Domination, forcing, array nonrecursiveness and relative recursive enumerability

Abstract

We present some abstract theorems showing how domination properties equivalent to being ${\overline{GL}}_2$ or array nonrecursive can be used to construct sets generic for different notions of forcing. These theorems are then applied to give simple proofs of some known results. We also give a direct uniform proof of a recent result of Ambos-Spies, Ding, Wang and Yu [2009] that every degree above any in ${\overline{GL}}_2$ is recursively enumerable in a 1-generic degree strictly below it. Our major new result is that every array nonrecursive degree is r.e. in some degree strictly below it. Our analysis of array nonrecursiveness and construction of generic sequences below $\mathbf{ANR}$ degrees also reveal a new level of uniformity in these types of results.