Intervals containing exactly one c.e. degree

Annals of Pure and Applied Logic 146 (1):91-102 (2007)

Abstract
Cooper proved in [S.B. Cooper, Strong minimal covers for recursively enumerable degrees, Math. Logic Quart. 42 191–196] the existence of a c.e. degree with a strong minimal cover . So is the greastest c.e. degree below . Cooper and Yi pointed out in [S.B. Cooper, X. Yi, Isolated d.r.e. degrees, University of Leeds, Dept. of Pure Math., 1995. Preprint] that this strongly minimal cover cannot be d.c.e., and meanwhile, they proposed the notion of isolated degrees: a d.c.e. degree is isolated by a c.e. degree if is the greatest c.e. degree below , and we also say that isolates . In [G. Wu, Bi-isolation in the d.c.e. degrees, J. Symbolic Logic 69 409–420], Wu extended Cooper–Yi’s notion and proved that there are intervals of d.c.e. degrees containing exactly one c.e. degree . Following Cooper and Yi’s notion, is called a bi-isolating degree. The bi-isolating degrees are dense in the high c.e. degrees. Arslanov asked whether the bi-isolating degrees occur in every jump class. In this paper, we prove that there are low bi-isolating degrees, providing a partial solution to Arslanov’s question
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DOI 10.1016/j.apal.2007.01.002
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The D.R.E. Degrees Are Not Dense.S. Cooper, Leo Harrington, Alistair Lachlan, Steffen Lempp & Robert Soare - 1991 - Annals of Pure and Applied Logic 55 (2):125-151.
A Minimal Pair of Recursively Enumerable Degrees.C. E. M. Yates - 1966 - Journal of Symbolic Logic 31 (2):159-168.
Systems of Logic Based on Ordinals.Alan Mathison Turing - 1939 - London: Printed by C.F. Hodgson & Son.

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