## Notre Dame Journal of Formal Logic

### $\Pi ^{0}_{1}$-Encodability and Omniscient Reductions

#### 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.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 60, Number 1 (2019), 1-12.

Dates
Accepted: 31 October 2016
First available in Project Euclid: 18 January 2019

https://projecteuclid.org/euclid.ndjfl/1547802298

Digital Object Identifier
doi:10.1215/00294527-2018-0020

Mathematical Reviews number (MathSciNet)
MR3911103

Zentralblatt MATH identifier
07060305

Subjects
Primary: 03D30: Other degrees and reducibilities
Secondary: 03D65: Higher-type and set recursion theory

#### Citation

Monin, Benoit; Patey, Ludovic. $\Pi ^{0}_{1}$ -Encodability and Omniscient Reductions. Notre Dame J. Formal Logic 60 (2019), no. 1, 1--12. doi:10.1215/00294527-2018-0020. https://projecteuclid.org/euclid.ndjfl/1547802298

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