S there exists sets having larger cardinals than S, in particular, the set U(S), whose elements are all subsets of S, is of larger cardinal than S. Sym-. bolically, U(S)~S. Reniarks: This assertion with respect to U(S) holds for finite sets as well as for infinite ones. It goes without saying that the null—set and S itself are included among the subsets of S forming the elements of U(S). Proof of Theorem.--First is the construction of a subset of U(S) equivalent to S. We may choose as the subsets in question the set whose elements are the sets c[~} where s runs over all the elements of S — that is to say, the subsets of S containing the single element. Second, we must show the impossibility of a one-to—one 5 6 correspondence between U(S) and S~ We must prove then that U(S) is not equivalent to S itself. Let~ denote any fixed representation between S and a subset U0 of U(S). By showing that this assumption neces sarily implies inequality U0 ≠ U(S) (that is, U0 to be a pro per subset of U(s~ ) we shall reach our aim, for this proves that the full set U(S) is not equivalent to S. ~ assigns to every element of S a certain subset of S; it will therefore suffice to construct a subset u (at least onel) of S that has no correspondence among the elements of S by virtue of in other words, to construct an element u of U(S) which is not contained in U0. For our purpose we shall take into consideration that, after having chosen~, we may classify the elements of S according to the following alternative: s is an element of the subset U3 corresponding to s by ~, or s is not an element of U5. (In either case U5 is, of course, an element of IJ~). According to this classification we shall speak of elements of the first kind and of the second kind. We do not require, however, that there exist elements of both kinds. We now denote by u* the set of all elements of the second kind. (If all elements of S should be of the first kind, u* will obviously be the null set). In any case u* is a subset of S and therefore an element of U(S). We shall now show that u* is not contained in the subset U0 of U(S) and thus the theorem. 7 If u* were an element of U0, and s* the element of S related to u* by~ , s* should be either of the first kind or the second kind. In the former case s* is an element of u* (by definition) but this contradicts the definition of u* which only contains elements of the second kind. (The as sumption here is that u* is contained in U0, implies its having an image s* in S. 3* may be an element of u* or not. Proof shows that either case involves a contradiction and hence the assumption has to be dropped). Since both sup-.
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