Re: Godel's Incompleteness and Nonmonotonic Logic

From: Aatu Koskensilta (aatu.koskensilta_at_xortec.fi)
Date: 08/25/04 

Date: Wed, 25 Aug 2004 23:13:08 +0300

Jamie Andrews; real address @ bottom of message wrote:
> 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.

This is nonsense. Gödel's completeness theorem does apply to infinite
sets of axioms and derivations from them. In particular, the deductive
closure of a set of axioms A is exactly the set D of its logical
consequences. This doesn't imply that the deductive closure of any
consistent set of axiom is complete in the sense that for every sentence
A, either A is in the deductive closure or the negation of A is. Some
are and some are not, but this has nothing to do with the completeness
theorem. 

-- 
Aatu Koskensilta (aatu.koskensilta@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
  - Ludwig Wittgenstein, Tractatus Logico-Philosophicus

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