The Journal of Symbolic Logic

Research Article

Maximal contiguous degrees

Peter Cholaka1, Rod Downeya2 and Stephen Walka3

a1 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

a2 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

a3 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

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.

(Received June 06 2000)

(Revised July 26 2001)

Key words and phrases

  • computably enumerable;
  • contiguous;
  • automorphism base;
  • array computable