Antibasis theorems for Π1 classes and the jump hierarchy
نویسنده
چکیده
We prove two antibasis theorems for Π1 classes. The first is a jump inversion theorem for Π1 classes with respect to the global structure of the Turing degrees. For any P ⊆ 2, define S(P ), the degree spectrum of P , to be the set of all Turing degrees a such that there exists A ∈ P of degree a. For any degree a ≥ 0, let Jump(a) = {b : b = a}. We prove that, for any a ≥ 0 and any Π1 class P , if Jump (a) ⊆ S(P ) then P contains a member of every degree. For any degree a ≥ 0 such that a is recursively enumerable (r.e.) in 0, let Jump ≤0′(a) = {b : b ≤ 0 ′ and b = a}. The second theorem concerns the degrees below 0. We prove that for any a ≥ 0 which is recursively enumerable in 0 and any Π1 class P , if Jump −1 ≤0′(a) ⊆ S(P ) then P contains a member of every degree.
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