COH, SRT22, and multiple functionals

Abstract

We prove the following result: there is a family R = R 0 , R 1 , . . . of subsets of ω such that for every stable coloring c : [ω] 2 → k hyperarithmetical in R and every finite collection of Turing functionals, there is an infinite homogeneous set H for c such that none of the finitely many functionals map R ⊕ H to an infinite cohesive set for R. This extends the current best partial results towards the SRT 2 2 vs. COH problem in reverse mathematics, and is also a partial result towards the resolution of several related problems, such as whether COH is omnisciently computably reducible to SRT 2 2 .

Lawvere-Tierney topologies for computability theorists

“…For instance, f is computable reducible to g in the sense of [14,19] if and only if f is one-query bilayered Turing reducible to (g | Advice N ) (see Definition 4.5) if we properly extend the above notions to the context of N N -computability. The notion of omniscient computable/Weihrauch reducibility [26,15,13] can also be explained in the bilayer context.…”