|
|
Re: Axiomization of Number Theory
Posted:
Jul 31, 2003 4:29 PM
|
|
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.)
|
|
