Date: Apr 13, 2013 2:31 PM
Author: Frederick Williams
Subject: Re: Matheology § 224

Nam Nguyen wrote:
> On 13/04/2013 9:57 AM, Frederick Williams wrote:

> > Nam Nguyen wrote:
> >

> >> But if GC is undecidable in PA, there's no proof left in FOL but
> >> _structure theoretically verifying_ the truth value of GC in
> >> this structure.

> >
> > If GC is undecidable in PA, then it's true.
> >

> I've already explained to Peter et al that this isn't true.

The reasoning is elementary. If GC is false then there is an even
number > 2 that is not the sum of two primes. Call that number n.
Consider each number k = 4, 6, 8, ... in turn. For each k, consider
each of the primes p < k and the numbers k - p. For each k - p test
whether it is prime. If it is, then k is a witness to GC being false.
The above algorithm will terminate because k is bounded by n n.

PA will prove ((k is an even number > 2) & (p and k - p are primes)).
I.e., PA decides in favour of ~GC.

We have proved that, if GC is false, PA decides it. Hence, if PA
doesn't decide GC, it is true.

Perhaps you'd like to spell out your proof of the opposite?

When a true genius appears in the world, you may know him by
this sign, that the dunces are all in confederacy against him.
Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting