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≥2 then it satisfies the jump property G(n-1)≡TG'⊕θ(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. © 2014 Elsevier B.V.
Cholak, P. A., Dzhafarov, D. D., Hirst, J. L., & Slaman, T. A. (2014). Generics for computable Mathias forcing. Annals of Pure and Applied Logic, 165(9), 1418–1428. https://doi.org/10.1016/j.apal.2014.04.011