$\Pi^0_4$ conservation of Ramsey's theorem for pairs

TL;DR

This paper proves that Ramsey's theorem for pairs and two colors is a conservative extension over a base system, refining previous results and advancing understanding of its logical strength.

Contribution

It improves upon prior results by Patey and Yokoyama, establishing $orall ext{Pi}^0_4$ conservation and contributing to the first-order part analysis of Ramsey's theorem.

Findings

Ramsey's theorem for pairs is a $orall ext{Pi}^0_4$ conservative extension.

The result refines previous work by Patey and Yokoyama.

Advances the understanding of the logical strength of Ramsey's theorem.

Abstract

In this article, we prove that Ramsey's theorem for pairs and two colors is a $\forall \Pi^0_4$ conservative extension of $\mathsf{RCA}_0 + \mathsf{B}\Sigma^0_2$, where a $\forall \Pi^0_4$ formula consists of a universal quantifier over sets followed by a $\Pi^0_4$ formula. The proof is an improvement of a result by Patey and Yokoyama and a step towards the resolution of the longstanding question of the first-order part of Ramsey's theorem for pairs.