> 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.