Generics for computable Mathias forcing

Abstract

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 $n\ge 2$ then it satisfies the jump property $G^{(n-1)}{\equiv }_T{G}^{\prime }\oplus {\varnothing }^{(n)}$. 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.