Complexity Classes as Mathematical Axioms

TL;DR

This paper explores the idea of adopting a major complexity class separation conjecture as an axiom, revealing unexpected links to three manifold topology and drawing parallels to large cardinal axioms in set theory.

Contribution

It introduces the novel concept of treating complexity class separations as axioms, bridging complexity theory and topology in a new foundational framework.

Findings

Implication of complexity class axioms in topology

Connection to large cardinal axioms in set theory

Potential new foundations for complexity theory

Abstract

Treating a conjecture, P^#P != NP, on the separation of complexity classes as an axiom, an implication is found in three manifold topology with little obvious connection to complexity theory. This is reminiscent of Harvey Friedman's work on finitistic interpretations of large cardinal axioms.