
Re: Axiomization of Number Theory
Posted:
Jul 31, 2003 5:57 PM


Mike Oliver wrote:
>Andrew Boucher wrote: > > > >>Well ok, I'm guess I'm nonstandard on this. Say Simpson doesn't put >>in the axioms of arithmetic, but just comprehension  I would call that >>secondorder logic. (Secondorder arithmetic minus the axioms of >>arithmetic is secondorder logic. Doesn't that sound logical ?) Is it >>really misleading you? Because if it does (and you are not alone), then >>obviously I should stop, just for the sake of clear communication. >> >> > >To me the important aspect of the "logic" is how inferences are >made, not the axioms. Hilbertstyle or Gentzenstyle derivations >or whatever fancy improvements on them there may have been, >all give you firstorder logic, no matter what axioms you feed >in the front end. > >Throw in omegarule and you're outside of firstorder logic but >you haven't gotten to secondorder yet. (omegarule is obviously >semantically valid  too bad journals are reluctant to publish >proofs that use it, paper prices being what they are today.) > > Well, I certainly don't want the omegarule !
Anyway, here's Boolos (p. 7, Logic, Logic, and Logics). "There is a standard extension of the proof theory for firstorder logic to secondorder logic. The notion of derivation is changed only by the addition of new axioms, most importantly by the scheme of comprehension..."

