Abstract
We prove two antibasis theorems for \({\Pi^0_1}\) classes. The first is a jump inversion theorem for \({\Pi^0_1}\) classes with respect to the global structure of the Turing degrees. For any \({P\subseteq 2^\omega}\), define S(P), the degree spectrum of P, to be the set of all Turing degrees a such that there exists \({A \in P}\) of degree a. For any degree \({{\bf a \geq 0'}}\), let \({\textrm{Jump}^{-1}({\bf a) = \{b : b' = a \}}}\) . We prove that, for any \({{\bf a \geq 0'}}\) and any \({\Pi^0_1}\) class P, if \({\textrm{Jump}^{-1} ({\bf a}) \subseteq S(P)}\) then P contains a member of every degree. For any degree \({{\bf a \geq 0'}}\) such that a is recursively enumerable (r.e.) in 0', let \({Jump_{\bf \leq 0'} ^{-1}({\bf a)=\{b : b \leq 0' \textrm{and} b' = a \}}}\) . The second theorem concerns the degrees below 0'. We prove that for any \({{\bf a\geq 0'}}\) which is recursively enumerable in 0' and any \({\Pi^0_1}\) class P, if \({\textrm{Jump}_{\bf \leq 0'} ^{-1}({\bf a)} \subseteq S(P)}\) then P contains a member of every degree.
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Çevik, A. Antibasis theorems for \({\Pi^0_1}\) classes and the jump hierarchy. Arch. Math. Logic 52, 137–142 (2013). https://doi.org/10.1007/s00153-012-0310-y
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DOI: https://doi.org/10.1007/s00153-012-0310-y