Pi01 encodability and omniscient reductions

TL;DR

This paper extends the concept of computable encodability to subsets of the Baire space, characterizes $\,\Pi^0_1$ encodable compact sets, and demonstrates a separation in computability reductions related to weak weak K"onig's lemma and Ramsey's theorem.

Contribution

It introduces a new characterization of $\,\Pi^0_1$ encodable compact sets via $\,\Sigma^1_1$ subsets and applies this to computability theory.

Findings

Characterization of $\,\Pi^0_1$ encodable compact sets as those with a non-empty $\,\Sigma^1_1$ subset

Proof that weak weak K"onig's lemma is not strongly computably reducible to Ramsey's theorem

Extension of computable encodability to subsets of the Baire space

Abstract

A set of integers $A$ is computably encodable if every infinite set of integers has an infinite subset computing $A$. By a result of Solovay, the computably encodable sets are exactly the hyperarithmetic ones. In this paper, we extend this notion of computable encodability to subsets of the Baire space and we characterize the $\Pi^0_1$ encodable compact sets as those who admit a non-empty $\Sigma^1_1$ subset. Thanks to this equivalence, we prove that weak weak K\"onig's lemma is not strongly computably reducible to Ramsey's theorem. This answers a question of Hirschfeldt and Jockusch.