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Complexity classes as mathematical axioms. (English) Zbl 1178.03069

Summary: Complexity theory, being the metrical version of decision theory, has long been suspected of harboring undecidable statements among its most prominent conjectures. Taking this possibility seriously, we add one such conjecture, \(\text{P}^{\#\text{P}}\neq \text{NP}\), as a new “axiom” and find that it has an implication in 3-dimensional topology. This is reminiscent of Harvey Friedman’s work on finitistic interpretations of large cardinal axioms.

MSC:

03E65 Other set-theoretic hypotheses and axioms
57M25 Knots and links in the \(3\)-sphere (MSC2010)
68Q15 Complexity classes (hierarchies, relations among complexity classes, etc.)
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References:

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