Nonisolated degrees and the jump operator
type: Item
instance of: scholarly article
title: Nonisolated degrees and the jump operator
zbMATH Open document ID: 1033.03027
DOI: 10.1016/S0168-0072(01)00117-8
author: Guohua Wu
published in: Annals of Pure and Applied Logic
publication date: 2002-12-02T00:00:00Z
review text: A Turing degree is called r.e.\ if it contains a recursively enumerable set and d.r.e.\ if it contains a set which is the set-theoretic difference of two r.e.\ sets. A d.r.e.\ degree \(\mathbf d\) is nonisolated if for any r.e.\ degree \({\mathbf a} < {\mathbf d}\) there is another r.e.\ degree \(\mathbf b\) properly between these two degrees: \({\mathbf a} < {\mathbf b} < {\mathbf d}\). It is shown that there are a low d.r.e.\ degree and a high nonisolated d.r.e.\ degree above it such that both bound the same r.e.\ degrees. More precisely, there is a low d.r.e.\ degree \(\mathbf c\) and a high nonisolated d.r.e.\ degree \(\mathbf d\) such that \({\mathbf c} < {\mathbf d}\) and \((\forall\) r.e.\ \({\mathbf a} < {\mathbf d})\,(\exists\) r.e.\ \({\mathbf b})\,({\mathbf a} < {\mathbf b} < {\mathbf c})\). Note that the degree \(\mathbf d\) can be taken to be properly d.r.e., that is, \(\mathbf d\) does not contain a recursively enumerable set. The degree \(\mathbf c\) is automatically properly d.r.e.\ and automatically nonisolated by the condition that it bounds all r.e.\ degrees below the nonisolated degree \(\mathbf d\). Ishmukhametov and the author proved the parallel result for isolated degrees: There is a high d.r.e.\ degree \(\mathbf d\) and a low r.e.\ degree \(\mathbf c\) such that every r.e.\ degree \({\mathbf a} \leq {\mathbf d}\) satisfies \({\mathbf a} \leq {\mathbf c}\) [\textit{Sh. Ishmukhametov} and \textit{G. Wu}, Arch. Math. Logic 41, 259--266 (2002; Zbl 1028.03035)].
reviewed by: Frank Stephan
Mathematics Subject Classification ID: 03D28
Mathematics Subject Classification ID: 03D25
zbMATH DE Number: 1838862
zbMATH Keywords: recursively enumerable sets
zbMATH Keywords: d.r.e.\ degrees
zbMATH Keywords: isolation
zbMATH Keywords: low degrees
zbMATH Keywords: high degrees
zbMATH Keywords: computably enumerable sets
zbMATH Keywords: Turing degree
MaRDI profile type: Publication
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cites work: Isolation and the jump operator
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description: scientific article; zbMATH DE number 1838862