On Thu, Sep 3, 2009 at 8:09 PM, James Propp<jpropp@cs.uml.edu> wrote:

I think I can derive the principle of induction for |N from the Dedekind-completeness of |R, via the supplementary assumption that 1 is the smallest positive integer, but I'm wondering if there's a subtle fallacy.  Or maybe this is a standard derivation.

I'd love to see the details of this - it sounds plausible to me but I haven't seen this argument before.

Also, I'm wondering if one can derive induction from completeness without using the assumption that 1 is the smallest positive integer. My guess is "no", but I don't see off-hand a non-standard model that makes this clear.

Wait a minute ... if the positive integers are closed under + and *, then if 1 is not the smallest positive integer there are infinitely many between 0 and 1. I'd be hard pressed to still call them "integers", but I guess that might be your nonstandard model -- say, all fractions whose denominators are powers of 2 or something like that? I'm not sure exactly what you're taking as your axioms here, so I'm not sure if I'm on the right track at all. --Joshua Zucker