Generics for computable Mathias forcing

Peter A Cholak,Damir D Dzhafarov,Jeffry L Hirst,Theodore A Slaman

doi:10.1016/j.apal.2014.04.011

Generics for computable Mathias forcing

Peter A Cholak, Damir D Dzhafarov + Show 2 more

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https://doi.org/10.1016/j.apal.2014.04.011

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Journal: Annals of Pure and Applied Logic	Publication Date: May 13, 2014
Citations: 25	License type: publisher-specific-oa

Affiliation: University of Notre Dame, University of Connecticut, Appalachian State University, University of California, Berkeley

#Mathias Forcing #Turing Reducibility + Show 8 more

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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≥2 then it satisfies the jump property G(n−1)≡TG′⊕∅(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.

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