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$K$-trivial degrees and the jump-traceability hierarchy


Authors: George Barmpalias, Rod Downey and Noam Greenberg
Journal: Proc. Amer. Math. Soc. 137 (2009), 2099-2109
MSC (2000): Primary 03D80
DOI: https://doi.org/10.1090/S0002-9939-09-09761-5
Published electronically: January 22, 2009
MathSciNet review: 2480292
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Abstract: For every order $h$ such that $\sum _n 1/h(n)$ is finite, every $K$-trivial degree is $h$-jump-traceable. This motivated Cholak, Downey and Greenberg to ask whether this traceability property is actually equivalent to $K$-triviality, thereby giving the hoped for combinatorial characterisation of lowness for Martin-Löf randomness. We show however that the $K$-trivial degrees are properly contained in those that are $h$-jump-traceable for every convergent order $h$.


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Additional Information

George Barmpalias
Affiliation: School of Mathematics, Statistics and Computer Science, Victoria University, P.O. Box 600, Wellington, New Zealand
Email: George.Barmpalias@mcs.vuw.ac.nz

Rod Downey
Affiliation: School of Mathematics, Statistics and Computer Science, Victoria University, P.O. Box 600, Wellington, New Zealand
MR Author ID: 59535
Email: downey@mcs.vuw.ac.nz

Noam Greenberg
Affiliation: School of Mathematics, Statistics and Computer Science, Victoria University, P.O. Box 600, Wellington, New Zealand
MR Author ID: 757288
ORCID: 0000-0003-2917-3848
Email: greenberg@mcs.vuw.ac.nz

Keywords: K-triviality, Turing degrees, jump traceability, superlinear orders
Received by editor(s): March 20, 2008
Received by editor(s) in revised form: September 11, 2008
Published electronically: January 22, 2009
Additional Notes: All authors were supported by the Marsden Fund of New Zealand.
Communicated by: Julia Knight
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.