If David Hilbert woke up in 1000 years he wouldn't ask "Does P=NP?" but "what progress have we made on P=NP?". Once the Riemann Hypothesis is done with, it's done with, but P=NP or P!=NP opens up whole new lines of discussion.

That said, P=NP could be independent of ZF[1], and if that was shown we'd be scratching our heads asking what P=NP actually means.

[1] http://www.scottaaronson.com/papers/pnp.pdf