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Intermediate β-r.e. degrees and the half-jump

Published online by Cambridge University Press:  12 March 2014

Steven Homer
Steven Homer*
Affiliation:
De Paul University, Chicago, Illinois 60614
*
Boston University, Boston, Massachusetts 02215
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Extract

In his thesis [2] Sy Friedman showed that for any inadmissible ordinal β there is an easily definable β-r.e. degree, 01/2, with β-degree strictly between 0 and 0′. In this paper we define, for some weakly admissible β, a β-r.e. degree 03/4 intermediate between 01/2 and 0′. This definition is given using the machinery developed by Maass [6], [7] to study the β-r.e. degrees.

A weak jump for β-recursion theory, the ½-jump, was first defined by Friedman [4] and has been studied by Sacks and Homer [5]. The degree 03/4 is of interest because of the way it interacts with the half-jump. In particular 03/4 forms the boundary between the half-jump of “hyperregular” and “nonhyperregular” β-r.e. degrees less than 01/2. One consequence of these results is that there are weakly inadmissible β for which either the jump theorem or the density theorem fails.

Much of this paper is based on earlier work in this area by Sy Friedman and Wolfgang Maass. I would like to thank Gerald Sacks for many helpful conversations and comments concerning this work.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 48 , Issue 3 , September 1983 , pp. 790 - 796
Copyright
Copyright © Association for Symbolic Logic 1983

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References

BIBLIOGRAPHY

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