On Mathias Generic Sets
نویسندگان
چکیده
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.
منابع مشابه
Generics for computable Mathias forcing
We study the complexity of generic reals for computable Mathias forcing in the context of computability theory. The n-generics and weak ngenerics 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 ≥ 3 then it satisfies the jump property G(n−1) ≡T G′ ⊕ ∅(n). We p...
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