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After examining the {\bf P} versus {\bf NP} problem against the Kleene-Rosserparadox of the $\lambda$-calculus [94], it was found that it represents acounter-example to NP-completeness. We prove that it contradicts the proof ofCook's theorem. A logical formalization of the liar's paradox leads to the sameresult. This formalization of the liar's paradox into a computable form is a2-valued instance of a fuzzy logic programming paradox discovered in the systemof [90]. Three proofs that show that {\bf SAT} is (NOT) NP-complete arepresented. The counter-example classes to NP-completeness are alsocounter-examples to Fagin's theorem [36] and the Immermann-Vardi theorem[89,110], the fundamental results of descriptive complexity. All these resultsshow that {\bf ZF$\not$C} is inconsistent.

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