The Kleene-Rosser Paradox, The Liar's Paradox & A Fuzzy Logic Programming Paradox Imply SAT is (NOT) NP-complete
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.
- Publication:
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arXiv e-prints
- Pub Date:
- June 2008
- DOI:
- 10.48550/arXiv.0806.2947
- arXiv:
- arXiv:0806.2947
- Bibcode:
- 2008arXiv0806.2947E
- Keywords:
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- Computer Science - Logic in Computer Science
- E-Print:
- Submitted to the ACM Transactions on Computation Theory