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From: mcmillan@eosp1.UUCP (John McMillan)
Newsgroups: net.ai
Subject: Re: Fermat's Last Theorem & Undecidable Propositions
Message-ID: <578@eosp1.UUCP>
Date: Mon, 13-Feb-84 10:50:48 EST
Article-I.D.: eosp1.578
Posted: Mon Feb 13 10:50:48 1984
Date-Received: Tue, 14-Feb-84 01:50:04 EST
References: <5683@mcvax.UUCP>
Organization: Exxon Office Systems, Princeton, NJ
Lines: 31

Paul Vitanyi has committed (it seems to me), a simple error of logic.

To Paraphrase him:

Suppose Fermat's last theorem were undecideable; then there can be
no counter examples; in which case the theorem is true.
If it is true, it is not undecideable.

(End paraphrase)

Please note that TRUE and PROVED are two entirely different things.

First, how can we tell whether there are no counterexamples?
If we can PROVE there are no counterexamples, we have proved the
theorem.  If we merely fail to find counter examples, we DON'T KNOW THERE
ARE NO COUNTEREXAMPLES.  Since we are dealing with an infinite set
(integers), there is no way to exhaustively search the set, looking for
counterexamples, without rogorously proving the theorem.

Now in addition, we must take into account what Goedel's theorem proved
for the set of integers and all sets embedding them -- it is possible for
a theorem to be true, and yet for the rigorous logical system containing the
theorm to be unable to formally prove the theorem.  One might think, in such
cases, that an axiom could be added to the system so that all such true
threorems can be proved, but Goedel showed that an inifinite number of axioms
would be required.
					- Toby Robison
					allegra!eosp1!robison
					decvax!ittvax!eosp1!robison
					princeton!eosp1!robison
					(NOTE! NOT McMillan; Robison.)