[WildUtah]

Everyone has been suspecting or assuming that P is a strict subset of NP for decades. Still, every time someone proposes a serious attempt at proving it, it's front page news and smart people will have to comb over the argument for months to have confidence that it's right.

The opposite proposition -- that P is equal to NP -- is widely doubted. But if there were a proof that P and NP were equal, anyone with a good data set could verify the proof in a few minutes. Just run the proof on a hard 3-SAT or whatever and observe the answer returning significantly before the heat death of the universe.

So what we think is true is insanely hard to verify but what we think is false is blazingly obvious to check. Chalk another one up for irony.

[Locke1689]

Actually that's only possible for one case -- a proof by counterexample. If, hypothetically, one were to prove that P=NP and only that an algorithm exists to solve NP-complete problems in P time, then that might be verifiable. However, even if that were the case, what if it turns out that the P-time algorithm complexity were O(n^100,000). That would be P-time, but not easily run for large NP-complete problems.

[Ramfjord]

P is a strict subset of NP. If you can solve the problem in polynomial time, then you can verify a solution simply by generating it.

[RiderOfGiraffes]

That means it's a subset and is trivial. To claim it's a strict subset means there's something in NP that's not in P.