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Transactions of the American Mathematical Society

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

 

 

Minimal covers and hyperdegrees


Author: Stephen G. Simpson
Journal: Trans. Amer. Math. Soc. 209 (1975), 45-64
MSC: Primary 02F30
DOI: https://doi.org/10.1090/S0002-9947-1975-0392534-3
MathSciNet review: 0392534
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Abstract: Every hyperdegree at or above that of Kleene's $ O$ is the hyperjump and the supremum of two minimal hyperdegrees (Theorem 3.1). There is a nonempty $ \Sigma _1^1$ class of number-theoretic predicates each of which has minimal hyperdegree (Theorem 4.7). If $ V = L$ or a generic extension of $ L$, then there are arbitrarily large hyperdegrees which are not minimal over any hyperdegree (Theorems 5.1, 5.2). If $ {O^\char93 }$ exists, then there is a hyperdegree such that every larger hyperdegree is minimal over some hyperdegree (Theorem 5.4). Several other theorems on hyperdegrees and degrees of nonconstructibility are presented.


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DOI: https://doi.org/10.1090/S0002-9947-1975-0392534-3
Keywords: Hyperdegrees, hyperjump, degrees of unsolvability, Turing degrees, constructible sets, sharp operation
Article copyright: © Copyright 1975 American Mathematical Society