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