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[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
Was this argument first made by Bourbaki?
In Patrick Suppes _Axiomatic Set Theory_ 2nd (?) edition New York: Dover Publiscations, Inc, 1972, ISBN 0486616304, 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 settheoretical 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>
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