Badness and jump inversion in the enumeration degrees

• Authors: Charles M. Harris
• Venue: Archive for Mathematical Logic
• Year: 2012

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 驴 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.