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Promptness does not imply superlow cuppability

Published online by Cambridge University Press:  12 March 2014

David Diamondstone
David Diamondstone*
Affiliation:
University of Chicago, Department of Mathematics, 5734 S. University Avenue Chicago, Illinois 60637, USA, E-mail: ded@math.uchicago.edu
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Abstract

A classical theorem in computability is that every promptly simple set can be cupped in the Turing degrees to some complete set by a low c.e. set. A related question due to A. Nies is whether every promptly simple set can be cupped by a superlow c.e. set, i.e. one whose Turing jump is truth-table reducible to the halting problem ∅′. A negative answer to this question is provided by giving an explicit construction of a promptly simple set that is not superlow cuppable. This problem relates to effective randomness and various lowness notions.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 74 , Issue 4 , December 2009 , pp. 1264 - 1272
Copyright
Copyright © Association for Symbolic Logic 2009

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References

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