Gödel proved that there are true sentences that cannot be proved.
Suppose I told you that the Goldbach conjecture is one of those. (The Goldbach conjecture supposes that every even integer number can be expressed as the sum of two odd primes.)
Is that logically possible? (And, no, I haven't proved it!)
(In reply to From a pure logic standpoint...
That's not completely true: for example, there are infinite even integers greater than 2, but we don't have to test each of one to see whether they are prime or not -- we can prove it otherwise. Why should the GC require thus?