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On the paradoxes of set theory
Prince Ilona Winston
Atlanta University
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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 f or
his assistance and advice in its development.
TABLE OF CONTENTS
Page
AC~OWL~G~NT.
. . . . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . .
ii
LIST OF FREQUENTLY OCCURRING SYMBOLS................. iv
Chapter
I.
II.
III.
IV.
INTRODUCTION,
• • • • • • • ,
• • • • • • • . • • .
, ,
. • • . . . . . . .
1
THEPARADOXESOFSETTHEORY......,.,,.,,,0..,
5
The Cantor Paradox.... . . . . .... . . . .. . . . . .. . . 5
The Russell Paradox. . . . . . . . . . . . . • • . . • . .
. .
8
The Burali— Forti Paradox......,....,.... 12
The Richard Paradox. ... ...... ...... .... .... 15
CAUSES OF AN) PROPOSES FOR SOLVING PARADOXES...
Axiomatic Set Theory.......................
Impredicative Definitions. .. .... . . . . ,......
Proposals of the Various Schools of
Mathematics
Logic isni. • . • . . . . . . . . . . . . . . . . . . . . . . . . . . .
Intuitionisni. . . . . . . . . . . . . . . . . . . . . . . . . . .
Fo~alis~. . . . • • • . . . . . • a • . • • • a a a a a a a . •
17
17
22
24
25
28
SUMMARY AND CONCLUSION.........,.....,......... 31
BIBLIOGRAPHY. .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iii
*
. . . . . . .
33
LIST OF FREQUENTLY OCCURRING SYMBOLS
AND NOTATIONS
U(S)
—
S
If S is~a set, then U(S) denotes the set of
subsets of S.
The cardinal number of the set S.
-
not equal to
inequality signs
G(S)
U
U
fl
fl
0
the sum of the sets belonging to the set S
whose members are sets.
-
is an element of
—
is not an element of
-
union
—
intersection
-
is contained in
-
is equivalent to
-.
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 f or 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
Jersey, 1958), p. 1.
1
Theory of Sets (New
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, f or 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
over all the elements of S
—
c[~}
where s runs
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
per subset of U(s~
)
≠
U(S) (that is, U0 to be a pro
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:
the subset U3 corresponding to s by
of U5.
~,
s is an element of
or s is not an element
(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~
the second kind.
,
s* should be either of the first kind or
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-.
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 cardinality 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).
U(S) ~ S.
Then it is clear that
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
But S is the set of all sets, so
a(s)
S
Sof 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.
not
Hence in particular
G(s)<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.
the latter example:
elements.
Russell’s paradox deals with
sets which do not have themselves as
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 argument, 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 symbol 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.
(a2)
{ ~:
.
.3’
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?
image
in
If so, m4iT, and
in
does not belong to its
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.
dox was not valid.
However, this para
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 ordinal number, 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
plies that
~
P
we have that C&,t 1 &~‘, which im
,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 Jules Richard 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 function.
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 enumeration 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
(0)
f (1)
-.~0
f
(o)
0
(2)
f
0
(3).
1
.f2 (0)
F~(i)
1
~
f2 (1)
f~’(2)
(3).
1
f2 (3).
.f3 (0)
f3. (i)
f3 (2)
£3 (3).
.f
.
.
.
(2) :t
.
.
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 natural number as value).
From the
specified enumeration of all the expressions in the English
language, by striking out those which do not define a numbertheoretic 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.
if we consider the enumeration, E0, E1, E2, E3,
...
For
which de
fines a number theoretic function, we can also write them in
a n~trix:
.E
.E
0
(o)
E
(0)
E
.E2 (0)
0
(i)
E
(1)
E
E2 (i)
.
.
.
(2)
(3).
0
E1 (3).
E2 (2)
E2 (3).
0
(2)
E
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.
E~ (n)
,~
It is clear that the new function
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)
by an expression In the English language.
,~
1
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 axioms 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
axiomatic system 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.
beside
There are two primitive (underfined) constants
““
and the other logical constant.
these is “&“ which is read
flj~
a member of.”
The first of
The second
19
constant is denoted,
“
all
.“
•.
such that
•••
..
:
...
and is read “the class of
It is the classifer.
In the axIom
system the term “class” does not appear in any definitions or
theorems, but the primary interpretatIon 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
describe the classifier.
“~
and now we must
The first blank in the classifier
constant is the variable and the second is the formula, for
example
4
X : X E,Y}
u £(X : XLY}
•
We ace ept as an axiom the statement:
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.
Axiom of Union.--.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
a set.
U
range f is
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,
and X u{x.~
&Y
0
£‘ 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 axiom of choice,
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 general set theory 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
member of M, and on the other hand the definition of
is a
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.
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 contradictions.
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.
criticized.
However, this method has been
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 Alfred North Whitehead.
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.
is as follows:
Roughly, this
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 classical logic, 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 proposition,
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.
the property not
-
P.
In other words every menber of D has
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 finite set.
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 infinite set 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
ies
—
-
including the antinom
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
consistency 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 “proof
theory.”
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.
not solve them.
This avoids the paradoxes but does
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.
Humanities Press, 1950.
New York:
The
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.
Holland Publishing Co., 1953.
Amsterdam:
Kattsoff, Louis. A Philosophy of Mathematics.
The Iowa State College Press, 1948.
Kelly, John. General Topo1o~y.
Inc., 1955.
New York:
North—
Ames, Iowa:
0. Van Nostrand Co.,
Kleene, Stephen. Introduction ~ Metamathematics.
0. Van Nostrand Co., Inc., 1952.
Rosser, J. Barkley. Logic
Mathematicians.
McGraw-Hill Book Co., 1953.
New York:
New York:
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.
33