I think you're mistaken in dismissing this as a "theorem". Once when I TA'd in a course "math for elementary school teachers" taught by Leon Henkin, he went over a proof of this theorem---I thought it was pretty revealing, (although I have to say it wasn't very appropriate for the future elementary school teachers, who didn't understand the point.)

I'm glad to hear that I was hasty in dismissing the theorematic nature of the claim. Such a nice claim should have a proof, even if it's rated M ("for (mathematically) Mature audiences only"). 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.) Jim