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Originally Posted by David Sklansky
Don't you mean "decidable" rather than "provable".
I meant "not proveable" as in independent. PA cannot prove it. PA cannot produce a counter example.
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If Goldbach can't be decided than there must no counterexample so Goldbach would be right.
Again, you have to specify "can't be decided" with respect to what axioms. The textbook model theory example is Goodstein's theorem. It is true for the standard integers, but independent of PA. So it is possible to construct a (nonstandard) model that satisfies the axioms of PA and produces a counterexample.
If this were the case for Goldbach, your statement "Goldbach would be right" is reasonable because Goldbach is a statement about the standard integers and not any model of PA. Of course for most "natural" conjectures" those are one in the same.
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However if you only can't prove that there is no counter example, then he may or may not be right.
Sure. That's more or less the situation we are actually in.
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(UNLESS there is some theorem that says that every conjecture of this ilk that is indeed true, has a proof of such. I was under the impression that I was told that there is such a theorem.)
Your impression certainly seems to be wrong. Unless "of this ilk" doesn't apply to certain Diophantine equations with no solutions but don't have proofs of this fact in ZF.