A Stronger Foundation for Computer Science and P=NP

TL;DR

This paper challenges foundational assumptions in computer science by constructing a machine for RE-complete problems, revealing hidden assumptions in ZFC, and proposing a new axiom to address these issues, ultimately impacting the P vs NP problem.

Contribution

It introduces a novel machine for RE-complete problems, identifies hidden assumptions in ZFC, and proposes a restriction on the axiom of substitution to address foundational issues.

Findings

Disproves the SPACE hierarchy theorem

Provides a method to solve P vs NP using a TIME-SPACE oracle

Reveals hidden assumptions in ZFC related to substitution

Abstract

This article describes a Turing machine which can solve for $\beta^{'}$ which is RE-complete. RE-complete problems are proven to be undecidable by Turing's accepted proof on the Entscheidungsproblem. Thus, constructing a machine which decides over $\beta^{'}$ implies inconsistency in ZFC. We then discover that unrestricted use of the axiom of substitution can lead to hidden assumptions in a certain class of proofs by contradiction. These hidden assumptions create an implied axiom of incompleteness for ZFC. Later, we offer a restriction on the axiom of substitution by introducing a new axiom which prevents impredicative tautologies from producing theorems. Our discovery in regards to these foundational arguments, disproves the SPACE hierarchy theorem which allows us to solve the P vs NP problem using a TIME-SPACE equivalence oracle.