8/10/2019 A.a.frankel Section4

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L O G I C A L P R O B L E M S O F F I N I T E N E S S A N D I N F I N IT Y

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Section 4 Logical Problems of Finiteness and Infinity

The Axiom of Choice

So far we have defined the finiteness of a

set

by means of the concept

of finite

number

(integer ; see p. 4). In this sense we shall more specifically

call a finite set, including the null set, an

inductive set

referring to mathe-

matical induction (sometimes called inference from n to n

),

which is characteristic of the theory of positive integers, and hence also

of inductive sets. Bertrand Russell has shown that the explicit reference

to integers can be eliminated

from this definition of finite sets, and Tarski

presented several definitions, seemingly quite different, which can be proved

to be logically equivalent to the definition of an inductive set and to

each other in an elementary sense, namely, without use of the axiom of

choice [see Tarski (1925)].

Yet there are other, fundamentally different definitions of finiteness and

infinity; we shall now examine one to which we have already referred on

p. 15. Surprisingly enough, this will lead us to a novel logico-mathematical

problem which for sixty years caused difficulties and heated discussions in

the world of science and which was finally solved, rather unexpectedly, in

1963.

We start with a theorem which looks quite harmless but which actually

conceals far-reaching depths:

T H E O R E M

1. Every infinite set I includes a denumerable subset.

Proof.

In accordance with the above concept of finiteness we here

consider infinite to mean noninductive. The theorem will be proved

by mathematical induction, as is usual in arithmetic.

1 The p rinciple of m athematical induct ion may be expressed briefly as fol lows : a set

which contains the member 1 and which, together with any element

n,

also contains

n 1,

contains

allpositive

ntegers . By replacing 1 by 0,

we ob tain the non-negative integers.)

2

Viz. by the following definitions a) a set of cardinals is called

hereditary

if the

fact that it contains

n

implies that it contains

n 1;

b) a cardinal is called

inductive

if it

belongs to every hereditary set that contains 1 ; c) a set is called

inductive

if i ts cardinal

is inductive. [Note that

n

1 in a) is taken in the sense of union, i.e ., of adding ele-

ments.]

Cf. the references in footnote 6 on p. 10.

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W e

start by choosing an arbitrary element i

of the nonempty set I,

and form the subset Si

= {i1

} of / which contains a single element. The

remainder, I = / S1

, is not empty, since otherwise / would contain one

element only and not infinitely many; therefore we may choose an arbi-

trary element i2

of I and form the subset

S2 = {i1 i2}

of I which contains

two elements. We may now inductively assume that after n such steps

(n denoting any positive integer) we have arrived at a subset

S

n

= {i i 2

i}

of / which contains n elements. Since the remainder, I

= /

Sn

,

is still

infinite, we can choose an arbitrary element i

n 4

.1

of /; by adding it to the

elements of S

n

we obtain a subset S

n+

= {i1

, i

2, , in, in 1}

3f

which

contains n 1 elements. Hence, by the induction principle, for

any

positive

integer n there exists a subset

S

of / which contains just n members; these

subsets form an infinite sequence (S

1

, S ,

, S

n, .

Moreover, they have

been chosen so that any

Sn

includes the preceding ones; that is to say,

for m