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

 

There is no degree invariant half-jump


Authors: Rodney G. Downey and Richard A. Shore
Journal: Proc. Amer. Math. Soc. 125 (1997), 3033-3037
MSC (1991): Primary 03D25, 03E60, 04A15; Secondary 03D30
MathSciNet review: 1401736
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Abstract: We prove that there is no degree invariant solution to Post's problem that always gives an intermediate degree. In fact, assuming definable determinacy, if $W$ is any definable operator on degrees such that ${\mathbf a<}W({\mathbf a)<{\mathbf a}}^{\prime }$ on a cone then $W$ is low$_2$ or high$_2$ on a cone of degrees, i.e., there is a degree ${\mathbf c}$ such that $W({\mathbf a)}^{\prime \prime }={\mathbf a}^{\prime \prime }$ for every ${\mathbf a\geq {\mathbf c}}$ or $W({\mathbf a)}^{\prime \prime }={\mathbf a}^{\prime \prime \prime }$ for every ${\mathbf a\geq {\mathbf c}}$.


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Additional Information

Rodney G. Downey
Affiliation: Department of Mathematics, Victoria University of Wellington, P. O. Box 600, Wellington, New Zealand
Email: rod.downey@vuw.ac.nz

Richard A. Shore
Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853
Email: shore@math.cornell.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-97-03915-4
PII: S 0002-9939(97)03915-4
Keywords: Post's problem, degree invariant operators, recursively enumerable degrees, computably enumerable degrees, Martin's conjecture, axiom of determinacy
Received by editor(s): February 16, 1996
Received by editor(s) in revised form: May 9, 1996
Additional Notes: The first author’s research was partially supported by the U.S. ARO through ACSyAM at the Mathematical Sciences Institute of Cornell University Contract DAAL03-91-C-0027, the IGC of Victoria University and the Marsden Fund for Basic Science under grant VIC-509.
The second author’s research was partially supported by NSF Grant DMS-9503503 and the U.S. ARO through ACSyAM at the Mathematical Sciences Institute of Cornell University Contract DAAL03-91-C-0027.
Communicated by: Andreas R. Blass
Article copyright: © Copyright 1997 American Mathematical Society