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The Complexity of Orbits of Computably Enumerable Sets

Published online by Cambridge University Press:  15 January 2014

Peter A. Cholak ,
Rodney Downey  and
Leo A. Harrington
Peter A. Cholak
Affiliation:
Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556-5683, USA E-mail: Peter.Cholak.1@nd.edu URL: http://www.nd.edu/~cholak
Rodney Downey
Affiliation:
School of Mathematics, Statistics and Computer Science, Victoria University, P.O. BOX 600, Wellington, New Zealand E-mail: Rod.Downey@vuw.ac.nz , URL: http://www.mcs.vuw.ac.nz/~downey
Leo A. Harrington
Affiliation:
Department of Mathematics, University of California, Berkeley, CA 94720-3840, USA E-mail: leo@math.berkeley.edu
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Abstract

The goal of this paper is to announce there is a single orbit of the c.e. sets with inclusion, ε, such that the question of membership in this orbit is complete. This result and proof have a number of nice corollaries: the Scott rank of ε is + 1; not all orbits are elementarily definable; there is no arithmetic description of all orbits of ε; for all finite α ≥ 9, there is a properly orbit (from the proof).

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Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 14 , Issue 1 , March 2008 , pp. 69 - 87
Copyright
Copyright © Association for Symbolic Logic 2008

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References

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