Jim Propp wrote: << Come to think of it, I was hasty in asserting that the axiom of infinity can't be proved. It seems to me that Cantor's construction of the counting numbers (where 0 is defined as the empty set, 1 is defined as {0}, 2 is defined as {0,1}, 3 is defined as {0,1,2}, etc.) deserves to be called a constructive proof of the existence of an infinite set. It certainly doesn't FEEL like circular reasoning, even if it's unclear how one would formalize it. (Someone who knows more about Bolzano, Dedekind et al. should chime in here.)
1. Which reasoning doesn't feel like circular reasoning? 2. Without the Axiom of Infinity, I don't see how the existence of the Cantorian natural numbers implies there's a set whose members they comprise. And I doubt one can prove even the existence of each such number without the A.I. though you can clearly prove that all numbers up to any N exist. 3. I'm not fond of the Axiom of Infinity, either, and it would be very nice if it could be inferred from more natural axioms. --Dan