Re: Who thinks Goldbach's Conjecture is unprovable?

From: Todd Trimble (trimble1_at_optonline.net)
Date: 09/26/04 

Date: Sun, 26 Sep 2004 00:23:58 +0000 (UTC)

On 25 Sep 2004, Stephen J. Herschkorn wrote:
>Brian Quincy Hutchings wrote:
>
>>these are silly conundrums. if it were impossible to prove,
>>a counterexample would be found "eventually, if not sooner."
>>
>>
>
>False. It is possible that Goldbach's Conjecture is independent of
>ZFC, i.e., that Con(ZFC) implies Con(ZFC+GC) and Con(ZFC) implies
>Con(ZFC+not GC). (Just as the Continuum Hypothesis is independent of
>ZFC.) I suspect most mathematicians "believe" that this independence is
>not the case.
>
>--
>Stephen J. Herschkorn herschko@rutcor.rutgers.edu

Maybe I'm confused, but suppose that (not GC) is unprovable
(say in PA). Then it would seem that each even constant > 2 is
in fact the sum of two primes, for if there were a counterexample,
then there would exist a finite calculation which showed that
(e.g. use Eratosthenes sieve to list all primes less than the
constant, and check that no two summed to the constant).
I.e. that a proof of unprovability of (not GC) would perforce
rule out the possibility of such a finite calculation, no matter
how lengthy.

Am I missing something?

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