## Notre Dame Journal of Formal Logic

### Enumeration $1$-Genericity in the Local Enumeration Degrees

#### 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 $\Delta$ such that, for any $A$, the set $\Delta^{A}$ 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 $0_{e}\lt b\lt a$. We conclude by proving the existence of both a nonzero low and a properly $\Sigma_{2}^{0}$ 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
Accepted: 9 May 2016
First available in Project Euclid: 13 October 2018

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

#### 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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