Converging Peano Arithmetic and SAT in the P vs NP Debate

Core Reduction via Cook’s Theorem

Cook’s theorem shows that every problem in NP can be transformed in polynomial time into an instance of Boolean satisfiability (SAT).

Because SAT is NP-complete, the question “P = NP?” becomes equivalent to “Is there a deterministic polynomial-time algorithm for SAT?”

This centralization means any arithmetic formalization of P vs NP can focus solely on the solvability of SAT within given time bounds.

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Encoding SAT and Time Bounds in PA

Peano Arithmetic can express statements about Turing machines, inputs, and runtimes by Gödel-encoding their descriptions as natural numbers.

One defines a predicate DecidesSAT(M, k, n) that asserts “Machine M correctly decides SAT on all inputs of length n within n^k steps.”

By quantifying over all machines and exponents, PA can articulate the absence of any polytime SAT solver.

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Formal Π₂ Statement of P ≠ NP in PA

Within PA, “P ≠ NP” is equivalent to the Π₂ sentence:

∀ M ∀ k ∃ x ¬DecidesSAT(M, k, |x|)
  

This reads: for every purported polynomial-time algorithm encoded by (M, k), there exists an input x on which it fails.

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Independence and Proof-Complexity Barriers

• Gödel’s incompleteness phenomenon suggests PA may not settle every true Π₂ statement, raising the possibility that P ≠ NP is independent of PA.
• Relativization and Natural-Proofs barriers further indicate that broad classes of proof techniques cannot resolve P vs NP, even in stronger systems like ZF.
• If P ≠ NP is independent, it mirrors set theory’s Continuum Hypothesis: a fundamental question beyond our standard axioms.

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Implications and Outlook

By funneling P vs NP into a single arithmetic claim about SAT, Cook’s theorem highlights exactly where PA’s proving power is tested.

A proof or disproof within PA would require novel, non-relativizing, non-natural arguments strong enough to overcome Π₂ incompleteness obstacles.

Whether future advances come from proof theory, alternate logics, or new complexity insights, the convergence of PA and SAT polynomials remains the crucible for resolving P vs NP.