
Re: Axiomization of Number Theory
Posted:
Jul 31, 2003 5:03 AM


Andrew Boucher wrote:
> My impression was that I was using standard terminology, e.g Simpson > speaks of "SecondOrder Arithmetic" and Shapiro makes a case for > "SecondOrder Logic". Boolos uses this terminology as well. While > "deductive" secondorder Peano Arithmetic *could* refer to any r.e. > way as you say, in practice it is not widely used that way, to my > knowledge anyway. Instead it refers (again in practice) to the > specific deductive system I have described, with little and > bigletters (i.e. two sorts), with comprehension.
Well, but you didn't actually specify a deductive system. You just gave the axioms, not the rules. I assume you mean that some standard Hilbert or Gentzentype thing is to be done with the axioms, but you didn't actually say so.
If you *are* doing the usual Hilbert or Gentzen thing, then it seems clear to me that this is *first*order logic, though I absolutely agree that it's secondorder *arithmetic*. It's secondorder arithmetic because the intended interpretation involves sets of naturals, not just naturals, but the *logic* is firstorder (though twosorted).
> In any case, I > can't think of any other interpretation of what people mean should > they ask whether PA2 can prove such or such theorem.
I agree that the term "PA2" is standardly used for the firstorder theory of secondorder arithmetic.

