What is the P versus NP question asking? Is the P versus NP question asking "P = NP" or "ZFC |- P = NP" (or "|- P = NP" for that matter)? Because if I say P = NP, then I will be asked to prove it. But if the goal is "ZFC |- P = NP" then the result will not be useful because of the set theoretic assumptions of ZFC. So you may say the third choice matches our intuition, but it's not a (complete) question. If P = NP, then we are asked to prove |- P = NP and if P != NP we are not asked ~(|- P = NP) but |- P != NP. So what is the P versus NP question asking?
Actually this question can be asked for any question, it's not about P versus NP alone.

For the $1 million P vs NP prize, the Clay Mathematics Institute problem description does not even talk about proofs explicitly:

Problem Statement. Does P = NP?

http://www.claymath.org/millennium/P_vs_NP/pvsnp.pdf
A proof is any completely convincing argument; set-theoretic foundations are just one tool that helps us study mathematical objects.
