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A 1-generic degree with a strong minimal cover

Published online by Cambridge University Press:  12 March 2014

Masahiro Kumabe
Masahiro Kumabe*
Affiliation:
University of the Air, Kanayama Study Center, 2-31-1, Osaka, Minami-Ku, Yokohama 232-0061, Japan E-mail: kumabe@u-air.ac.jp
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Extract

We consider a set generic over the arithmetic sets. A subset A of the natural numbers is called n-generic if it is Cohen-generic for n-quantifier arithmetic. This is equivalent to saying that for every -set of strings S, there is a string σ ⊂ A such that σ ∈ S or no extension of σ is in S. By degree we mean Turing degree (of unsolvability). We call a degree n-generic if it has an n-generic representative. For a degree a, let D(≤ a) denote the set of degrees which are recursive in a.

We say a is a strong minimal cover of g if every degree strictly below a is less than or equal to g. In this paper we show that there are a degree a and a 1-generic degree g < a such that a is a strong minimal cover of g. This easily implies that there is a 1-generic degree without the cupping property. Jockusch [7] showed that every 2-generic degree has the cupping property. Slaman and Steel [17] and independently Cooper [3] showed that there are recursively enumerable degrees a and b < a such that no degree c < a joins b above a. Take a 1-generic degree g below b. Then g does not have the cupping property.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 65 , Issue 3 , September 2000 , pp. 1395 - 1442
Copyright
Copyright © Association for Symbolic Logic 2000

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References

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