Enumeration 1-genericity in the Local Enumeration Degrees

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 ∆ is enumeration 1-generic and has the same jump complexity as A. We also prove that every nonzero ∆ 2 degree bounds a nonzero enumeration 1-generic ∆ 2 degree. We deduce from these results and the properties of good degrees that, not only does every degree a bound an enumeration 1-generic degree b such that a = b, but also that, if a is good and 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 Σ02 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.