Recent results on initial segments of the Turing degrees are presented, and some conjectures about initial segments that have implications for the existence of nontrivial automorphisms of the Turing degrees are indicated.

We study the complexity of generic reals for computable Mathias forcing in the context of computability theory. The n -generics and weak n -generics form a strict hierarchy under Turing reducibility, as in the case of Cohen forcing. We analyze the complexity of the Mathias forcing relation, and show that if G is any n -generic with n≥2n≥2 then it satisfies the jump property G≡TG′⊕∅G≡TG′⊕∅. We prove that every such G has generalized high Turing degree, and so cannot have even (...) Cohen 1-generic degree. On the other hand, we show that every Mathias n-generic real computes a Cohen n-generic real. (shrink)

We show that there is a generalized high degree which is a minimal cover of a minimal degree. This is the highest jump class one can reach by finite iterations of minimality. This result also answers an old question by Lerman.