I think that whether P is NP or not could turn out to be undecidable and in fact one could add either equality or inequality as an independent axiom. The theory of computational complexity is about asymptotic results as the size of the instances goes to infinity and involves `there exists` statements whose meaning when applied to infinite sets is typically ambiguous. For instance, it turned out to be a matter of choice whether or not you want the set of all subsets of real numbers between zero and one to equal or to be strictly larger than the set of so called measurable sets. And whether or not a set is measurable can be characterized in a very concrete way about the possibility of approximating it by unions of intervals. So the question of whether or not there exist non measurable sets turned out to depend on `what you mean by set`in a way which people hadn´t thought about before. I suspect the question whether or not P is NP will depend on what we mean by P and NP in ways which so far no one thought about.