On finite cohesiveness principle

TL;DR

The paper investigates the finite cohesiveness principle $\mathrm{fin}$-$\mathrm{COH}$ within reverse mathematics. It proves that over $\mathrm{RCA}_0^*$, $\mathrm{fin}$-$\mathrm{COH}$ implies $\mathrm{I}\Sigma_1^0$, resolving a question and showing that $\mathrm{fin}$-$\mathrm{COH}$ is not provable from $\mathrm{WKL}_0$; it further shows $\mathrm{fin}$-$\mathrm{COH}$ is provable over $\mathrm{RCA}_0+\mathrm{B}\Sigma_2^0$ and identifies a condition under which it coincides with the full $\mathrm{COH}$. The work also analyzes $\Sigma_2^0$-separation principles, showing $\mathrm{fin}$-$\Sigma_2^0$-separation is not $\Pi_1^1$-conservative over $\mathrm{RCA}_0^*$ and discussing their relation to $\mathrm{COH}$, with several open questions about implications in weaker bases and the extent of nonconservativity.

Abstract

We examine the cohesiveness principle applied to finite sequences of sets, referred to as $\mathrm{fin}$-$\mathrm{COH}$. We investigate its strength over different base theories: We show that over $\mathrm{RCA}^*_0$, $\mathrm{fin}$-$\mathrm{COH}$ implies $\mathrm{I}Σ_1^0$, which answers a question by Fiori-Carones, Kołodziejczyk and Kowalik. We also show that $\mathrm{fin}$-$\mathrm{COH}$ is not provable over $\mathrm{WKL}_0$.

Theorems & Definitions (16)

• Lemma 1: folklore
• Lemma 2: Chong--Mourad Coding Lemma art:CM
• Theorem 3
• proof
• Proposition 4
• proof
• Theorem 5: Fiori-Carones et al. art:weakercousins
• Theorem 6
• proof
• Corollary 7