The Turing degrees of infinite levels of the Ershov hierarchy were studied by Liu and Peng [8]. In this paper, we continue the study of Turing degrees of infinite levels and lift the study of density property to the levels beyond ω2. In doing so, we rely on notations with some nice properties. We introduce the concept of normalizing notations and generate normalizing notations for higher levels. The generalizations of the weak density theorem and the nondensity theorem are proved for (...) higher levels in the Ershov hierarchy. Furthermore, we also investigate the minimal degrees in the infinite levels of the Ershov hierarchy. (shrink)

We show that for any ω-r.e. degree d and n-r.e. degree b with d

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Say that a d.c.e. degree d is nonisolated if for any c.e. degree a

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We prove that there is no maximal low d.c.e degree.

Examining various kinds of isolation phenomena in the Turing degrees, I show that there are, for every n>0, (n+1)-c.e. sets isolated in the n-CEA degrees by n-c.e. sets below them. For n≥1 such phenomena arise below any computably enumerable degree, and conjecture that this result holds densely in the c.e. degrees as well. Surprisingly, such isolation pairs also exist where the top set has high degree and the isolating set is low, although the complete situation for jump classes remains unknown.