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Article 5453 of comp.ai.philosophy:
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>From: jcollier@ariel.its.unimelb.EDU.AU (John Donald Collier)
Newsgroups: comp.ai.philosophy
Subject: Re: Goedel's theorem proof without self-referencing?
Message-ID: <2358@ariel.its.unimelb.EDU.AU>
Date: 7 May 92 11:38:55 GMT
References: <1992Apr23.183732.25378@kum.kaist.ac.kr> <1992May4.214051.16767@hellgate.utah.edu> <76781@netnews.upenn.edu>
Organization: University of Melbourne, Australia
Lines: 34

In <76781@netnews.upenn.edu> weemba@sagi.wistar.upenn.edu (Matthew P Wiener) writes:

>This is nonsense.  Goedel's theorem is the assertion that a certain formal
>system (PA=Peano Arithmetic) is incomplete.  Goedel's proof was both
>ingenious and significant, but that does not make it the only proof.  Any
>independence result from PA will suffice.

>Gentzen's proof was proof-theoretic.  He analyzed the logical complexity
>of PA-based proofs, and showed that they only went so far.  So statements
>beyond that complexity were independent--no self-reference.

>Kirby and Paris gave a model-theoretic proof.  They analyzed non-standard
>models of arithmetic, and showed which properties of their models correspond
>to which fragments of PA.  When they went far enough, they had independence
>results.

I am not familiar with all the work you cite, and it has been over ten
years since I last studied mathematical logic, but I have to ask, do
any of the proofs you mention not rely on self reference in any the
theorems they rely on?

One of the things I thought I learned from Thompson and Boolos when
I was doing philosophy at MIT some twenty years ago or more was that
the only way you can get a handle on the difference in relative
sizes of the infinities involved is through an arguemnt involving
a self-referential statement. I would be really surprised to find
there is some other way to do this, to say the least. I am not, I 
hasten to add, an expert in mathematical logic, just someone who
uses it extensively.

-- 
John Collier 			Email: jcollier@ariel.ucs.unimelb.edu.au
HPS -- U. of Melbourne		  	Fax:   61+3 344 7959
Parkville, Victoria, AUSTRALIA 3052


