Π10-Encodability and Omniscient Reductions

Benoit Monin; Ludovic Patey

doi:10.1215/00294527-2018-0020

2019

\Pi ^{0}_{1}

-Encodability and Omniscient Reductions

Benoit Monin, Ludovic Patey

Notre Dame J. Formal Logic 60(1): 1-12 (2019). DOI: 10.1215/00294527-2018-0020

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 article, 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 which admit a nonempty $\Sigma^{1}_{1}$ -subset. Thanks to this equivalence, we prove that weak weak König’s lemma is not strongly computably reducible to Ramsey’s theorem. This answers a question of Hirschfeldt and Jockusch.

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