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by digama
3866 days ago
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If P=NP is independent, then we already know that a complete search for a polynomial algorithm for 3SAT will fail (because if there were such an algorithm it would prove P=NP, so it wouldn't be independent). Adding P=NP as an axiom doesn't change this; instead it effectively augments the system with certain "nonstandard" natural numbers (which are larger than any concretely describable term but "look" like regular natural numbers from within the system), and this mythical polynomial algorithm for 3SAT will have nonstandard length (meaning for pratical purposes that it is infinitely long). |
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A search for a provably polynomial algorithm will fail, but that doesn't mean that an unprovably polynomial time algorithm couldn't exist. We might come up with strong arguments for why a particular algorithm would be polynomial in practice. Still, I'd say "independent but false" is more likely than "independent but true."