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Topic: [HM] Axiom of Infinity
Replies: 5   Last Post: Apr 11, 2006 1:44 PM

 Messages: [ Previous | Next ]
 James Landau Posts: 4 Registered: 12/24/05
[HM] Axiom of Infinity
Posted: Feb 26, 2006 7:26 PM

Several weeks ago Alexander Zenkin (alexzen@com2com.ru), in a post to HM
which unfortunately I did not keep, said that Bourbaki stated the need for
Axiom of Infinty: There exists an infinite set

In Patrick Suppes _Axiomatic Set Theory_ 2nd (?) edition New York: Dover
Publiscations, Inc, 1972, ISBN 0-486-61630-4, there appears on page 138 the
following

<begin quote>
Because we cannot prove the existence of the set of natural numbers or of
any other infinite set, we cannot define the standard binary operations of
srithmetic as proper set-theoretical functions. <snip> both for the theory
of denumerable sets and for the theory of the real numbers in the next
chpter, the existence of the set of natural numbers is essential.<snip>
We introudce at this point the axiom of infinity
(there exists A) [0 is an element of A & (for every B) (B is an element of
A - - >
B union {B} is an element of A)].

The attempt to prove the existence of an infinite set of objects has a
rather bizarre and sometimes tortured history. Proposition No. 6 of
Dedekind's famous Was sind und was sollen die Zahlen?, first published in
1888, asserts that there is an infinite system. (Dedekind's systems
correspond to our sets.)
[footnote] A similar argument is to be found in Bolzano [Paradoxien des
Unendlichen, Liepsiz 1851] section 13
<end quote>

Greetings from southern New Jersey, where winter has finally settled in.
- James A. Landau
Test Engineer
Northrop-Grumman Information Technology
8025 Black Horse Pike, Suite 300
West Atlantic City New Jersey USA 08232

Date Subject Author
2/26/06 James Landau
3/6/06 Martin Davis
3/18/06 Yaakov S. Kupitz
3/19/06 Jon Borwein
3/14/06 James Landau
4/11/06 Jose Ferreiros