We know $NP \neq P$ from a lot of point of view like empirical reason,or theoretical reasons such as finite model theory or descriptive complexity.Although we find so many reasons to believe $NP \neq P$,we have not found any proof for it now.

So any reason to believe that $NP \neq P$ is unprovable in ZFC?Any result about unprovability of $NP \neq P$ in ZFC or weaker system than ZFC.

Bulletin of the EATCS81 (2003): 109-136. (PDF download link). Or better:Quantum Computing Since Democritus. $\endgroup$ – Joseph O'Rourke Sep 11 '14 at 0:48