Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Posting-Version: version B 2.10 5/3/83; site eosp1.UUCP Path: utzoo!linus!decvax!mcnc!unc!ulysses!princeton!eosp1!mcmillan 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.)