Notre Dame Journal of Formal Logic
- Notre Dame J. Formal Logic
- Volume 59, Number 4 (2018), 461-489.
Enumeration -Genericity in the Local Enumeration Degrees
We discuss a notion of forcing that characterizes enumeration -genericity, and we investigate the immunity, lowness, and quasiminimality properties of enumeration -generic sets and their degrees. We construct an enumeration operator such that, for any , the set is enumeration -generic and has the same jump complexity as . We deduce from this and other recent results from the literature that not only does every degree bound an enumeration -generic degree such that , but also that, if is nonzero, then we can find such satisfying . We conclude by proving the existence of both a nonzero low and a properly nonsplittable enumeration -generic degree, hence proving that the class of -generic degrees is properly subsumed by the class of enumeration -generic degrees.
Notre Dame J. Formal Logic, Volume 59, Number 4 (2018), 461-489.
Received: 26 June 2015
Accepted: 9 May 2016
First available in Project Euclid: 13 October 2018
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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