Constructibility, computational complexity and P versus NP

TL;DR

This paper explores the relationship between constructibility, computational complexity, and the P versus NP problem through a metamathematical framework involving G"odel numbers and provability conditions, proposing a principle (IPL) that suggests NP is not contained in P.

Contribution

It introduces a new metamathematical principle (IPL) linking provability bounds to complexity class separations, offering a novel approach to the P versus NP question.

Findings

Defines a set G of G"odel numbers related to provability

Proposes the IPL thesis connecting bounds to NP problems

Argues that under IPL, NP is not contained in P

Abstract

A set G consisting of the G\"odel numbers of PA formulas expressing provability conditions for a wide class of formal theories, is defined. By combining universality properties of G with G\"odel's second incompleteness theorem, an argument is given for the validity of a metamathematical principle stating the existence of an integer m representing a specific upper bound condition concerning the set G, along with the impossibility of constructing an arithmetical sound formal theory in which the value of m can be decided. Assuming this principle, which is referred to as the IPL thesis, the integer m may be used for constructing a decision problem D which is in NP, yet which is algorithmically unsolvable. Hence under a constructive interpretation of algorithmic complexity classes, IPL implies that NP is not contained in P. Known proofs concluding that NP is contained in larger complexity classes such as EXPTIME tacitly use an assumption which, although it is not satisfied for D, clearly can be assumed for all NP problems typically encountered in practice. This assumption can easily be added to the proofs in question, establishing refined versions of the classical results.