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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Discrete $ \omega $-sequences of index sets


Author: Louise Hay
Journal: Trans. Amer. Math. Soc. 183 (1973), 293-311
MSC: Primary 02F25
MathSciNet review: 0349365
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Abstract: We define a discrete $ \omega $-sequence of index sets to be a sequence $ {\{ \theta {A_n}\} _{n \geq 0}}$, of index sets of classes of recursively enumerable sets, such that for each n, $ \theta {A_{n + 1}}$ is an immediate successor of $ \theta {A_n}$ in the partial order of degrees of index sets under one-one reducibility. The main result of this paper is that if S is any set to which the complete set K is not Turing-reducible, and $ {A^S}$ is the class of recursively enumerable subsets of S, then $ \theta {A^S}$ is at the bottom of c discrete $ \omega $-sequences. It follows that every complete Turing degree contains c discrete $ \omega $-sequences.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1973-0349365-8
PII: S 0002-9947(1973)0349365-8
Keywords: Index sets, classes of recursively enumerable sets, one-one reducibility, complete degrees
Article copyright: © Copyright 1973 American Mathematical Society