>Andrew Boucher wrote: > > > >>Well ok, I'm guess I'm non-standard on this. Say Simpson doesn't put >>in the axioms of arithmetic, but just comprehension - I would call that >>second-order logic. (Second-order arithmetic minus the axioms of >>arithmetic is second-order 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. Hilbert-style or Gentzen-style derivations >or whatever fancy improvements on them there may have been, >all give you first-order logic, no matter what axioms you feed >in the front end. > >Throw in omega-rule and you're outside of first-order logic but >you haven't gotten to second-order yet. (omega-rule 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 omega-rule !
Anyway, here's Boolos (p. 7, Logic, Logic, and Logics). "There is a standard extension of the proof theory for first-order logic to second-order logic. The notion of derivation is changed only by the addition of new axioms, most importantly by the scheme of comprehension..."