I am aware of foundational axiomatic systems for mathematics such as ZFC, though I am unsure which to use / which would be accepted as underpinning foundational axioms of which P=NP, P!=NP, or it's undecidability, could be proven. For instance, if I just came up with my own axioms that do not stem from a foundational system (that I'm aware of using), it becomes almost trivial to prove that P!=NP, at least in my experience (I could be fooling myself here).
I would like to pick axioms that the mathematics community would largely accept as reasonable (or, at least, better my chances of that), and I think I may need an appropriate foundational system of which my axioms can be founded upon.
I understand the difficulty of this problem to the degree that I know I will most likely not solve it, however, I'd be remiss if I didn't give it a try and get familiar with the intricacies of it. So, please, no comments on that.
Please point out the flaws in my questioning here.
Thanks!