Abstract
This paper continues the investigation into the relationship between good approximations and jump inversion initiated by Griffith. Firstly it is shown that there is a \({\Pi^{0}_{2}}\) set A whose enumeration degree a is bad—i.e. such that no set \({X \in a}\) is good approximable—and whose complement \({\overline{A}}\) has lowest possible jump, in other words is low2. This also ensures that the degrees y ≤ a only contain \({\Delta^{0}_{3}}\) sets and thus yields a tight lower bound for the complexity of both a set of bad enumeration degree, and of its complement, in terms of the high/low jump hierarchy. Extending the author’s previous characterisation of the double jump of good approximable sets, the triple jump of a \({\Sigma^{0}_{2}}\) set A is characterised in terms of the index set of coinfinite sets enumeration reducible to A. The paper concludes by using Griffith’s jump interpolation technique to show that there exists a high quasiminimal \({\Delta^{0}_{2}}\) enumeration degree.
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Research supported by EPSRC research grant No. EP/G000212, Computing with Partial Information: Definability in the Local Structure of the Enumeration Degrees.
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Harris, C.M. Badness and jump inversion in the enumeration degrees. Arch. Math. Logic 51, 373–406 (2012). https://doi.org/10.1007/s00153-012-0268-9
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DOI: https://doi.org/10.1007/s00153-012-0268-9