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Article 5272 of comp.ai.philosophy:
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>From: torkel@sics.se (Torkel Franzen)
Subject: Re: Self-reference in Goedel's theorem
In-Reply-To: tomh.bbs@cybernet.cse.fau.edu's message of 24 Apr 92 21:36:56 GMT
Message-ID: <1992Apr27.075556.11677@sics.se>
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Date: Mon, 27 Apr 1992 07:55:56 GMT
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In article <0J9qJB1w164w@cybernet.cse.fau.edu> tomh.bbs@cybernet.cse.fau.edu 
writes:

   >I think self-reference is required in a Goedel type statement;
   >in order to create a sentence that is true but not provable one
   >needs to jump outside the system, so to speak, to create a
   >statement that has semantic content.  Any purely syntactic
   >statement that's true will be reachable from the axioms.

  This doesn't make a whole lot of sense. We know that any sound arithmetical
axiomatic theory we come up with will have infinitely many undecidable
statements of the form "the Diophantine equation P(x1,..xn)=0 has no
solution". The intended import of your distinction between statements
of this form that are "purely syntactic" and statements that have
"semantic content" is obscure.

  On the other hand, if we consider existing proofs of the
incompleteness of formalizations of arithmetic, it may be argued that
they all involve diagonalization of some kind.


