Degrees bounding minimal degrees

A set is called n -generic if it is Cohen generic for n -quantifier arithmetic. A (Turing) degree is n -generic if it contains an n -generic set. Our interest in this paper is the relationship between n -generic (indeed 1-generic) degrees and minimal degrees, i.e. degrees which are non-recursive and which bound no degrees intermediate between them and the recursive degree. It is known that n -generic degrees and minimal degrees have a complex relationship since Cohen forcing and Sacks forcing are mutually incompatible. The goal of this paper is to show.