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 $a\text{'}=b\text{'}$, but also that, if $a$ is nonzero, then we can find such $b$ satisfying $0_e<b<a$. 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.