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Warrigal comments on When does an insight count as evidence? - Less Wrong
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Sure it makes sense. Something is falsifiable if if it is false, it can be proven false. It's not obvious, given that P != NP, that there is a proof of this; nor is it obvious, given that P = NP, that for one of the polynomial-time algorithms for an NP-complete problem, there is a proof that it actually is such a thing. Though there's certainly an objective truth or falsehood to P = NP, it's possible that there is no proof of the correct answer.
Isn't this true of anything and everything in mathematics, at least in principle? If there is "certainly an objective truth or falsehood to P = NP," doesn't that make it falsifiable by your definition?
It's not always that simple (consider the negation of G).
(If this is your first introduction to Gödel's Theorem and it seems bizarre to you, rest assured that the best mathematicians of the time had a Whiskey Tango Foxtrot reaction on the order of this video. But turns out that's just the way it is!)
I know they get overused, but Godel's incompleteness theorems provide important limits to what can and cannot be proven true and false. I don't think they apply to P vs NP, but I just note that not everything is falsifiable, even in principle.