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Minimal complementation below uniform upper bounds for the arithmetical degrees

Published online by Cambridge University Press:  12 March 2014

Masahiro Kumabe
Masahiro Kumabe*
Affiliation:
University of the Air, Kanagawa Study Center, 2-31-1, Ohoka, Minami-Ku, Yokohama, 232, Japan, E-mail: [email protected]
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Extract

This paper was inspired by Lerman [15] in which he proved various properties of upper bounds for the arithmetical degrees. We discuss the complementation property of upper bounds for the arithmetical degrees. In Lerman [15], it is proved that uniform upper bounds for the arithmetical degrees are jumps of upper bounds for the arithmetical degrees. So any uniform upper bound for the arithmetical degrees is not a minimal upper bound for the arithmetical degrees. Given a uniform upper bound a for the arithmetical degrees, we prove a minimal complementation theorem for the upper bounds for the arithmetical degrees below a. Namely, given such a and b < a which is an upper bound for the arithmetical degrees, there is a minimal upper bound for the arithmetical degrees c such that b ∪ c = a. This answers a question in Lerman [15]. We prove this theorem by different methods depending on whether a has a function which is not dominated by any arithmetical function. We prove two propositions (see §1), of which the theorem is an immediate consequence.

Our notation is almost standard. Let A ⊕ B = {2n∣n ∈ A} ∪ {2n + 1∣n + 1∣n ∈ B} for any sets A and B. Let ω be the set of nonnegative natural numbers.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 61 , Issue 4 , December 1996 , pp. 1158 - 1192
Copyright
Copyright © Association for Symbolic Logic 1996

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