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Conservation theorems for the Cohesiveness Principle

David R. Belanger
Dec 2022
摘要
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 $n \geq 2$ byrecursion-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 suitablemachinery including a version of the Friedberg jump-inversion theorem. The maintheorem is obtained when we combine these with known results about WKL. In anappendix we give a proof of the $\Pi^1_1$ conservativity of WKL over $RCA_0$ byway of the Superlow Basis Theorem and a new proof of a recent jump-inversiontheorem of Towsner.
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