The high/low hierarchy in the local structure of the ω-enumeration degrees

This paper gives two definability results in the local theory of the ω-enumeration degrees. First, we prove that the local structure of the enumeration degrees is first order definable as a substructure of the ω-enumeration degrees. Our second result is the definability of the classes Hn and Ln of t...

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Bibliographic Details
Published in:Annals of pure and applied logic 2012-05, Vol.163 (5), p.547-566
Main Authors: Ganchev, Hristo, Soskova, Mariya
Format: Article
Language:English
Subjects:
[formula omitted]-enumeration degrees
Definability
Enumeration degrees
Local theory
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Summary:This paper gives two definability results in the local theory of the ω-enumeration degrees. First, we prove that the local structure of the enumeration degrees is first order definable as a substructure of the ω-enumeration degrees. Our second result is the definability of the classes Hn and Ln of the highn and lown ω-enumeration degrees. This allows us to deduce that the first order theory of true arithmetic is interpretable in the local theory of the ω-enumeration degrees. ► We prove that each nonzero Δ20 e-degree is cupped by a member of a K-pair, splitting 0e′. ► We characterize the K-pairs in the local structure of the ω-enumeration degrees, Gω. ► We prove that for every n, the degree on is first order definable in Gω. ► We prove that an isomorphic copy of the Σ20 e-degrees is definable in Gω. ► We prove that every level of the high/low jump hierarchy is definable in Gω.
ISSN:0168-0072
DOI:10.1016/j.apal.2010.10.004