> 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.)