Two theorems about the P versus NP problem

TL;DR

This paper presents two theorems related to the P versus NP problem, showing independence results and limitations of proof within ZFC and for polynomial-time Turing machines.

Contribution

It introduces new theoretical results demonstrating independence and unprovability aspects related to P versus NP within formal systems.

Findings

Existence of a language independent of ZFC for P membership

Existence of an NP language undecidable by any polynomial-time Turing machine

Highlights limitations of formal proof systems in resolving P vs NP

Abstract

Two theorems about the P versus NP problem be proved in this article (1) There exists a language $L$, that the statement $L \in \textbf{P}$ is independent of ZFC. (2) There exists a language $L \in \textbf{NP}$, for any polynomial time deterministic Turing machine $M$, we cannot prove $L$ is decidable on $M$.