Generics for computable Mathias forcing |
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Authors: | Peter A Cholak Damir D Dzhafarov Jeffry L Hirst Theodore A Slaman |
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Institution: | 1. Department of Mathematics, University of Notre Dame, 255 Hurley Building, Notre Dame, IN 46556, USA;2. Department of Mathematics, University of Connecticut, 196 Auditorium Road, Storrs, CT 06269, USA;3. Department of Mathematical Sciences, Appalachian State University, 342 Walker Hall, Boone, NC 28608, USA;4. Department of Mathematics, University of California, Berkeley, 970 Evans Hall, Berkeley, CA 94720, USA |
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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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Keywords: | 03D80 03E40 03D32 03E75 |
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