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Relatively computably enumerable reals

Relatively computably enumerable reals A real X is defined to be relatively c.e. if there is a real Y such that X is c.e.(Y) and $${X \not\leq_T Y}$$ . A real X is relatively simple and above if there is a real Y < T X such that X is c.e.(Y) and there is no infinite set $${Z \subseteq \overline{X}}$$ such that Z is c.e.(Y). We prove that every nonempty $${\Pi^0_1}$$ class contains a member which is not relatively c.e. and that every 1-generic real is relatively simple and above. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Archive for Mathematical Logic Springer Journals

Relatively computably enumerable reals

Anderson, Bernard
Archive for Mathematical Logic , Volume 50 (4) – Nov 19, 2010
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References (12)

Publisher
Springer Journals
Copyright
Copyright © 2010 by Springer-Verlag
Subject
Mathematics; Mathematics, general; Algebra; Mathematical Logic and Foundations
ISSN
0933-5846
eISSN
1432-0665
DOI
10.1007/s00153-010-0219-2
Publisher site
See Article on Publisher Site

Abstract

A real X is defined to be relatively c.e. if there is a real Y such that X is c.e.(Y) and $${X \not\leq_T Y}$$ . A real X is relatively simple and above if there is a real Y < T X such that X is c.e.(Y) and there is no infinite set $${Z \subseteq \overline{X}}$$ such that Z is c.e.(Y). We prove that every nonempty $${\Pi^0_1}$$ class contains a member which is not relatively c.e. and that every 1-generic real is relatively simple and above.

Journal

Archive for Mathematical Logic – Springer Journals

Published: Nov 19, 2010

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