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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 Crummer 

Fortunately (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