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Ask HN: Is it possible that p != np as well as p = np
In a sensea russell-paradox
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- twangist 4y agoThe 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.
- mardiyah 4y agoNO as each is complementary other
- DougMerritt 4y agoIn 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.