On Mathias Generic Sets

Abstract

We present some results about generics for computable Mathias forcing. The n-generics and weak n-generics in this setting form a strict hierarchy 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 ≥ 3 then it satisfies the jump property G (n − 1) = G′ ⊕ ∅ (n). We prove that every such G has generalized high degree, and so cannot have even Cohen 1-generic degree. On the other hand, we show that G, together with any bi-immune A ≤ T ∅ (n − 1), computes a Cohen n-generic.