On mathias generic sets

  • Authors:

  • Peter A. Cholak;Damir D. Dzhafarov;Jeffry L. Hirst

  • Affiliations:

  • Department of Mathematics, University of Notre Dame, Notre Dame, IN, United States of America;Department of Mathematics, University of Notre Dame, Notre Dame, IN, United States of America;Department of Mathematical Sciences, Appalachian State University, Boone, NC, United States of America

  • Venue:
  • CiE'12 Proceedings of the 8th Turing Centenary conference on Computability in Europe: how the world computes
  • Year:
  • 2012

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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.