Enumeration 1-Genericity in the Local Enumeration Degrees
Liliana Badillo, Charles M. Harris, Mariya I. Soskova · Notre Dame Journal of Formal Logic · 2018
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