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Maximal contiguous degrees

Published online by Cambridge University Press:  12 March 2014

Peter Cholak ,
Rod Downey  and
Stephen Walk
Peter Cholak
Affiliation:
Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556-5683., USA, E-mail: Peter.Cholak.l@nd.edu, URL: http://www.nd.edu/~cholak
Rod Downey
Affiliation:
Department of Mathematics, Victoria University of Wellington, Wellington, New Zealand, E-mail: rod.downey@vuw.ac.nz, URL: http://www.mcs.vuw.ac.nz/people/Rod-Downey.shtml
Stephen Walk
Affiliation:
Department of Mathematics, Ecc 139, 720 4th Avenue South, St. Cloud State University, St. Cloud, Minnesota 56301-4498, E-mail: smwalk@stcloudstate.edu, URL: http://tigger.stcloudstate.edu/~swalk
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Abstract

A computably enumerable (c.e.) degree is a maximal contiguous degree if it is contiguous and no c.e. degree strictly above it is contiguous. We show that there are infinitely many maximal contiguous degrees. Since the contiguous degrees are definable, the class of maximal contiguous degrees provides the first example of a definable infinite anti-chain in the c.e. degrees. In addition, we show that the class of maximal contiguous degrees forms an automorphism base for the c.e. degrees and therefore for the Turing degrees in general. Finally we note that the construction of a maximal contiguous degree can be modified to answer a question of Walk about the array computable degrees and a question of Li about isolated formulas.

Keywords

computably enumerablecontiguousautomorphism basearray computable
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 67 , Issue 1 , March 2002 , pp. 409 - 437
Copyright
Copyright © Association for Symbolic Logic 2002

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