The Kleene-Rosser Paradox, The Liar's Paradox & A Fuzzy Logic Programming Paradox Imply SAT is (NOT) NP-complete

TL;DR

This paper argues that the P versus NP problem is influenced by logical paradoxes like the Kleene-Rosser and liar's paradox, challenging the assumption that SAT is NP-complete and suggesting foundational inconsistencies in set theory.

Contribution

It introduces paradox-based counter-examples to NP-completeness and related theorems, questioning established complexity class assumptions and foundational set theory.

Findings

Counter-examples to NP-completeness of SAT

Implications for Fagin's and Immermann-Vardi theorems

Indication of inconsistency in ZF$ ot$C set theory

Abstract

After examining the {\bf P} versus {\bf NP} problem against the Kleene-Rosser paradox of the $\lambda$-calculus [94], it was found that it represents a counter-example to NP-completeness. We prove that it contradicts the proof of Cook's theorem. A logical formalization of the liar's paradox leads to the same result. This formalization of the liar's paradox into a computable form is a 2-valued instance of a fuzzy logic programming paradox discovered in the system of [90]. Three proofs that show that {\bf SAT} is (NOT) NP-complete are presented. The counter-example classes to NP-completeness are also counter-examples to Fagin's theorem [36] and the Immermann-Vardi theorem [89,110], the fundamental results of descriptive complexity. All these results show that {\bf ZF$\not$C} is inconsistent.