To Michael Blackmon (in comment on Can ZFC prove "false theorems", and still be consistent? (was "Junk Theorems" follow up) , sorry I don't have the cookie to add a new comment to that post, maybe someone with enough points can transfer this):
Yes, the idea is that if there are two models of PA (one that affirms Goldbach's conjecture and one that refutes it), it's conceivable that ZFC disproves Goldbach's conjecture even though the conjecture is true in the standard integers. That just means that the $\omega$ in every model of ZFC turns out to be nonstandard, i.e. ZFC itself is unsound (though still consistent) and proves theorems that are false for the standard integers. This seems like an unlikely situation, but I don't see how it's nonsensical. Am I missing something?