On the difficulty of proving P equals NP in ZFC

TL;DR

This paper explores the deep connection between the P vs. NP problem and large cardinal axioms, suggesting that proving P=NP within ZFC may be inherently impossible due to their equivalence with certain large cardinal statements.

Contribution

It extends Friedman's results to construct theorems that link P=NP with large cardinal hypotheses, highlighting the potential unprovability of P=NP in ZFC.

Findings

Constructs theorems linking P=NP to large cardinal axioms

Suggests P=NP may be unprovable in ZFC due to this connection

Highlights the complexity of proving P=NP within standard set theory

Abstract

Harvey Friedman, in his remarkable paper Finite functions and the necessary use of large cardinals, Ann. Math. 148:803-893, 1998 and in a technical report, Applications of large cardinals to graph theory, Ohio State University, 1997, presents numerous combinatorial statements with clear geometric meaning that are proved using large cardinals and shown to require them. By slightly extending some of Friedman's results, we construct an infinite class of structurally similar theorems which can be proved using the same large cardinals but also can be proved using the statement "subset sum is solvable in polynomial time." This curious connection between the P vs. NP problem and the theory of large cardinals seems to suggest that either P=NP is false or otherwise not provable in ZFC.