It so happens that I asked an analogous questionan analogous question at CS Theory StackExchange. A brief informal summary is this. First, it is undecidable to determine if a given theorem is provable in ZFC, and this would not change if we knew P=NP. Second, for other mathematical theorems (I used Goldbach's conjecture), if P=NP, and if the theorem has a "short proof," then not only could we determine whether the theorem is true or false, we could also find a proof quickly. More precisely, a proof could be found "in time polynomial in the length of the statement and the length of the shortest proof" (to quote one respondent) using Levin's universal search algorithm.
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It so happens that I asked an analogous question at CS Theory StackExchange. A brief informal summary is this. First, it is undecidable to determine if a given theorem is provable in ZFC, and this would not change if we knew P=NP. Second, for other mathematical theorems (I used Goldbach's conjecture), if P=NP, and if the theorem has a "short proof," then not only could we determine whether the theorem is true or false, we could also find a proof quickly. More precisely, a proof could be found "in time polynomial in the length of the statement and the length of the shortest proof" (to quote one respondent) using Levin's universal search algorithm.
It so happens that I asked an analogous question at CS Theory StackExchange. A brief informal summary is this. First, it is undecidable to determine if a given theorem is provable in ZFC, and this would not change if we knew P=NP. Second, for other mathematical theorems (I used Goldbach's conjecture), if P=NP, and if the theorem has a "short proof," then not only could we determine whether the theorem is true or false, we could also find a proof quickly. More precisely, a proof could be found "in time polynomial in the length of the statement and the length of the shortest proof" (to quote one respondent) using Levin's universal search algorithm.
It so happens that I asked an analogous question at CS Theory StackExchange. A brief informal summary is this. First, it is undecidable to determine if a given theorem is provable in ZFC, and this would not change if we knew P=NP. Second, for other mathematical theorems (I used Goldbach's conjecture), if P=NP, and if the theorem has a "short proof," then not only could we determine whether the theorem is true or false, we could also find a proof quickly. More precisely, a proof could be found "in time polynomial in the length of the statement and the length of the shortest proof" (to quote one respondent) using Levin's universal search algorithm.