Conservation theorems for the Cohesiveness Principle

Abstract

We prove that the Cohesiveness Principle (COH) is ${\Pi }_1^1$ conservative over $RC{A}_0+I{\Sigma }_n^0$ and over $RC{A}_0+B{\Sigma }_n^0$ for all $n\ge 2$ by recursion-theoretic means. We first characterize COH over $RC{A}_0+B{\Sigma }_2^0$ as a `jumped' version of Weak König'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 $RC{A}_0$ by way of the Superlow Basis Theorem and a new proof of a recent jump-inversion theorem of Towsner.