The Solution of the P vs NP. Why the P=NP cannot be Proved in Peano Arithmetic and Why Eventually, We can Prove the P≠NP in 1st order ZFC Set Theory

Abstract

The millennium problem P vs NP problem, has resisted any solution more than half a century now. This is not without a good reason. In this work we prove that P ≠NP in 1^st order FC set theory. In order to prove it, we use the method of Martin Davis in the solution of the 10^th Hilbert problem about the Diophantine equations, where he derives a non-provability in the logic of Peano axiomatic system (which is that there is a Diophantine equation which has no solution, but this is not provable), from a non-recursive enumerability of non-solvable Diophantine equations. We transfer this method in the case of the non-recursive enumerability, that the Rice-Shapiro theorem or other methods gives. Then we derive easily the above mentioned result.