Developing Takeuti-Yasumoto forcing

TL;DR

This paper reformulates Takeuti-Yasumoto forcing for bounded arithmetic using two-sort methods, linking it to P=NP, and demonstrates its ability to satisfy or falsify dual weak pigeonhole principles.

Contribution

It introduces a new formulation of Takeuti-Yasumoto forcing with two-sort bounded arithmetic and connects it to fundamental complexity and logical principles.

Findings

Generic extensions relate to P=NP problem

Forcing constructions can be expressed as Takeuti-Yasumoto forcing

Dual weak pigeonhole principles can be satisfied or falsified

Abstract

In late 90's G.Takeuti and Y.Yasumoto gave forcing constructions for bounded arithmetic. We will reformulate their constructions using two-sort bounded arithmetic and prove the followings. 1. Generic extensions are related with P=NP problem. 2. J.Krajicek's forcing constructions can be given as Takeuti-Yasumoto forcing. 3. We can either satisfy or falsify the dual weak pigeonhole principles in generic extensions.