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