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Home , General set theory, Jules Richard, Paradoxes of set theory

I’

ON THE OF 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 ...... 5 The Russell Paradox...... • • . . • . . . . 8 The Burali— Forti Paradox...... ,....,.... 12 The Richard Paradox...... 15 III. CAUSES OF AN) PROPOSES FOR SOLVING PARADOXES... 17 Axiomatic ...... 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 of S. S The 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 of

— is not an element of

U U - 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 (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

“ 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 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, 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 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 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 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 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,--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 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 s* in S. 3* may be an element of u* or not. Proof shows that either case involves a and hence the assumption has to be dropped). Since both sup-. positions lead to a contradiction, no element s* of S can be a mate of u* L,U(s), and hence u* does not belong to U0,

Q.E.D.

We have now proved that U(S)) S. A paradox occurs in Cantor’s theorem if we consider the set consisting of the set of all sets.

Proof.-—Consider the set of all sets. Call it S. By the theorem just proved, the set of subsets of a given set has greater than the given set. Since S is the set of all sets and if we consider U(S) as a set of sets (namely, the set of subsets of S). Then it is clear that

U(S) ~ S. From the Bernstein equivalence theorem: If M’-’N1~N and Nt’M1~M we can extract the following corollary: If M ~N, then IVI ~ N. Hence since U(S) ~ S then U(s) ~ S.

Q.E.D. 8 I~ow contrary to Cantor’s theorem that U(s)>S, we have proved by setting S as the set of all sets, that US~S. We can also arrive at the Cantor paradox thusly:

Proof.——Consider the same set S (the set of all sets), To each a of 5, that is, to any set a, by Cantor’s theorem, there is another member s~’of 3, namely U(s) such that, ~ But we know that if S is the set of all sets, and to each member a of S there is another member s’of S such that

~CS then~dT~) (a(s) is the sum of the sets belonging to

the set S whose members are sets) for every member S Sof S. But S is the set of all sets, so a(s) is one of its members. Taking the a in the inequality just proved to be this member, we have d(S1<~(~J. But the definition of cardinal number says for any set S not ~ but SS. Hence in particular not G(s)

Q,.E.D

The Russell Paradox

The Russell paradox (1902-3) was discovered Independently by Zermelo. It deals with the set of all sets which are not members of themselves.

Since generally, the elements of a set S may be sets themselves, the possibility arises that S may happen to be 9 an element of itself. For example, it has been suggested:

the set of all abstract ideas; such a set is certainly an ele— ment of itself if we grant that this is itself an abstract idea. Again, the set of all sets is itself a set, but the set of all stars is not a star. Russell’s paradox deals with the latter example: sets which do not have themselves as elements. Let us call this set T. The question arises and answers to it leads to the Russell paradox: Is T a member of itself?

Proof.-—Let us assume, for the sake of , that T is a member of itself; that is in symbols T&T. The as sumption says that T is a member ofT, that is, T is a member of the set of all sets which is not a member of itself, that is in T4.T. This contradicts the assumption that TE.iT. Thus far we have no paradox as the contradiction between TCiT and T~iT has arisen only under the assumption that T&T. By reductio ad absurdum, we conclude that the assumption is false. Thus we have proved that T4~T.

From the established result T4≠T, we can argue further. The result says that T is not a member of the set of all sets which are not members of themselves, that is, T is not a set which is not a member of itself, that is, T is a set which is a member of itself. In symbols, T€-T. Now we have both es tablished that T4T and T ~,T, so that we have the paradox. 10 The Russell paradox can also be extracted from Cantor’s,, If we prescribe (a1) and (a2) as admissible elements, so that

sets have only sets as members, then when S is the set of all

sets, U(s)=s and the set T of Russell’s paradox is obtained. Proof._...First: Consider

(a1) =4:0, 1, 2. . .3’ (a2) { ~: the null set Then we have b~4:the set of subsets of (a1) and (a2)}3

Second: If we admit (b) as an admissible ele ment (that is, treat (b) as an element) then 8, the set of all admissible elements, (this consists of admissible ele ments and subsets of admissible elements) contains only ad— mis sible elements.

Third: Consider the subsets of S, namely U(s), However, we have assumed the subsets of admissible elements to be admissible, that is, U(s) contains only admissible ele ments. Consequently, U(S)8, In other words, they are the same set.

But this is impossible as we shall show below, by de— riving a paradox set T which is exactly Russell’s paradox set T. If it were true that U(S).s then there must be an identical one—to—one correspondence between the elements.

We now construct the set T in the following manner: 11 Since U(S) has the 1—1 identical correspondence to 3, any element a~S corresponds to b~iU(S).

If a belongs to b, we let a4iT. If a does not belong to b, then we put a (iT. Then T is obviously also a subset. The question we now raise is does any element cor

respond to T? If so, m4iT, and in does not belong to its

image in hence does not belong to T, hence a belongs to its

image and we get a contradiction. If not, then there is no element corresponding to T. However, T is a subset of 3, this is also a contradiction.

This is a paradox. In other words T is the set of all sets which is not a member of itself.

Comments: This is because the subset of admissible elements is admissible and because of identical 1-1 cor respondence.

Q.E.D

The Buralj-Fortj Paradox

Burali-.Forti, Italian logician, was the first to dis cover a paradox in the theory of sets. However, this para dox was not valid. He discovered another paradox in this theory and it was published in 1897. 12 This paradox came about as a result of the theorems below. These theorems are the basic principles upon which the theory of ordinal numbers is developed. Theorem A.--Every well-ordered series has an ordinal numb e r. Theorem B.--The series of ordirials up to and including a given , say~, has ordinal number ~ 1.

Theorem C.--Tho series of all ordinal numbers is well ordered and hence, has an ordinal number, say.fl. The Burali.-Forti paradox accerbs the incompatibility of the above three theorems. in order to deal with his paradox of the greatest ordinal number some definitions are necessary.

These include:

Definition 1.—— A set M is called ordered if there exists a rule which tells us that for each two distinct elements in M which one precedes the other.

Definition 2,~.— A~ ordered set is said to be well-ordered if every non-empty subset has a first element.

Definition 3.——The “ordinal type” is a symbol associated with a given similarity class.

Definition 4.-— The “ordinal type” of a well-ordered set is called its ordinal number.

Burali—Forti’s paradox now arises in the following way: Consider the setPof all ordinal numbers arranged ac— 13 cording to magnitude. By theorem C, the set must be well- ordered andhence has ordinal number, say,CQ) • But by theorem B, the series of ordinalsupto and including &)has ordinal number(&,L1. But since &J is the ordinal number of the set of all ordirials P we have that C&,t 1 &~‘, which im plies that ~ ,L i<&. Hence we arrive at Burali—Forti’s paradox which in ax— sence i~: The well—ordered series of all ordinal numbers de.. tines a new ordinal number which is not one of the ~ This paradox can also be demonstrated with cardinal numbers. We can show that the sum of all cardinal numbers of an aggregate of cardinal numbers is a cardinal greater than any in the aggregate.

Consider the aggregate K of all cardinal numbers, and take the sum Z of all cardinals in K. Hence ~. defines another cardinal number which is greater than any cardinal in K. But by definition, K is the set of all cardinal numbers.

Hence ~. itself is in K. Hence we get ~ Thus the paradox.

The Richard Paradox

The Richard paradox was discovered by in

1905. It involves the whole concept of definability as well as Cantor’s diagnol method. The Richard paradox is es 14 pecially interesting for its implIcation concerning languages, and because it runs so close to Cantor’s proof of the non— enumerability of the number - theoretic . The paradox as given by Richard was in a form relating to the definition of a real number, paralleling Cantor’s proof of the non enumerability of the real numbers. Since the Richard paradox concerns the concept of de-. finability, for definiteness let us refer to a given language, say the English language, with a preassigned alphabet, dic tionary and grammar. The alphabet we may take as consisting of the blank space (to separate words), the twenty-six Latin letters, and the comma. By an “expression” in the language we may understand simply any finite sequence of the twenty-eight symbols not beginning with the blank space. The expressions in the English language can then be enumerated thus:

Suppose the is of the expressions: fo ~ ~i (n), f2 (n), 1~3 (n),... Write the sequence of the values of the successive functions one below the other as the rows of a matrix:

.f (0) f (1) f (2) f (3). . 0 -.~0 0 0 .f F~(i) ~ (2) :t (3). . 1 (o) 1 1 .f2 (0) f2 (1) f~’(2) f2 (3).

.f3 (0) f3. (i) f3 (2) £3 (3).

. . . 15 Now let us consider a definite expression of the above enumerated expressions. Let this expression define a number- theoretic function of one variable (that is, a function of a natural nu~iber taking a as value). From the specified enumeration of all the expressions in the English language, by striking out those which do not define a number- theoretic function, we obtain an enumeration say E0, E1, E2, ~ of those defined respectively by

F0 (n), f1 (n), f2 (n), f3 (n). . Now consider the following ~the fun~tion whose value, for any given natural number as argument, is equal to one more than the value, for the given natural number as argu ment, of the function defined by the expression which cor responds to the given natural number in the last described enumer ati on.” In the quoted expression we refer to the above described enumeration of the expressions in the English language de fining a number-theoretic function, without defining it. For if we consider the enumeration, E0, E1, E2, E3, ... which de fines a number theoretic function, we can also write them in a n~trix: .E E (i) E (2) E (3). 0 (o) 0 0 0 .E (0) E (1) E (2) E1 (3). .E2 (0) E2 (i) E2 (2) E2 (3).

. . . 16

If the function ~those value, for any given natural number as argument, is equal to one more than the real value for the natural number as argument, we will have a functIon not in the enumeration. It is clear that the new function

E~ (n) ,~ 1 differs at least in the nth decimal place. We could easily have considered the totality of the ex

pression in the English language. Then by defining the funct ion for a particular value, by adding one to this value we

will have before us a definition of the function f~ (n) ,~ 1 by an expression In the English language. Here we have an enumeration of the expressions in the English language de fining a function without defining it, and thus, the Richard paradox. CHAPTER III

SOME CAUSES OF AND PROPOSES FOR SOLVING

THE PARADOXES

Ideas for solving the paradoxes of set theory came to mind on first considering them. Since their discovery, a great deal of literature has appeared on the subject and numerous attempts at a solution have been offered. It is the belief of ~some mathematicians that a set theory based on is the answer. Others think that the answer lies in impredicative definitions. Exponents of the three schools of mathematics have offered solutions. These proposals will be given below and the criticisms that have been found of each.

Axiomatic Set Theory

Reconstruction of set theory can be given, placing around the notion of set as few restrictions to exclude too large sets as appears to be required to forestall the known antinonjes. A~ was apparent in the Burali-Forti, Cantor, and Russell paradoxes, one may propose that the error is in using too large sets, such as the set of all sets, or in per mitting sets to be considered as members of themselves vthich

17 18 again argues against the set of all sets. Since the free use of our conceptions in constructing sets ui~.er Cantor’s definition led to disaster, It is felt that governing the notions of the theory by axioms would alleviate this dif ficulty. The ffrst system of axiomatic set theory was Zermelo’s

(1908). Axiomatic set theory is perhaps the simplest basis set up since the paradoxes for the deduction of existing mathematics. S~iie very interesting discoveries have been made in connection with axiomatic set theory, notably by Skolem (1922-3) and G~del (1938, 1939, 1940). It is assun~d that the reader has an adequate knowledge of elementary set theory as developed by Cantor. The follow ing is a system of axioms under a variant of systems by

Skolem and A. P. Morse who owe much to the Hilbert-Bernays - von Newniann systems as formulated by G~de1. To present the in detail would entail more than the scope of this paper will allow; hence the axioms are merely stated with hope that the reader Is able to see the relationship of this to Cantor’s system. As Gb’del’s axioms are stated, there are three primitive notions. There are two primitive (underfined) constants beside ““ and the other logical constant. The first of these is “&“ which is read flj~ a member of.” The second 19

constant is denoted, “ .. : ... and is read “the class of all •. such that ••• .“ It is the classifer. In the system the term “class” does not appear in any definitions or theorems, but the primary of these statements is an assertion about class. The following are the axioms:

I~ Axiom of Extent,--For each X and each Y it is true that X~Y if and only if for each Z, Z LX when and only when ZLY.

Thus, two c2asses are identical if every member of each is a member of tI~ other. It is well to note here that it is feasible to make this axiom the definition of equality. However, then there would be no unhi~nited substitution rule for equality and one would have to assume as an axIom: If X&Z and YX then Y~Z.

We have now described the notion “~ and now we must describe the classifier. The first blank in the classifier constant is the variable and the second is the formula, for example 4 X : X E,Y} • We ace ept as an axiom the statement: u £(X : XLY} if u is a set and u&Y.

This axiom scheme is precisely the usual intuitive con struction of classes except for the requirement “u is a set.” This requirement is very evidently unnatural and is intui tively quite undesirable. However, without it a contradiction 20 may be construction simply on the basis of the axiom of ex tent.

II. Axiom of Subset3.-—If X is a set there is a set Y such that for each Z, if ZCX, then Z&Y,

III. .--.If X is a set and Y is a set so is XL)Y. Under ~ system, this axiom includes theorems concerning ordered pairs: relations, and functions. The two following axioms further delineate the class of all sets,

IV. Axiom of Substjtutjon.-~..If f is a function and do-. main f is a set, then range f is a set.

V. If X is a set, so is U~. These two axioms may be replaced by the single axiom: If f is a function and domain f is a set then U range f is a set. In Cantor’s system there axioms include theorems under well ordering. Under the sixth axiom the ordinal numbers are defined. It is a priori possible that there are classes X and Y such that X is the only member of X. The following axiom denies this possibility by requiring that each nonvoid class Z have 21 at least one member whose elements do not belong to A.

VI. Axiom of Re~ularjty.--If X ~ 0 then there is a member Y of X such that XflY=$. In considering the integers under this system they are

defined and Peano’s postulates are derived as theorems. Hence another axiom is needed.

VII • Axiom ~ Infinitv.-..For some Y, Y is a set, 0 £‘ Y, and X u{x.~ & Y whenever X E.Y. The choice axiom is the eighth and last axiom. In

tuitively, a choice function is a simultaneous selection of a member from each set belonging to domain C. The following is a strong form of Zermelo’s postulate or the ,

VIII. There is a choice function C,whose domain u~i{Ø)

G-6de1’s axiom system is extremely powerful. From it, with appropriate definItions, the usual classical analysis and much of can be deduced. However, the axiomatic set theory has been criticIzed as merely avoiding the paradoxes. Certally, the axioms are set up so that pre vailing paradoxes will be excluded, It does, by no means, explain the paradoxes and their existence. Moreover, this procedure carries no guarantee that ui~er its system other kinds of paradoxes will not come up in the future. 22

Impredicative Definitions

There is another procedure which apparently both ex plains and avoids the known paradoxes. When a set M and a particular object m are so defined that on one hand in is a member of M, and on the other hand the definition of in de pends on M we say that the definition is impredicative. If examined carefully, it will be seen that the Cantor, Burali-Forti, Russell, and the Richard paradoxes involve an impredicative definition. in Cantor’s paradox the set S of all sets includes as m~nbers of the set U(s) and a(s) defined from S. The impredicative procedure in the Russell paradox stands out when the set T is elaborated thus: If we divide the set M of all sets into two parts, the first comprising those members which contain themselves and the second, those which do not, then we put T (defined by this division of M into two parts) back into M to ask which part of M it falls in. In the Richard paradox the totality of expressions in the English language which constitutes definitions of a function is taken as including the quoted expression, which refer to that totality. Poinoare judged the cause of the paradoxes to lie in these impredicative definitions;1 and Russell enunciated the

1Stephen Kleene, Introduction ~ Metomathematics (New York, 1952), p. 42. ______striction of the concept of set. Cantor had attempted to

23 same explanation in his Vicious Circle Principle: No to tality can contain members definable only in terms of this

totality or presupposing the totality.1 Thus it might ap pear that we have a sufficient solution and adequate insight into the paradoxes, except for one circumstance: Parts of

mathematics we want to retain also contains impredicative de finitIons. The impredicative definition principle amounts to a re

give the concept of set a very general meaning. A~ has been

shown, the theory of sets constructed on Cantor’s ger~ral con ception of set leads to . If the notion of set is restricted by Russell’s Vicious Circle Principle the re

sulting theory avoids the known antinomies. This n~ans not only seeks a method of avoiding the para doxes but explains them also. However, this method has been

criticized. In order to have set theory at all, there must be theorems about sets, and all sets constitute a set under Cantor’s definition. If not so, it must be said what other

definition shall be incorporated instead or Cantor’s de finition must be supplemented with some further criterion to determine ~then a collection of objects as described in his

definition shall constitute a set.

Howard Eves, An Introduction To The Foundations and Fundamental Concepts ~ MathematicsTNew.York, 1950), p. 284. 24

Logicism

The chief expositors of the logistic school are Ber— trand Russell and . The logistic

thesis is that mathematics is a branch of logic. Since the theory of classes is an essential part of logic, the idea of reduci~g mathematics to logic certainly suggests itself.

The main literature of the logistics is the Principia Mathematica0 This work contributes proposals f©r solutions

to the paradoxes of set theory. To avoid the contradictions of set theory, Principia Mathematics employs a “theory of types.” This “theory of types” was established for the pur pose of excluding impredicative definitions. Roughly, this is as follows: The primary elements constitute those of type 0; classes of elements of type 0 constitute those of type 1; classes of elements of type 1 constitute those of type 2; and so on. in applying the theory of types, one follows the rule that all the elements of any class must be of the same type. Adherence to this rule precludes impredicative definitions and thus avoids the paradoxes of set theory. In order to obtain the impredica~ive definitions needed to establish analysis, an “axiom of ~educibility” had to be introduced. It states in essence that to any property belonging to an order above the lowest, there is a coextensive property (one 25 possessed by exactly t1~ same objects) of order 0.1 If only definable properties are considered to exist, then the axiom means that to every impredicative definition within a given type there is an equivalent predicative one.

Whether or not this thesis has been established seems to be a matter of opinion. Although some accept the pro gram as satisfactory others have fou~ many objects to it. The “axiom of reducibility” drew forth severe criticism be

cause of its non-primitive and arbitrary character. Much

of subsequent refinement of the logistic program lies in the attempt to devise some method of avoiding the disliked “axiom of reducibility.” Also, the logistic thesis can be

questioned on the ground that the systematic development of logic presupposes mathematical Ideas in its formulation, such as the fundamental idea of interatjon whIch must be used in

describing the theory of types.

Intuitionjsm

The intuitjonjst school (or a school) originated about 1908 with Dutch mathematicians L. E. J. Brouwep. For the intuitionist, an entity whose existence is to be proved must be shown to be constructible in a finite number of steps; it is not sufficient to show that the assumption of the entity’s

1Stephen Kleene, op. cit., pp. 44-45, 26 nonexistence leads to a contradiction. This means that many existence proofs fcund in current mathematics are not acceptable to the intuitionists. An important instance upon the intuitionists’ insistence upon constructive procedure is in the theory of sets. For the intuitionists, a set can not be thought of as a ready made collection, but must be considered as a law by means of which the elements of the set can be constructed in a step— by-step fashion. This concept rules out the possibility of such contradictory sets as “the set of all sets.”

There is another remarkable consequence of the intui tionists’ insistence upon finite constructibility, and this is the denial of the universal acceptance of the law of ex cluded middle. This principle of , valid in reasoning about finite sets is not accepted by Brouwer for infinite sets. The law in general form says, for every pro position A, either A or not A. Now let A be the , there exists a member of the set D having the property P. Then, not A, is equivalent to every member of I) that does not have the property P. In other words every menber of D has

the property not - P. The law applies to this A, hence gives either there exists a member of D having the property P, or

every member of D has the property not - P.

For definiteness, let us specify P to be a property such that, for any given member of 0, we can determine whether 27

that member has the property P or does not. Now suppose D is a . Then we could examine

every member of D and thus find a member having property P,

or verify that all members have the property not - P. There might be practical difficulties if 0 is a very large set having say a million members. But the possibility of com pleting the search exists in principle. It is this pos sibility which Brouwer makes the law of the excluded middle a valid principle for reasoning with finite sets 0 and pro perties P of the kind specified.

For an 0, the situation is fundamentally different. It is no longer possible in principle to search through the entire set D. Brouwer blames this state of af

fairs on the sociological development of logic. The laws of logic emerged at a time in man’s evolution when he had a good language for dealing with finite sets, he then later made the mistake of applying these laws to the infinite sets of mathematics, with the result that the paradoxes arose.

AccordIng to Weyl, 1946, “Brouwer made it clear, ..., that there is no evidence supporting the belief in the ex istential character of the totality of all natural numbers....

The sequence of numbers which grow beyond any stage already

reached by passing to the next number, is a manifold of pos sibilities open toward infinity; it remains forever in the 28 status of creation, but is not a closed realm of things existing in themselves. That we blindly converted one into the other

is the true source of our difficulties - including the antinom

ies — a source of a more fundamental nature than Russell’s Vicious Circle Principle indicated... ~

However, there is this question: How much of existing mathematics can be built within the intuitionists restrictions? If all of it can be so rebuilt without too great an increase in the labor required, then the present dilemma of the foundations of mathematics would appear to be solved. Intuitionist mathe— natics is considered to be less powerful and in many ways it is more complicated to develop. Another fault found is that too much that is dear to most mathen~ticjang is sacrificed. Meanwhile, in spite of present objections raised toward their method, it is generally conceded that its methods do not lead to contradic tions.

Formalism

The formalist school was founded by David Hubert. Its thesis is that :i~athematics is concerned with formal symbolic system. It asserts that the ultimate base of mathematics does not lie in logic but only in a collection of prelogical symbols

1lbid,, pp. 48-49. 29 and in a set of operations with these. The formalist point

of view was developed by Hubert to meet the crisis caused by the paradoxes in set theory.

In order to salvage classical mathematics from the critici-. sms proposed by the intuitionists Hubert proposed a program

which we state: classical mathematics shall be for~nulated as a formal axiomatic theory, and this theory shall be proved to

be consistent, that is, free from contradioticins. It is with proofs that the formalists oroposed to solve the existing paradoxes.

Hubert asserted that freedom from contradictions is guaranteed only by consistency proofs, and the older consis tency proofs based upon interpretations and models usually merely shift the question of consistency from one domain of mathematics to another.1 Hubert, therefore, conceived a new direct approach to the consistency problem. Much as one may prove, by the rules of a game, that certain situations cannot occur within the game, Hubert hoped to prove that a contra dictory formula can never occur. In logical notat~9n, a con tradictory formula is any formula of the type F1\F where F is some acceptable formula of the system. If one can show that no such contradictory formula is possible, then one has es tablished the consistency of the system.

‘Howard Eves, op. cit., p. 290. 30 The above development is called, by Hubert, the “.” A detailed exposition of this theory was to have been given in Hubert and Bernays Grundla~en der Mathematik. However, unforeseen difficulties arose, and it was not possible for them to complete it. As of yet the problem of consistency remains refractory. As a matter of fact, the Hubert program, at least as originally envisioned by Hilbert, appears to be doomed to failure. This was brought out by Kurt Godel in 1931. By un impeachable methods Godel showed that it is impossible to prove consistency of the system by ri~thods belonging to the system. Hence, it is seen, that such a system, one that lacks com plete evidence of proof would be totally invalid and inadequate to apply to the solutions of the paradoxes. Hilbert’s ideas about consistency proof seemed acceptable enough but lack of formal proofs invalidates its use. CHAPTER IV

SUMMARY AND CONCLUSIONS

In set theory a field of inquiry was entered - a field which today places the various areas of mathematical study on a firmer foundation, and which also has enriched them. The

~ to bring to light new concepts was made use of in this penetrating creation of the human mind. But in this freedom of creation care was not taken in keeping the paradoxes from appearing. More and more direct attack on the paradoxes was replaced by a thorough investigat-. ion of the foundations of logic and mathematics. The fact that fundamentally the concern is with a problem belonging to logic necessitates an investigation of logic. This was especially true because actually paradoxes were not new in logical theory. It has been demonstrated that there is a belief that an axiom which excludes such a paradoxical aggregate need only to be inserted in mathematics. This avoids the paradoxes but does not solve them. For the logistic approach, which reduces mathematics to logic, there is needed some explanation of types of predicates or a new definition of set. Russell’s theory of types is an attempt to reduce logic of these illigitimate to— talities.

31 32 Another method of solution lIes in the attempt to set up a definition of set which will not lead to contradictions. This becomes the problem of consistency to which Formalists direct their attention. Since some aggregates which lead to paradoxes are infinite aggregates, it is possible to avoid them by denying the ap plicability of the method of proof by means of which paradoxes are deduced, to infinite aggregates. Such is the position of the Intuitiorijsts.

Weyl has said that mathematics is the science of the infinite.]- It has been felt that the theory of the infinite has led to a precision in the problem of the nature of the in finite. But the precision seems to have generated problems in the very roots of the foundations of mathematics. The para doxes that have been uncovered in the theory of sets have proved to be a deterrent to the program of the theory toward its present acceptance. A final clarification of these pro— blems does not exist to this day.

]-Loui~ 0. Kattsoff, A Philosophy of Mathematics (Ames, 1948), p. 92. BIBLIOGRAPHY

Black, Max. The Nature ~ Mathematics. New York: The Humanities Press, 1950. Breuer, Joseph. introduction ~ ~Theory of S5t5. Engle-. wood Cliffs, New Jersey: Prentice-Hall, Inc., 1958. Cantor, Georg. Contributions to the Founding of the Theory of Transfinite Number. New York: Dover Publications, (n.d.). Eves, Howard. An Introduction ~ ~ Foundations ~ Funda mental Concepts of Mathematics. New York: Rinehard and Co., Inc., 1958. Fraenkel, Abraham. Abstract ~ Theory. Amsterdam: North— Holland Publishing Co., 1953. Kattsoff, Louis. A Philosophy of Mathematics. Ames, Iowa: The Iowa State College Press, 1948. Kelly, John. General Topo1o~y. New York: 0. Van Nostrand Co., Inc., 1955. Kleene, Stephen. Introduction ~ Metamathematics. New York: 0. Van Nostrand Co., Inc., 1952. Rosser, J. Barkley. Logic Mathematicians. New York: McGraw-Hill Book Co., 1953. Wilcox, L, H., and Kreshner, R. B. ~ Anatomy of Mathematics. New York: The Roland Press Co., 1950. Wilder, Raymond. Introduction ~ the FOundations ~ Mathematics. New York: John Wiley and Sons, Inc., 1952.

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