Some results concerning the SRT 2 2 vs. COH problem

Abstract

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THE REVERSE MATHEMATICS OF ${\mathsf {CAC\ FOR\ TREES}}$

“…, g k , and every computable instance X of RT n k , there exists a solution Y such that at least two among the hyperimmune functions remain Y -hyperimmune, but that it is not the case for computable instances of RT n k`1 . This property fails when considering the star operator, as the following lemma shows: Lemma 1.2 (Cholak et al [4]). There exist 4 hyperimmune functions g 0 , .…”

Weihrauch Complexity and the Hagen School of Computable Analysis

“…This implies, in particular, C N < W L. There are many further studies, e.g., on Nash equilibria by Arno Pauly [89,91] that can also be described with the help of choice, on probabilistic versions of choice, for instance, by the author and Pauly [25], the author, Gherardi and Hölzl [16], Bienvenu and Kuyper [3], on degrees defined by jumps of choice, e.g., by the author, Gherardi and Marcone [17], non-standard degrees obtained by Ramsey's theorem, by Dorais, Dzhafarov, Hirst, Mileti and Shafer [31], Hirschfeldt and Jockusch [56,57], the author and Rakotoniaina [27], Dzhafarov, Goh, Hirschfeldt, Patey and Pauly [33], Cholak, Dzhafarov, Hirschfeldt and Patey [29], Marcone and Valenti [75], and Dzhafarov and Patey [34].…”

COH, SRT22, and multiple functionals

“…We assume familiarity with computability theory and reverse mathematics, and refer to Soare [20] and Simpson [19], respectively, for background on these subjects. We also assume the basics of Weihrauch reducibility and computable reducibility, and refer, e.g., to Brattka, Gherardi, and Pauly [1] for a detailed survey, or, e.g., to Cholak, Dzhafarov, Hirschfeldt, and Patey [3,Section 1] for an introduction aimed more specifically at the kinds of questions we will be dealing with here. Definition 1.1.…”

“…We return to this below. We follow the now-standard practice of regarding Π 1 2 statements of second-order arithmetic as problems, equipped with a set of instances, and for each instance, a set of solutions, all coded or represented by subsets of 2 ω (see [3], Definition 1.1). This facilitates their study both in the framework of reverse mathematics and in terms of Weihrauch and computable reducibilities.…”

“…The relationship between the stable Ramsey's theorem and the cohesiveness principle is the focus of a longstanding and ongoing investigation (see, e.g., [2,3,6,8,9,10,12,13,14,15,16]). We refer the reader to [3, Section 1] for a discussion of some of the history of these principles, and their larger significance in the exploration of the logical strength of combinatorial principles.…”