Agreed, definitely off topic. Fun to consider anyway :) Without considering many facts specific to P=NP, it could be feasible for it to neither true nor false

Agreed, definitely off topic. Fun to consider anyway :) Without considering many facts specific to P=NP, it could be feasible for it to neither true nor false in ZFC (i.e., some model of ZFC has P=NP, and some other has P!=NP), in which case you could take P=NP as another axiom. Other statements that work this way include the continuum hypothesis. If I'm not mistaken (and misstatements regarding P=NP are common, so don't take this blindly), P=NP has a substantial amount of additional structure to it due to it being a low-complexity arithmetical statement. In particular, I believe you're correct that in ZFC it is actually either true or false in all models, but not necessarily provably so from within ZFC. In this case you could try to take it as an axiom (e.g. ZFC+PNP), but you'd either wind up with ZFC itself or an inconsistent axiom set. As a related (maybe equivalent?) statement, P=NP has the property that if A is an axiom independent from ZF and P=NP (resp P!=NP) is provable in ZF+A then P=NP (resp P!=NP) is provable in ZF, again due to its nature as a low-complexity arithmetical statement, so either P=NP is unprovable in any meaningful extension of ZF, or it's provable in all of them.