Ah. ZFC is a particular collection of axioms for set theory, which, due to the foundational role of set theory in modern mathematics, means it serves as a particular foundational theory for mathematics as a whole. One could, of course, pick a different set of starting axioms to work with, and various people do, but none are nearly as popular as ZFC. Among the modern day mathematical community, ZFC is pretty much taken as the gold standard for mathematical proof; i.e., a statement is considered proved, by the mainstream mathematical community, when it’s been given a proof which can be carried out from the axioms of ZFC.
However, like any other formal theory, if ZFC is consistent, then there are statements which it neither proves nor disproves. (This is the essence of Goedel’s Incompleteness Theorem, although, technically, Goedel’s original result required further assumptions which were only later shown to be unnecessary by Rosser). And, for all we know, P = NP could be one of those; indeed, a great many people believe, for various reasons, that it is. So it is possible that there are no proofs or disproofs of P = NP which would be accepted by the current mathematical community according to their current standards.
The tricky thing about P != NP is that even a counterexample would require some proof to demonstrate that it is a counterexample. That is, suppose P does equal NP, so that there is some polynomial-time computer program which solves the Traveling Salesman Problem. God could present us with the source for such a program and the polynomial describing how long it takes to run, but in itself, this would not be enough; we would still have to be convinced that the program always gave the correct answer. We would still need a kind of proof.
It is conceivable that the independence of P = NP from ZFC could be proven from ZFC (modulo the assumption of ZFC’s consistency) or from some other set of immediately plausible axioms, thus providing God with a path to bestow upon us clear evidence of this fact. But it’s also conceivable that there might not be any such path. There might be A) no proof from immediately plausible axioms of P = NP, and B) no proof from immediately plausible axioms of Fact A, and C) no proof from immediately plausible axioms of Fact B, etc. God might know the truth value of P = NP, and have no simple evidence to provide us with for it, and no way of accounting for this lack of simple evidence, other than his own assertion.
So that’s why I don’t think the truth value of P = NP is a great example of verifiable information God could give us. Among mathematical statements, something like the Goldbach Conjecture would be better: if it’s false, then God could provide a counterexample which we could easily verify, and if it’s true, then at least we could gain some inductive confirmation of this by brute-forcely checking up to larger and larger bounds. And something like “Which, if any, side can force a win in chess?” would be even better, in a way: it would take us a damn long time to check, but whatever answer God gives, we could eventually exhaustively confirm.
Of course, extramathematical statements would be even better. “Hey, I lost my glasses. Could you tell me where they are?”. And demonstrations better than that: “Hey, I lost my glasses. Could you make me a new pair?”. But God is probably too lazy.
No problem. Nothing makes me happier than talking to people about this stuff.