Re: (probably stupid) question on Fermat decidability

From: Ted Hwa (hwatheod_at_xenon.Stanford.EDU)
Date: 11/20/04 

Date: Sat, 20 Nov 2004 04:31:30 +0000 (UTC)

Mike Ferenduros <mike_ferenduros@hotmail.com> wrote:
: I was reading Simon Singh's book on Fermat's last theorem, and got a
: big confused by one passage - he mentions the possibility that the
: theorem was undecidable (although obviously it turned out not to be).

: What confused me was this: The theorem couldn't be false but
: undecidable, since falsehood implies the existance of a definite
: counter-example. So if it's undecidable then it must be true, which
: contradicts it being undecidable. So you get a contradiction,
: therefore it cannot be undecidable.

: Would anyone care to point out where I'm going wrong here?

There is a difference between "truth" in an absolute system, and
"decidability" (synonymous with "provability" in this context)
in a first-order axiom system. See an old post of mine

http://www.math.niu.edu/~rusin/known-math/97/goedel

for a more detailed discussion. Usually "decidability" refers
to a first-order axiom system, where as "truth" in the 'absolute'
natural numbers is defined by a second-order set of axioms. So
a given statement could be true in the natural numbres (provable
by second-order axioms that define the natural numbers) but not
decidable (provable by the first-order axioms we are working with).

Ted