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OBJ (Application/Pdf) OIT TIIE AXIOM 01* ABSTRACTIOK Ain) RUS3ELL»S PARADOX V A THESIS SUBMITTED TO THE EACUDTY OE ATLAHTA UITIVERSITY IN PARTIAL PUIPILLMENT OP THE REQUIRMENTS POR THE DEGREE OP TIASTER OP SCIENCE BY RALPH SHIRLEY HAYNES DEPARTMENT OP LM.THEMATICS ATLANTA, GEORGIA AUGUST I960 Ip^ LcC \ SjO' 2^^ ACKIIOWLEDGEI-IEITTS The writer expresses sincere appreciation to hr. Lonnie Cross for suggesting this problem and for his helpful suggestions and counsel during its development. ii TABLE OP C0LTEET3 Page ACKIIOV/LEDGLIEIITS ii Chapter I. IETPlODUCTIOIT 1 II. JIATHEI.IATICAL LOC-IC AED ITOTATIOE 5 Sets and Subsets 5 Propositions and Basic Operations .... 6 Quantifiers 9 Bound and Free Yariables 10 III. TtlE AXIOM AEL TEE PARADOX 12 Axioms in Cantor^^s Set Theory 12 The Russell Paradox . »15 Symbolic Representation of the Axiom and the Paradox 15 lY. ON THE EOE-EXISTEECE OP THE PARADOXES IE TI-IE THEORY OP SETS OR LOGIC 16 Lemma and Corollary 17 Application of Lemma to Russell^s Paradox 18 Applicability of the Lemma to Other Paradoxes 19 BIBLIOGRAPHY iii CHAPTER I INTRODUCTIOH Mathematics is one of the most interesting and challeng¬ ing subjects known to mankind. This is due primarily to the fact that it involves logic, logic as a science was developed by Aristotle (384-322 B. C.) and the scholastic philosophers of the Mddle Ages. Aristotle was considered more as a logician than a mathematician because of his gneat interest in logic and his attempt to systematize classical logic. Probably because of the development of logic by philosophers, many leading colleges and universities have traditionally taught it in the philosophy department of the school. It has in recent years, come more and more to be included in mathematics courses. This is due largely to its usefulness to the subject. The inclusion of logic as a phase of mathematics has been a slow process. However, much progress has been made in recent years as a result of the combined efforts of logicians and mathematicians. Leibniz was the first to attempt to create a symbolic logic. This was between 1666 and 1690. Little was done to continue this development of symbolic logic until the latter part of the nineteenth centuiy when interest in mathematical logic was renewed.^ ^Howard Eves and Carroll Hewsom, An Introduction To The Foundations and Fundamental Concepts of Mathematics (New York. 1958), p.’26Q 1 2 All those persons who made contributions to the field can not he discussed here. The writer would like, however, to mention three or four outstanding men whose contributions to mathematical logic deserve to be mentioned in any work on logic. These persons are (1) George Boole (1815-1864) for whom Boolean algebra is named, (2) Augustus BeMorgan (1806- 1871) a contemporary of Boole, who did extensive work in the logic of relations, (3) Gottlob Prege (1848-1925) a German logician, who because of his desire to place mathematics on 2 a more sound foundation gave us a modern approach to logic. .George Cantor (1845-1918) introduced into mathematics the theory of sets. The discovery of paradoxes in Cantor*s theory of sets contributed more to the development of mathe¬ matical logic than any other discovery of a mathematical nature. Por centuries before Cantor, logicians had engaged themselves in the construction of paradoxes of a different nature. These were in many instances created merely for enjoyment or to determine to what extent logical systems were valid. In 1897, an Italian mathematician, Cesare Burali-Porti, found it difficult to accept one of Cantor's basic definitions and thus published the first paradox of set theory. This ^Ibid.,p. 261. 3 paradox, as first published, involved many technical terms. The technical implications of this paradox are not within the scope of this paper and will not be discussed here. In 1903» a paradox of a less technical nature was discovered by Bertr^d Russell (1872- ). His contradiction dealt only with the concept of set and was obtained by a purely logical deduction. It is not surprising, then, that the first reaction of some mathematicians to this matter was that it concerned 'logicians*, not 'mathematicians* so let the logicians find what was wrong and set their house in order! But this attitude v/as short lived, since, if mathematics is to use logic, any defect of logic is of concern to mathematics ■—especially when the defect is found in a concept of logic that is so frequently used in mathematical definition and proof! The natural result was a movement among mathe¬ maticians, slow at first but accelerating in recent years, toward the study of logic. And, whereas logic was traditionally a cut- and-dried rehash of Aristotle—with great concentration on the syllogism, etc., — it has today become a live and growing field of investigation, Imown under the name of symbolic logic or mathematical logic.^ Thus it is seen as a result of the concern over the Russell paradox, logic tended not to be associated solely with philosophy but with mathematics and other sciences. Since the discovery of the Russell paradox, many of the greatest mathe¬ maticians have devoted much time to the study of logic and set theory. Their efforts have in many instances been rewarding, for they have resulted in the publication of a vast and 5 •'^Raymond L. Wilder, The Foundations of Mathematics (London, 1952), p. 56. f 4 informative amount of literature in the field. It is the purpose of tliis paper to examine the Axiom of Abstraction, v/hose denial by Russell resulted in the discovery of the paradox. The v/riter will consider briefly the elemen¬ tary operations involving propositions which he feels are essential to the symbolic formulation of the axiom and paradox. ITo attempt will be made to explore the whole field of logic. The writer will endeavor to give a general idea of the natujre of mathematical-logic as it applies to sets and propositions of which we shall be concerned. After having derived symbol¬ ically the axiom and the paradox, a lemma will be stated and proved.^ This lemma and a corollary will be applied to the popularized version of the Russell paradox. The assertion will be that the Russell paradox fails to exist because it violates a basic lemma. '^This lemma, and its proof-was discovered by a Hungarian Mathematician, Tiber Rado. ^ CH4PTER II I^THEIIA.TICAL LOGIC AND ROTATION Sets and Subsets One of the most important terms found in tv\fentieth- century mathematics and logic is that of set or class. No attempt will he made to distinguish between set and class. They will be used throughout this discussion as synon3n]ious terms, ^he concept of set is not something new for we work with sets and often speak of sets in our everyday experiences. In paying income taxes, attention is given to particular sets of people; in obtaining welfare benefits from certain govern¬ mental agencies, attention is given to particular sets of people; in census work much use is made of the concept of set as people are grouped into sets on the basis of similar characteristics. Prom observation it is seen that in each instance the set under consideration was composed of a col¬ lection of objects. Thus vie arrive at a definition of a set as being a collection of objects. The objects which consti¬ tute the set are called elements or members of the set. This concept of set is very fundamental to mathematics. A set may be named in one of two ways. Pirst, a set may be described by listing its elements. Por example, the set consisting of the moon, a river and a baby or the set consisting of the letters of the alphabet, a, b, c, d and e. 5 6 Secondly, a set may be named by considering a condition wMch is satisfied by the elements of the set and no other elements. For example, the set S, of all integers greater than or equal to one but less than or equal to ten, is written: ^ ^ x/l ^ x ^ 10, x an integer^ , where x is a 5 representation for each element of the set. The reason for this notation is quite obvious, it facilitates the naming of sets when the number of elements is very large. An element is said to belong * € * to a set if it is an element of the set. If an element is not a member of the set then the element does not belong '^ ' to the set. In the set just described, 7®S but 12^3. A set *A* is said to be a subset of a set if and only if every element of ^A* is also an element of *B'. If B = £ x/l£ x^lO, X an integer | , then A x/4-x^9» x an integer 5 is a subset of B. This will be denoted by A<^ B. It follows that every set is a subset of itself. If a subset A of a set B is not the entire set we say that A is a proper subset of B. If a set has no members it is said to be empty or null. This is denoted by Thus the set of all human beings with three heads is a null set. Propositions and Basic Operations (5 ^This type notation will be employed in describing sets through out this paper. It is read: "The set S of all x's such that X is greater than or equal to one but less than or equal ten. 7 In order to develop symbolically in the next chapter the axiom of abstraction and the Rnssell paradox with some degree of brevity, a few logical symbols and a very brief dis¬ cussion of their meaning will be presented.
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