Conservation theorems for the Cohesiveness Principle

2022/12/26 by David Belanger, Belanger, David R. · 2 citations
Mathematics · #03F35 (Primary) 03D99 (Secondary) #Advanced Operator Algebra Research #Advanced Topology and Set Theory #FOS: Mathematics #Logic (math.LO) #Markov Chains and Monte Carlo Methods

paper · pdf · doi:10.48550/arxiv.2212.13011

openalex publication_date 2022/12/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that the Cohesiveness Principle (COH) is Π¹₁ conservative over RCA₀ + IΣ⁰_n and over RCA₀ + BΣ⁰_n for all n ≥ 2 by recursion-theoretic means. We first characterize COH over RCA₀ + BΣ⁰₂ 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 Π¹₁ conservativity of WKL over RCA₀ by way of the Superlow Basis Theorem and a new proof of a recent jump-inversion theorem of Towsner.

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• The reverse mathematics of bounded Ramsey's theorem for pairs
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