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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Splitting an $ \alpha $-recursively enumerable set

Author: Richard A. Shore
Journal: Trans. Amer. Math. Soc. 204 (1975), 65-77
MSC: Primary 02F27
MathSciNet review: 0379154
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Abstract: We extend the priority method in $ \alpha $-recursion theory to certain arguments with no a priori bound on the required preservations by proving the splitting theorem for all admissible $ \alpha $. THEOREM: Let $ C$ be a regular $ \alpha $-r.e. set and $ D$ be a nonrecursive $ \alpha $-r.e. set. Then there are regular $ \alpha $-r.e. sets $ A$ and $ B$ such that $ A \cup B = C,A \cap B = \phi ,A,B{ \leq _\alpha }C$ and such that $ D$ is not $ \alpha $-recursive in $ A$ or $ B$. The result is also strengthened to apply to $ { \leq _{c\alpha }}$, and various corollaries about the structure of the $ \alpha $ and $ c\alpha $ recursively enumerable degrees are proved.

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Keywords: $ \alpha $-recursion theory, admissible ordinals, $ \alpha $-recursively enumerable, priority argument, splitting theorem, $ \alpha $-degrees, $ \alpha $-calculability degrees
Article copyright: © Copyright 1975 American Mathematical Society