[FOM] From theorems of infinity to axioms of infinity (Michael Detlefsen)

[MartDowd at aol.com]

Dr. Ferreiros,
 
The existence of infinite sets is in fact a question which arises with  
regard to the natural numbers.  ZFC - infinity has the hereditarily finite  
sets as a model.  It does not have a finite model, analogously to PA, or  even 
Q (see 
_http://mathoverflow.net/questions/51703/is-robinson-arithmetic-biinterpretable-with-some-theory-in-lst_ 
(http://mathoverflow.net/questions/51703/is-robinson-arithmetic-biinterpretable-with-some-theory-in-lst) ).   The 
universe of discourse of arithmetic is infinite and contains infinite  
subsets, whose properties are readily studied using PA.  An axiom must be  added to 
set theory, so that infinite sets are elemens of the universe of  discourse.
 
- Martin Dowd
 
 
In a message dated 3/11/2013 3:16:11 P.M. Pacific Daylight Time,  
josef at us.es writes:

One can  actually show that the existence of infinite sets, and furthermore 
the  power set axiom, are intimately linked with the real numbers 
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