On the jump classes of noncuppable enumeration degrees

We prove that for every ${\mathrm{\Sigma }}_{2}^{0}$ enumeration degree b there exists a noncuppable ${\mathrm{\Sigma }}_{2}^{0}$ degree a > 0 e such that b′ ≤ e a′ and a″ ≤ e b″. This allows us to deduce, from results on the high/low jump hierarchy in the local Turing degrees and the jump preser...

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Bibliographic Details
Published in:The Journal of symbolic logic 2011-03, Vol.76 (1), p.177-197
Main Author: Harris, Charles M.
Format: Article
Language:English
Subjects:
Approximation
Archives
Degree of unsolvability
Index sets
Mathematical functions
Mathematical logic
Mathematical theorems
Oracles
Property replacement
Recursion theory
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Summary:We prove that for every ${\mathrm{\Sigma }}_{2}^{0}$ enumeration degree b there exists a noncuppable ${\mathrm{\Sigma }}_{2}^{0}$ degree a > 0 e such that b′ ≤ e a′ and a″ ≤ e b″. This allows us to deduce, from results on the high/low jump hierarchy in the local Turing degrees and the jump preserving properties of the standard embedding l: D T → D e , that there exist ${\mathrm{\Sigma }}_{2}^{0}$ noncuppable enumeration degrees at every possible—i.e., above low₁—level of the high/low jump hierarchy in the context of D e .
ISSN:0022-4812
1943-5886
DOI:10.2178/jsl/1294170994