
Re: Axiomization of Number Theory
Posted:
Jul 31, 2003 4:29 PM


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.)

