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Article 5467 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: <77079@netnews.upenn.edu>
Date: 7 May 92 20:59:38 GMT
References: <1992Apr23.183732.25378@kum.kaist.ac.kr> <1992May4.214051.16767@hellgate.utah.edu> <76781@netnews.upenn.edu> <2358@ariel.its.unimelb.EDU.AU>
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Reply-To: weemba@sagi.wistar.upenn.edu (Matthew P Wiener)
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In-reply-to: jcollier@ariel.its.unimelb.EDU.AU (John Donald Collier)

In article <2358@ariel.its.unimelb.EDU.AU>, jcollier@ariel (John Donald Collier) writes:
>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?

Not in any obvious way that I remember.

>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.

If you identify a property that countable sets must have and that the
the unit interval [0,1] does not, you have shown that the reals are
uncountable.  I wouldn't call this self-reference.

The most well-known such property is zero measure (as a subset of [0,1]).

Gentzen's proof for the incompleteness of PA uses a more subtle property
to distinguish Con(PA) from PA-provable results.
-- 
-Matthew P Wiener (weemba@sagi.wistar.upenn.edu)


