In this paper, I prove that P NP and NP=EXPTIME in the context of Zermelo-Frankel set theory and deterministic Turing computers in a very short but definitive proof. The complex consequences of tackling the P versus NP dilemma in various axiomatic systems are also discussed. The current paper’s findings definitively resolve the third Clay Millennium problem of P versus NP in a clear and transparent manner that the broader scientific community, as well as experts in the field, can follow, understand, and hence accept. The hierarchy theorem is a more sophisticated finding than the P against NP problem, and there should be a proof for the P versus NP problem that is not much more complicated than the proof for the hierarchy theorem. The demonstration of the P vs NP problem in the direction P NP also implies that the current technique of creating passwords on the internet is secure.

**Author (S) Details**

**Konstantinos E. Kyritsis**

Department of Accounting-Finance, University of Ioannina, Greece.

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