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Topic: Axiomization of Number Theory
Replies: 52   Last Post: Aug 4, 2003 12:18 AM

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 Andrew Boucher Posts: 221 Registered: 12/3/04
Re: Axiomization of Number Theory
Posted: Jul 31, 2003 5:57 PM

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..."

Date Subject Author
7/25/03 Charlie-Boo
7/25/03 Charlie Johnson
7/25/03 Arief
7/25/03 Jeffrey Ketland
7/28/03 Charlie-Boo
7/28/03 William Elliot
7/28/03 Charlie-Boo
7/28/03 Andrew Boucher
7/29/03 Andrew Boucher
7/29/03 Andrew Boucher
8/4/03 Charlie-Boo
7/29/03 Pete Moore
7/29/03 Robin Chapman
7/29/03 David C. Ullrich
7/29/03 Robin Chapman
7/29/03 David C. Ullrich
7/29/03 Robin Chapman
7/30/03 David C. Ullrich
7/30/03 Robin Chapman
7/30/03 David C. Ullrich
7/31/03 Robin Chapman
7/29/03 George Cox
7/29/03 Per Eriksson
7/30/03 Charlie-Boo
7/30/03 Per Eriksson
7/31/03 Robin Chapman
7/30/03 David C. Ullrich
7/30/03 Per Eriksson
7/30/03 Mike Oliver
7/31/03 David C. Ullrich
7/31/03 Charlie-Boo
7/31/03 tchow@lsa.umich.edu
7/31/03 Per Eriksson
7/29/03 Arief
7/30/03 Keith Ramsay
7/30/03 David C. Ullrich
7/30/03 Andrew Boucher
7/30/03 David C. Ullrich
7/30/03 Andrew Boucher
7/30/03 Mike Oliver
7/31/03 Andrew Boucher
7/31/03 Mike Oliver
7/31/03 Aatu Koskensilta
7/31/03 Andrew Boucher
7/31/03 Mike Oliver
7/31/03 Andrew Boucher
7/31/03 Mike Oliver
7/31/03 Andrew Boucher
7/31/03 Mike Oliver
7/31/03 Aatu Koskensilta
7/31/03 Andrew Boucher
7/31/03 Keith Ramsay
7/31/03 Andrew Boucher