Re: Godel's Incompleteness and Nonmonotonic Logic

From: Stephan Lehmke (Stephan.Lehmke_at_ls1.cs.uni-dortmund.de)
Date: 08/26/04 

Date: 26 Aug 2004 11:55:31 GMT

In article <2p47gfFgp0b4U1@uni-berlin.de>,
        me@privacy.net (Jamie Andrews; real address @ bottom of message) writes:
>
> This *completeness* result does not extend to arithmetic
> because no *finite* set of axioms characterizes arithmetic.
> Induction on the integers is an axiom *schema*, not a single axiom.

I think it's exactly the other way round: `Formalizing' arithmetic
with an axiom schema which stands for a (countable) infinity of axioms
leads to a first order `formalization' which does not axiomatize the
naturals up to isomorphism.

The formalization leading to incompleteness is the second order one
with a single (second order) axiom.

regards
Stephan

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