|
|
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 "Second-Order Arithmetic" and Shapiro makes a case for
> "Second-Order Logic". Boolos uses this terminology as well. While
> "deductive" second-order 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
> big-letters (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 Gentzen-type 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 second-order *arithmetic*. It's second-order arithmetic
because the intended interpretation involves sets of naturals, not
just naturals, but the *logic* is first-order (though two-sorted).
> 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 first-order
theory of second-order arithmetic.
|
|
