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Π11 Sets,ω-Sets, and metacompleteness

Published online by Cambridge University Press:  12 March 2014

James C. Owings Jr.
James C. Owings Jr.*
Affiliation:
University of Maryland
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Extract

An ω-set is a subset of the recursive ordinals whosecomplement with respect to the recursive ordinals is unbounded and has ordertype ω. This concept has proved fruitful in the study of sets in relation to metarecursion theory. We provethat the metadegrees of the setscoincide with those of the meta-r.e. ω-sets. We then show that, given any set, ametacomplete set can befound which is weakly metarecursive in it. It then follows that weakrelative metarecursiveness is not a transitive relation on the sets,extending a result of G. Driscoll [2, Theorem 3.1]. Coincidentally, wediscuss the notions of total and complete regularity. Finally, we solvePost's problem for the transitive closure of weak relativemetarecursiveness. We recommend the reader look at pp. 324–328 of thefundamental article [6] of Kreisel and Sacks before proceeding. He will findthere a proof of the following very basic fact: a subset of the integers is iff it ismetarecursively enumerable (metafinite).

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 34 , Issue 2 , 25 July 1969 , pp. 194 - 204
Copyright
Copyright © Association for Symbolic Logic 1969

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Footnotes

1

Most of the material in this paper is taken from the author's Ph.D. thesis (Cornell University, 1966), supervised by Gerald E. Sacks and supported by a N.S.F. Graduate Fellowship. The author is beholden to Sacks for developing and popularizing the beautiful intricacies of metarecursion theory. This work was also supported by NSF Contract GP 6897.

References

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