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.
The basic raw material of logic is formed from statements considered to be either true or false. A statement of this nature is called a proposition. Propositions will be denoted by the letters P, Q, R, ...... Examples of propositions are: (1) "Tennessee is one of the fifty states of the United
States" and (2) "The number seven is smaller than the niimber three". The first is true and the second is false. Let the first proposition be denoted by P and the second denoted by Q.
It is possible to combine P and Q to read thusly; "Tennessee is one of the fifty states of the U.S. and the number 7 is smaller than the number 3". Such a combination is called a conjunction and written P^Q. Such a proposition is true of course when and only when both P and Q are. true.
The propositions P and Q may be combined to form one proposition by use of the word or. Por example, "Tennessee is one of the fifty states of the U.S. or the number 7 is smaller than the number 3". This form of combination is called disjunction and is vnritten PVQ. Such a proposition is false only when P and Q are both false.
By using the v/ords and then a new proposition may be formed from P and Q. Such a proposition would read*. "If
Te:^essee is one of the fifty states of the U.S., then the 8
nimber seven is smaller than the number three". This is written sjrmbolically as P—> Q meaning if P then Q. Such a proposition is false when and only when P is true and Q is false. This is called implication.
Another combination of P and Q can yeild still another proposition by use of the words ^ and only if. Por example,
"Tennessee is one of the fifty states of the TJ. 3. if and only if the number 7 is smaller than the number 3". This is re¬ presented symbolically as P<—>Q and read "P if and only if
Q". It is called equivalence and represents the proposition
that is true when and only when both P and Q are true or both
P and Q are false.
Any single proposition may be changed to form a new proposition by use of the word not. The proposition P be¬
comes "Tennessee is not one of the fifty states of the U.ST."
This is written -P and is called the negative of P, It is read* "Pt is not the case that P"' or "It is false that P".
It represents the proposition that is false if P is true and ture if Pis false.^
It seems necessary at this point to call attention to
the fact that the logical meanings of conjunction, disjunc¬
tion, implication and equivalence may differ slightly from
g Symbols for conjunction, disjunction, implication, negation and equivalence may not appear the same in all dis cussions on logic. 9
7 the meaning usually associated with them.
Quantifiers
The concept of a quantifier will be introduced in order to facilitate the symbolic formulation of the axiom. Consider the statements "All v/omen are beautiful"; "Some women are in¬ telligent"; "No women are ignorant". Such propositions refer to collections or parts of collections of women. As previous¬ ly stated, such collections form "sets". In the examples just given, women form a set and statements made about portions of the set involved the words "all", "some", and "no". Such words are called quantifiers since they tell how many. Exam¬ ples of two mathematical propositions which involve quantifiers are the following (1) a^ - b^ = (a-b)(a+b) and (2) m^ - 3m 2 2 -4 = 0* The first says for all numbers a and b, a - b = 2 (a-b)(a+b). The second says for some numbers m, m - 3m - 4
= 0. The use of a quantifier should always enable one to answer questions of the type "all what" or "some what". It 2 does not suffice to say, for example, "For all m, m is positive". This is true only with certain restrictions on m.
Thus it may be said "Eor all m, m G E, m^^ 0, m is positive."
Use will be made of only two quantifiers, namely, the imiversal quantifier meaning "for all x" denoted by (V x)
7 Eor further discussion see Eves Howard and Newson Carroll, An Introduction to the Foundations and Fundamental Concepts of laaxnemaoics, CTTSW Yorh^ '195ts) p« ' 10
and the existential quantifier meaning "for some x" written
S3niiholically ( 3 x). The scope of a q.uantifier means the quantifier itself together with the smallest proposition
O following the quantifier. This smallest proposition will be enclosed in parentheses. The proposition (1) above may be written as follows; (Va)(V b)(a^-b^) = (a+b)(a-b), while
(2) is written C 3 m)(m^-3m-4 = 0).
Bound and Bree Variables
Sometimes it is necessary to analyze the character of the propositions involved. The form of a proposition is important g in the case of propositional functions. "X is a dog" or "M is the sister of N" are not'.called propositions since they contain variables. Every variable has associated with it a valid domain of values. A propositional function is defined as a statement containing one or more variables which become propositions when specific values are substituted for the variable. A variable occurring in a statement may be boimd or free.
An occurrence of a variable in a formula is bound if and only if this occurrence is within the scope of a quantifier using this variable. An occurrence of a variable in a formula is free if not bound. Finally, a variable is a
8 _Patrick Suppes, Axiomatic Set Theory, (Hew Jersey, I960) pp. 4-5. 9 A function is a relation R whereby to each object there is at most one . 0,'feject y such that x has the relation R to y. 11
■bound varia'ble in a formula if and only if at least one occurrence is bound; it is a free variable in a formula ifQand only if at least one occurrence is free.
Consider the statement ( V x) ( 3 y) (x-
^*^Patrick Suppes, Axiomatic Set Theory, (New Jersey, I960) pp. 4-5. CHA.PTER III
THE AXIOM AlTD THE PAHADOX
Mathematicians had used collections in their work long before Cantor’s time but not in the manner in which Cantor used them. Cantor v/as dissatisfied with the concept of the infinite. It v/as because of this dissatisfaction that he de¬ vised the transfinite numbers. His work was not published until ten years later but as a result of the publication the infinite was understandable.
Today we laiow that Cantor as Hilbert has said, thereby ’created one of the most fertile and powerful branches of mathematics; a paradise from v/hich no one can drive us out! The theo¬ ry of sets stands as one of the boldest and most beautiful creations of the human mind; its construction of concepts and its methods of proof have reanimated and revitalized all branches of mathematical study. The theory of sets, indeed, is the most impressive example of the validity of Cantor's statenent that essence of mathematics lies in its freedom’.
Axioms in Cantor’s Set Theory
If the proofs of the theorems in Cantor’s initial work in set theory are carefully analyzed, this analysis will in¬ dicate that the theorems proved can be derived from three axioms.The first of these is,proposed by Zermel (1904),
12 Joseph Breuer, Introduction to the Theory of Sets, (Hew Jersey, 1958) p. 2.
^^Patrick Suppes, Axiomatic Set Theory, (Hew Jersey, I960) p. 5.
12 13
:lie axiom of choice which asserts that: given a class of lutually disjoint non-empty classes there exists a classY inch that for eachtf< inthe class oC 0 V* has exactly one lemher. I'he second, referred to as the axiom of extension-
,lity, asserts that: two sets are identical if they have the
;ame members. The third is "the axiom of abstraction which states that given any property there exists a set whose mem- sers are just those entities having that property"
Although the axiom of choice is still opposed by some secause it assumes the existence of the class V' without any lethod of constructing such a class, it is with the axiom of ibstraction that the writer is concerned. It is this axiom rhich led to the discovery of the paradox.
The Russell Paradox
All sets may be divided into two sets, those which con-
:ain themselves as elements and those which do not contain
:hemselves as elements. Por example, the set of all abstract
.deas is itself an abstract idea and hence contains itself as n element. The set of all teachers is itself not a teacher nd hence does not contain itself as a member. Consider the let X of all sets which do not contain themselves as elements,
'or example, let X contain as its elements sets like the set
^"^Ibid.., p. 5. 14
’ all teachers, of all automobiles, of all fish, etc. The lestion arises as to the type of the set X. Is X a member
' itself?
Assume X is a member of itself, i.e., X € X. Then by le assumption X is a member of the set of all sets which
'e not members of themselves. That is, X ^ X. This con-
’adicts the assumption that X 6 X. However, this contra-
.ction was arrived at only under the assumption that X e X. lus, it has not yet been established that a contradiction cists bety/een X G X and X 4* X. Since a contradiction appears r the assumption is allowed, the conclusion is reached that le assumption made was false. Hence X ^ X. How, since
^ X, X is not a member of the set of all sets v/hich are
)t members of themselves, i.e., X is not a set which is not
member of itself, hence X must be a set which is a member r itself, that is, Xe X and hence the paradox is obtained.
The Russell paradox has been popularized in many forms. One of the best known of these forms was given by Russell himself in 1919 and concerns the plight of the barber of a certain village who has enun¬ ciated the principle that he shaves all those persons and only those persons of the villagd who do not shave themselves. The paradoxical nature of this situation is realized when we try to answer the question, *Does the barber shave himself?’ If he does shave himself, then he shouldn't according to his principle; if he doesn't shave himselfthen he should according to his principle. ^
^^Howard Eves and Carroll Hewsom, An Introduction To The )undations and Fundamental Concepts of Ifethematics, (Hew Tork, i3Q) p. ^^85. 15
S^nubolic formulation Of The Axiom
And The Paradox
Let 0 (x) be a proposition V\rhere the variable x is not free, for example, let 0 (x) be the proposition that "x is not a member of itself". Then according to the axiom there exists some y for every x such that x belongs to y if and only if X is not a member of itself i.e.; (3y)(Vx)(xey
*c—y 0 (x)). This is a symbolic representation of the axiom of abstraction.
If 0 (x) represents the proposition asserting that x is not a member of itself, i.e., -(xex), an illustration of the axiom of abstraction is (3 y) (Vx) (xe y <—>--(x6x)).
In order to obtain the paradox all that is necessary is to take X = y. This implies that (y€y)^—> -(yey). This is eq.uivalent to the paradox (y e y) ^-(y£ y). Thus the axiom of abstraction and the Russell paradox are symbolically represented.^^
^^Patrick Suppes, Axiomatic Set Theory, (New Jersey, I960) p.6. CHAPTER IV
OH TEE IIOII-EXISTEHCE OP PARADOXES IH THE TtlEORY OP
SETS OR LOGIC
Paradoxes in mathematical logic are generally of three types, those purely mathematical, those that are logical and those due to linguistic tricks.Belonging to the first type is the paradox of Burali-Porti concerning the greatest ordinal number. The second type involves predicates not predicable of themselves. The third type was loaown to the
Greeks and concerns the liar. If a man says, ”I am lying”, if he is really lying his statement is true and he is there¬ fore not lying. On the other hand if he is not lying, then, he is lying when he says he is lying. Either assumption im¬ plies a contradiction. It is easily seen that the logical and mathematical paradoxes or contradictions are not distinguish¬ able.
Literature on paradoxes is plentiful. Although paradoxes seemingly have existed for centuries, the subject is still controversial. A recent discovery by a prominoit research professor of mathematics, Tiber Rado, of Ohio State University may possibly settle the controversial question of the exist¬ ence of paradoxes in the theory of sets which date back to
1897.
17 Por further discussion on types the reader is referred to Principles of Mathematics by BeJtrand Russell.
16 17
Lemma and Corollary
Assime there exists a non-empty set S. Let p he a bi¬ nary relation defined on S. This means that for every ordered pair (a,h) of elements belonging to 3 it can be determined whether or not the relation holds. (It is assumed that any binary relation either holds or does not hold). In case there exists elements, a,b,c, in the set such that apa, bpb, cpc, the elements are said to be reflexive with respect to the relation p . Por example, con¬ sider the set 3 of real numbers and the relation "=" (equal to), then a=a, b=b, c=c, for all a,b,c 3. A relation which is not reflexive is said to be irreflexive.
Examples of irreflexive relations are " ^ " (less than), •'> "
(greater than), "father of" and "in love with". Now let T denote the negative of the relation ^ , If /O is the relation " ^ " then ^ is the relation " ". Let p" X, xe3^. Now let Xq be a fixed element of 3, then consider ^ x/x p Xq, Xge 33 . 3uch a set is called a p-set. The following definition can. be made: A set AC 3 is said to be a p-set if and only if there exists some XqC 3 such that A = ^x/xp Xq, xe S J . Lemma. ZI i ^x/xp'x, xG 3^ is not a yo-set. (This means that 21 cannot be written as ^x/xpXo» XqC 3^ » where Xq is a fixed element of 3. 18
Proof (By contradiction). Assime that E is a p-set.
This implies that ZT = J^x/x p Xq^ for some fixed Xq e S.
Observe that by definition Eis a subset of S. Hence every element belonging to T. is either contained in E or is not con¬ tained in E. . Thus we have two cases, (1) where XpC E and
(2) where Xq^ ^ •
Case I. Xq e E , If E , then according to the assump¬ tion Xq p Xq. At the same time, by definition of E , Xq /S' Xq .
This is a contradiction. Hence E is not a f^-set if XoGZI .
Case II. Xq then by definition of E, Xq must be reflexive, i.e., Xq p Xq. By the assumption, x^ will be irreflexive if Xq^ E, i.e., Xq p" Xq. Again this is a contradiction. Hence E is not a /®-set when x^^ E . Since a contradiction occurs in both cases it follows that E is nc^a
p -set and hence the lemma is established.
The following corollary exists as an immediate conse¬ quence of the preceding lemma:
Corollary. ^ x, x e S 1 x/x/o Xq, Xq any element of S, X6 s3 .
Since it has been proved that E is not a /^-set the ineq.uality is established betweenE and the ^-set.
Application of Lemma and Corollary to
Bussell^s Paradox 19
Applying the lemma and corollary to the popularized
version of the Russell paradox, the non-existence of the
paradox will he shown.
Let S = ^ Set of all men in the village 3 • ^’or purpose
of brevity let all men in the village be denoted by x’s. Thus
S = ^ x/x is a man in village.} . Let ^ be the relation "is shaved by". Let the barber of the village be the fixed ele¬ ment of S. Denote this by Xq* Let ^ = ^ x/x P X, X € s} .
The set of all persons in the village shaved by the barber is
£ x/x Xq , X e S J . Row by the corollary it is seen that £" x./x~^ x, x €
^ x/x ^Xq, xes} . As a result of the inequality of these
two sets it is seen that the barber paradox does not exist
since it does not violate the lemma. This paradox exist if
and only if the lemma is violated.
Applicability of Lemma to Other Paradoxes
The Russell paradox is one of the most popular paradoxes in set theory or logic, hov/ever, other paradoxes just as important and as valid as Russell’s have been discovered.
Among these are the Burali-Porti paradox concerning the largest ordinal number. Cantor’s paradox concerning transfinite numbers,
Richard’s paradox which deals with the notion of finite de¬ finability, and Berry’s paradox pertaining to the division of whole numbers into two classes, definable in words containing 20 a finite number of syllables. The lemma and corollary can be applied to these paradoxes to show their non-existence just as in the case of the Russell paradox.
Although attempts have been made for several years to find a solution to the paradoxes, no one has been successful in finding a solution that has been imlversally accepted. It is believed by some that Tibor Rado has found not the answer to the paradox, but the startling fact that no paradox exists in the theory of sets. It will be interesting to note the reactions of renowned mathematicians to his findings when they are published. BIBLIOGRAPHY
Allendoerfer, C, B. and Oakley, C. 0. Princinles of Kath- ematlcs.. New York; McGraw-Hill Book Co., 1X95^1?) •
Breuer, Joseph. Introduction To The Theory of Sets. New Jersey: Prentico-Hall, Inc., {19^o).
Christian, Robert R., Introduction to Lo^i^ic and Sets. Pre¬ liminary ed. Boston: Ginn and Co.., (195^S'). Church, Alonzo.. Introduction to Mathematical Lo.^ic. Vol. I New Jerseys Princeton University Press, (1956).
Eves, Howard and Newsom, Carroll B., An Introduction to the Foundations and Fundamental Concents of Mathematics .■ New York: Rinehart and Co., (195b).
Kattsoff, Louis 0. A Philosophy of I'lathematlcs.- Iowa: Iowa State College Press, IT9^B).
Kleene, Stephen C.- Introduction to Metamathematics.- New Jerseys D. Van Nostrand Co., Inc., (19^2).
Rosenbloom, Paul. The Elements of Mathemtlcal Logic. New York: Dover Publications, Inc., (195377
Russell, Bertrand. Principles of Mathematics. Vol. I. London: W. W. Norton Co., (190377
Suppes, Patrick. Axiom-atic Set Theory. New Jersey: D, Van Nostrand Co., Inc., (I960).
Wilder, Raymond L. The Foundations of I-fathematics. London: Chapman and Hill, Limited, (195277