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