From: Eliezer S. Yudkowsky (sentience@pobox.com)
Date: Tue Apr 03 2001 - 01:26:28 MDT
hal@finney.org wrote:
>
> Goldbach's Conjecture is that every even number is the sum of two primes.
> If it is false, there must be an even number which is not the sum of
> two primes. If such a number exists, then by showing it we could show
> that GC is false and GC would not be unprovable (that is, it would be
> provably false). For GC to be unprovable, therefore, would mean that
> no such number exists. But this is simply the statement of GC itself,
> so to show it to be unprovable (undecidable) is to show it to be true.
> Hence you can't show GC to be unprovable.
And did not Godel's Theorem prove the Godel Statement, then? (Techically,
no; Godel proved the Godel Statement was true if and only if mathematics
is consistent, a proposition which itself cannot be formally proven.)
What you really need is for someone to say "I have proved the GC is
*Godel-undecidable*" - not "unproveable", "Godel-undecidable" - which
would be accepted as a proof of the Goldbach Conjecture.
Much more interesting would be a proof that, *if* GC is true, it must be
unproveable - not just Godel-undecidable, but Chaitlin-unproveable, which
means you can't even prove it by showing it to be undecideable. Has
anyone ever succeeded in showing this for any finite mathematical
statement, or, conversely, in proving that no proof of a finite
statement's "conditional Chaitlin-unproveability" is possible?
Hey, maybe someone could show that the statement "If Goldbach's Conjecture
is true, it must be Chaitlin-unproveable" is Godel-undecidable...
-- -- -- -- --
Eliezer S. Yudkowsky http://singinst.org/
Research Fellow, Singularity Institute for Artificial Intelligence
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