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Inherent enumerability of strong jump-traceability


Authors: David Diamondstone, Noam Greenberg and Daniel D. Turetsky
Journal: Trans. Amer. Math. Soc. 367 (2015), 1771-1796
MSC (2010): Primary 03D25, 03D28, 03D32
DOI: https://doi.org/10.1090/S0002-9947-2014-06089-3
Published electronically: November 12, 2014
MathSciNet review: 3286498
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Abstract: We show that every strongly jump-traceable set obeys every benign cost function. Moreover, we show that every strongly jump-traceable set is computable from a computably enumerable strongly jump-traceable set. This allows us to generalise properties of c.e.strongly jump-traceable sets to all such sets. For example, the strongly jump-traceable sets induce an ideal in the Turing degrees; the strongly jump-traceable sets are precisely those that are computable from all superlow Martin-Löf random sets; the strongly jump-traceable sets are precisely those that are a base for $ \textup {Demuth}_{\textup {BLR}}$ randomness; and strong jump-traceability is equivalent to strong superlowness.


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Additional Information

David Diamondstone
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Email: ddiamondstone@gmail.com

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

Daniel D. Turetsky
Affiliation: Kurt Gödel Research Center, University of Vienna, 1090 Vienna, Austria
Email: turetsd4@univie.ac.at

DOI: https://doi.org/10.1090/S0002-9947-2014-06089-3
Received by editor(s): October 25, 2011
Received by editor(s) in revised form: January 27, 2013
Published electronically: November 12, 2014
Additional Notes: All authors were supported by the Marsden Fund of New Zealand, the first and the third as postdoctoral fellows. The second author was also supported by a Rutherford Discovery Fellowship.
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.