Conservation theorems for the Cohesiveness Principle
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 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 $RCA_0$ by way of the Superlow Basis Theorem and a new proof of a recent jump-inversion theorem of Towsner.
- Publication:
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arXiv e-prints
- Pub Date:
- December 2022
- DOI:
- 10.48550/arXiv.2212.13011
- arXiv:
- arXiv:2212.13011
- Bibcode:
- 2022arXiv221213011B
- Keywords:
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- Mathematics - Logic;
- 03F35 (Primary) 03D99 (Secondary)