Re: (probably stupid) question on Fermat decidability

From: Arturo Magidin (magidin_at_math.berkeley.edu)
Date: 11/22/04 

Date: Mon, 22 Nov 2004 21:05:24 +0000 (UTC)

In article <cnm6ph$kmq$1@newslocal.mitre.org>,
Keith A. Lewis <lewis@PROBE.MITRE.ORG> wrote:
>mike_ferenduros@hotmail.com (Mike Ferenduros) writes in article <e6b5e269.0411191540.5a65594e@posting.google.com> dated 19 Nov 2004 15:40:20 -0800:
>>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?
>
>If you could prove that FLT was undecidable, that would prove it true, which
>would actually be a contradiction.

No. Because "undecidable" should really be "formally undecidable"; the
proof that it is true (in the standard model based on the fact that it
is formally undecidable) is not formalizable within the model, so
there is no contradiction. 

-- 
======================================================================
"It's not denial. I'm just very selective about
 what I accept as reality."
    --- Calvin ("Calvin and Hobbes")
======================================================================
Arturo Magidin
magidin@math.berkeley.edu

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