I believe the answer is no, but Scott Aaronson on his blog just gave in interesting argument to the contrary. This is in connection with the now famous paper Undecidability of the Spectral Gap, and what it would mean for a Hamiltonian describing an actual physical system to be mathematically undecidable.
"This is a case, in my opinion, where you should have the courage of your intuitions! :) Imagine, for example, that God assured us that Goldbach’s Conjecture was independent of ZF. Even then, I would say: Goldbach’s conjecture is true in the real world, and what I mean by that, is that every even integer greater than 2 really is a sum of two primes. (Indeed, I would know that, since if there were a counterexample, then it wouldn’t be independent of ZF)... suppose you reject this viewpoint as “Platonism,” and you say: no, for me there’s no fact of the matter about anything until it’s proven or disproven in ZF. In that case, I reply: why should you even say there’s a fact of the matter about whether something is or isn’t provable in ZF, or about whether ZF itself is consistent? You seem caught in an infinite regress, where the only way out is to admit that, while you might or might not have any intuition about what transfinite sets are that’s conceptually prior to axioms, at any rate, you do have such an intuition about the positive integers".
I believe "infinite regress" is not the right word, but what Scott seems to mean is that proof constructions in ZF (or other formal theory) are essentially equivalent to arithmetic. Let's grant that God is not a deceiver. Scott's ZF fanatic appears to be in a predicament. She already accepted God's assurance that there is a fact of the matter concerning provability of the Goldbach’s Conjecture. Is she now compelled to accept that there is a fact of the matter concerning its truth as well? Is her only way out to doubt God?