```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 evennumber > 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, considereach of the primes p < k and the numbers k - p.  For each k - p testwhether 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 PAdoesn'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
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