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Re: Axiomization of Number Theory
Posted:
Jul 31, 2003 5:57 PM
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Mike Oliver wrote:
>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..."
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