You could perhaps have a program that for every input you've tried gives the correct answer in polynomial time, but you haven't yet proved that it always will.

You could perhaps have a program that for every input you've tried gives the correct answer in polynomial time, but you haven't yet proved that it always will. Nevertheless, I don't see how the answer to P!=NP could depend on the choice of axioms for set theory. Programs, their inputs and the state of a machine after executing n steps can all be encoded as integers, so P!=NP can be expressed as a statement about integers, and we know what the integers are: we don't need any dodgy set theory axioms for that. If that's wrong, someone please explain how. There are lots of interesting unsolved questions about integers (Goldbach's conjecture, ...) but people don't usually suggest that the answer to those questions might depend on the Axiom of Choice. Or do they?