Conservation theorems for the Cohesiveness Principle

TL;DR

This paper establishes conservation results for the Cohesiveness Principle (COH) over certain subsystems of second-order arithmetic, using recursion-theoretic methods and characterizations related to Weak König's Lemma.

Contribution

It characterizes COH as a jumped version of WKL over specific systems and develops new machinery including a jump-inversion theorem, advancing the understanding of COH's logical strength.

Findings

COH is $ ext{Pi}^1_1$ conservative over $RCA_0 + I ext{-} ext{Sigma}^0_n$ and $RCA_0 + B ext{-} ext{Sigma}^0_n$ for all $n \\geq 2$

Characterization of COH as a jumped version of WKL over certain systems

Development of a new jump-inversion theorem and proofs of conservativity results

Abstract

We prove that the Cohesiveness Principle (COH) is $\Pi^1_1$ conservative over $RCA_0 + I\Sigma^0_n$ and over $RCA_0 + B\Sigma^0_n$ for all $n \geq 2$ by recursion-theoretic means. We first characterize COH over $RCA_0 + B\Sigma^0_2$ as a `jumped' version of Weak K\"{o}nig's Lemma (WKL) and develop suitable machinery including a version of the Friedberg jump-inversion theorem. The main theorem is obtained when we combine these with known results about WKL. In an appendix we give a proof of the $\Pi^1_1$ conservativity of WKL over $RCA_0$ by way of the Superlow Basis Theorem and a new proof of a recent jump-inversion theorem of Towsner.