Ask HN: Is it possible that p != np as well as p = np

In such paradoxes, it's not that case that the final conclusion is that true = false. Any time that seems to be proven, it indicates that an earlier assumption was false.

Necessarily so. If you build a formal system where true = false (say, as one of the axioms, or in a proof), you can literally prove anything, including things that are obviously false in normal mathematics, and then immediately also disprove that very same thing -- so it's not useful to go in that direction.

Probably the closest thing is "nonmonotonic systems", including "nonmonotonic logic", where true or false is not a timeless absolute, but instead can be advanced or retracted over time, you might say. The late great John McCarthy worked in that area for decades.

The notions involved are all “absolute”, denote the same sets of objects in every standard model of mathematics (= model of ZFC — quite unlike “the set oll subsets of the integers” which varies from model to model.

NO as each is complementary other