Cone avoiding closed sets

TL;DR

This paper proves a general result about cone avoiding sets in computability theory and applies it to show separations between various logical principles and randomness notions.

Contribution

It introduces a new cone avoiding theorem for subtrees of 2^{<ω} and applies it to establish separations in reverse mathematics and randomness.

Findings

RT_2^2 does not imply WWKL_0

DNR is strictly weaker than WWKL_0

Martin-Löf random sets contain infinite subsets not computing any ML-random set

Abstract

We prove that for an arbitrary subtree $T$ of $2^{<\omega}$ with each element extendable to a path, a given countable class $\mathcal{M}$ closed under disjoint union, and any set $A$, if none of the members of $\mathcal{M}$ strongly $k$-enumerate $T$ for any $k$, then there exists an infinite set contained in either $A$ or $\bar{A}$ such that for every $C\in\mathcal{M}$, $C\oplus G$ also does not strongly $k$-enumerate $T$. We give applications of this result, which include: (1) $\mathsf{RT_2^2}$ doesn't imply $\mathsf{WWKL_0}$; (2) (Ambos-Spies et al.2004) $\mathsf{DNR}$ is strictly weaker than $\mathsf{WWKL_0}$; (3) (Kjos-Hanssen 2009) for any Martin-L\"{o}f random set $A$ either $A$ or $\bar{A}$ contains an infinite subset that does not compute any Martin-L\"{o}f random set; etc. We also discuss further generalizations of this result.