Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Posting-Version: version B 2.10.1 6/24/83 (MC840302); site mcvax.UUCP Path: utzoo!watmath!clyde!floyd!cmcl2!philabs!mcvax!paulv From: paulv@mcvax.UUCP (Paul Vitanyi) Newsgroups: net.ai Subject: Fermat's Last Theorem & Undecidable Propositions Message-ID: <5683@mcvax.UUCP> Date: Wed, 8-Feb-84 11:46:08 EST Article-I.D.: mcvax.5683 Posted: Wed Feb 8 11:46:08 1984 Date-Received: Sat, 11-Feb-84 07:18:57 EST Organization: CWI, Amsterdam Lines: 33 ***** QUOTE net.ai 365 ***** From: Charlie CrummerFortunately (or unfortunately) puzzles like Fermat's Last Theorem, Goldbach's conjecture, the 4-color theorem, and others are not in the same class as the geometric trisection of an angle or the squaring of a circle. The former class may be undecidable propositions (a la Goedel) and the latter are merely impossible. Since one of the annoying things about undecidable propositions is that it cannot be decided whether or not they are decidable, (Where are you, Doug Hofstader, now that we need you?) people seriously interested in these candidates for undecidablilty should not dismiss so-called theorem provers like A. Arnold without looking at their work. Is the 4-color theorem undecidable or not? --Charlie ***** UNQUOTE ***** Fermat's Last Theorem, Goldbach's conjecture, the Four Color Theorem (or Conjecture) and others of that type which can be disproved by a single counterexample can not be undecidable propositions in mathematics. Suppose Fermat's Last Theorem were undecidable. Then there cannot be a counterexample (four positive integers x,y,z > 0 and n > 2) such that x**n + y**n = z**n. Consequently Fermat's Last Theorem is true. Thus the assumption of undecidability of this individual statement implies that the statement is true. The assumption that Fermat's Last Theorem is undecidable and false is a contradiction. The assumption that Fermat's Last Theorem is decidable does not contradict either the falsehood or the truth of the Theorem. Paul Vitanyi