Enumeration 1-Genericity in the Local Enumeration Degrees

Abstract

We discuss a notion of forcing that characterises enumeration 1genericity and we investigate the immunity, lowness and quasiminimality properties of enumeration 1-generic sets and their degrees. We construct an enumeration operator ∆ such that, for any A, the set ∆ 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 0e < b < a. We conclude by proving the existence of both a nonzero low and a properly Σ 0 2 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.

Pa Relative to an Enumeration Oracle

“…There is a slightly different notion of genericity that plays better with enumeration reducibility. Enumeration 1-genericity was introduced by Badillo and Harris [3] and further studied by Badillo, Harris, and Soskova [4]. It is straightforward to check that every 1-generic set is enumeration 1-generic.…”