Separating principles below

Abstract

In this paper, we study the Ramsey‐type weak Kőnig's Lemma, written sans-serifRWKL, using a technique introduced by Lerman, Solomon, and the second author. This technique uses iterated forcing to construct an ω‐model satisfying one principle T1 but not another T2. The technique often allows one to translate a “one step” construction (building an instance of T2 along with a collection of solutions to each computable instance of T1) into an ω‐model separation (building a computable instance of T2 together with a Turing ideal where T1 holds but this instance has no solution). We illustrate this translation by separating d-DNR from sans-serifDNR (reproving a result of Ambos‐Spies, Kjos‐Hanssen, Lempp, and Slaman), and then apply this technique to separate sans-serifRWKL from sans-serifDNR (which has been shown separately by Bienvenu, Patey, and Shafer).

The Strength of the Tree Theorem for Pairs in Reverse Mathematics

“…This strict weakening of WKL 0 is sufficient in most applications of WKL 0 involved in proofs of RT 2 2 . The statement RWKL has been later studied by Bienvenu, Patey and Shafer [1] and by Flood and Towsner [11].…”

Iterative forcing and hyperimmunity in reverse mathematics

“…They successfully used their framework for separating the Erdős-Moser theorem (EM) from the stable ascending descending sequence principle (SADS) and separating the ascending descending sequence (ADS) from the stable chain antichain principle (SCAC). Their approach has been reused by Flood & Towsner [5] and the author [20] on diagonal non-computability statements.…”

Ramsey-type graph coloring and diagonal non-computability

“…Flood and Towsner [6] asked for which functions h the statement h-DNR implies RWKL over RCA 0 . In the case of functions whose range is bounded by some constant k, h-DNR implies k-DNR and therefore is equivalent to WKL 0 .…”

“…When considering orders h, that is, non-decreasing and unbounded functions, hbounded d.n.c. functions are known to form a strict hierarchy within reverse mathematics [1,6]. Recently, Bienvenu and the author [2] proved that witnesses to the strictness of this hierarchy can be constructed by probabilistic means.…”

“…[3] studied extensively variants of RWKL and constructed an ω-model of DNR -and even of the weak weak König's lemma (WWKL 0 ) -which is not a model of RWKL. Flood & Towsner [6] reproved the existence of an ω-model of DNR which is not a model of RWKL using the techniques developped by Lerman & al. in [17]. They asked in particular for which functions h the principle h-DNR implies RWKL.…”