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2018 Enumeration 1-Genericity in the Local Enumeration Degrees
Liliana Badillo, Charles M. Harris, Mariya I. Soskova
Notre Dame J. Formal Logic 59(4): 461-489 (2018). DOI: 10.1215/00294527-2018-0008

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.

Citation

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Liliana Badillo. Charles M. Harris. Mariya I. Soskova. "Enumeration 1-Genericity in the Local Enumeration Degrees." Notre Dame J. Formal Logic 59 (4) 461 - 489, 2018. https://doi.org/10.1215/00294527-2018-0008

Information

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

zbMATH: 06996539
MathSciNet: MR3871896
Digital Object Identifier: 10.1215/00294527-2018-0008

Subjects:
Primary: 03D30
Secondary: 03D28

Rights: Copyright © 2018 University of Notre Dame

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Vol.59 • No. 4 • 2018
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