Degrees bounding minimal degrees

Abstract

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.