Notre Dame Journal of Formal Logic

Enumeration 1-Genericity in the Local Enumeration Degrees

Liliana Badillo, Charles M. Harris, and Mariya I. Soskova

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Abstract

We discuss a notion of forcing that characterizes enumeration 1-genericity, and we investigate the immunity, lowness, and quasiminimality properties of enumeration 1-generic sets and their degrees. We construct an enumeration operator Δ such that, for any A, the set ΔA is enumeration 1-generic and has the same jump complexity as A. We deduce from this and other recent results from the literature that not only does every degree a bound an enumeration 1-generic degree b such that a'=b', but also that, if a is nonzero, then we can find such b satisfying 0e<b<a. We conclude by proving the existence of both a nonzero low and a properly Σ20 nonsplittable enumeration 1-generic degree, hence proving that the class of 1-generic degrees is properly subsumed by the class of enumeration 1-generic degrees.

Article information

Source
Notre Dame J. Formal Logic, Volume 59, Number 4 (2018), 461-489.

Dates
Received: 26 June 2015
Accepted: 9 May 2016
First available in Project Euclid: 13 October 2018

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1539396032

Digital Object Identifier
doi:10.1215/00294527-2018-0008

Mathematical Reviews number (MathSciNet)
MR3871896

Zentralblatt MATH identifier
06996539

Subjects
Primary: 03D30: Other degrees and reducibilities
Secondary: 03D28: Other Turing degree structures

Keywords
enumeration reducibility degrees genericity

Citation

Badillo, Liliana; Harris, Charles M.; Soskova, Mariya I. Enumeration $1$ -Genericity in the Local Enumeration Degrees. Notre Dame J. Formal Logic 59 (2018), no. 4, 461--489. doi:10.1215/00294527-2018-0008. https://projecteuclid.org/euclid.ndjfl/1539396032


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