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On Mathias generic sets

In S. Barry Cooper (ed.), How the World Computes. pp. 129--138 (2012)

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  1. Generics for computable Mathias forcing.Peter A. Cholak, Damir D. Dzhafarov, Jeffry L. Hirst & Theodore A. Slaman - 2014 - Annals of Pure and Applied Logic 165 (9):1418-1428.
    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 (...)
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