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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The Slaman-Wehner theorem in higher recursion theory


Authors: Noam Greenberg, Antonio Montalbán and Theodore A. Slaman
Journal: Proc. Amer. Math. Soc. 139 (2011), 1865-1869
MSC (2010): Primary 03C57, 03D60; Secondary 03D45
Published electronically: November 23, 2010
MathSciNet review: 2763773
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Abstract: Slaman and Wehner have independently shown that there is a countable structure whose degree spectrum consists of the nonzero Turing degrees. We show that the analogue fails in the degrees of constructibility. While we do not settle the problem for the hyperdegrees, we show that every almost computable structure, in the sense of Kalimullin, has a copy computable from Kleene's $ \mathcal{O}$.


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Additional Information

Noam Greenberg
Affiliation: School of Mathematics, Statistics and Operations Research, Victoria University, Wellington, New Zealand
Email: greenberg@msor.vuw.ac.nz

Antonio Montalbán
Affiliation: Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, Illinois 60637
Email: antonio@math.uchicago.edu

Theodore A. Slaman
Affiliation: Department of Mathematics, University of California Berkeley, Berkeley, California 94720-3840
Email: slaman@math.berkeley.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10693-7
PII: S 0002-9939(2010)10693-7
Received by editor(s): May 12, 2010
Published electronically: November 23, 2010
Additional Notes: The first author’s research was partially supported by the Marsden fund of New Zealand.
The second author’s research was partially supported by NSF grant DMS-0901169
The third author’s research was partially supported by NSF award DMS-1001551 and by the John Templeton Foundation.
Communicated by: Julia Knight
Article copyright: © Copyright 2010 American Mathematical Society