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Article 5434 of comp.ai.philosophy:
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>From: weemba@sagi.wistar.upenn.edu (Matthew P Wiener)
Newsgroups: comp.ai.philosophy
Subject: Re: Goedel's theorem proof without self-referencing?
Message-ID: <76916@netnews.upenn.edu>
Date: 6 May 92 17:03:09 GMT
References: <1992Apr23.183732.25378@kum.kaist.ac.kr> <1992May4.214051.16767@hellgate.utah.edu> <76781@netnews.upenn.edu> <1992May5.154030.18664@sics.se>
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In-reply-to: torkel@sics.se (Torkel Franzen)

In article <1992May5.154030.18664@sics.se>, torkel@sics (Torkel Franzen) writes:
>The nugget of sense consists in the observation that some form of
>diagonalization is used in all proofs of Godel's theorem.

That can get pretty loose, so I won't argue the point....

>							   Indeed, in
>the case of the specific undecidable statement of Kirby and Paris,
>they proved it to be undecidable in PA by proving that it implies the
>consistency of PA,

I gather (from your mention of "specific") that you are thinking of the
Paris-Harrington theorem.  Kirby and Paris proved several undecidability
results (one of which inspired Harrington), and their methods were model
theoretical.  As for PH, Solovay and Ketonen showed that the modified
Ramsey numbers grow at the omega_1 level of the Grzegorszyk hierarchy,
and hence are PA independent via the results of Gentzen.

>		    and the proof of Godel's second theorem uses the
>Godel sentence.

Goedel's proof of G2I does.  There are other proofs, for example, the
model-theoretic one in Smorynski's HANDBOOK OF MATHEMATICAL LOGIC article,
or Gentzen's proof-theoretic analysis of consistency proofs.
-- 
-Matthew P Wiener (weemba@sagi.wistar.upenn.edu)


