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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 whose complement with respect to the recursive ordinals is unbounded and has order type ω. This concept has proved fruitful in the study of sets in relation to metarecursion theory. We prove that the metadegrees of the sets coincide with those of the meta-r.e. ω-sets. We then show that, given any set, a metacomplete set can be found which is weakly metarecursive in it. It then follows that weak relative metarecursiveness is not a transitive relation on the sets, extending a result of G. Driscoll [2, Theorem 3.1]. Coincidentally, we discuss the notions of total and complete regularity. Finally, we solve Post's problem for the transitive closure of weak relative metarecursiveness. We recommend the reader look at pp. 324–328 of the fundamental article [6] of Kreisel and Sacks before proceeding. He will find there a proof of the following very basic fact: a subset of the integers is iff it is metarecursively enumerable (metafinite).

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

[1] Dekker, J. C. E., A theorem on hypersimple sets, Proceedings of the American Mathematical Society, vol. 5 (1954), pp. 791–796.Google Scholar
[2] Driscoix, G. C. Jr., Contributions to metarecursion theory, Ph.D. thesis, Cornell University, Ithaca, New York, 1965.Google Scholar
[3] Driscoll, G. C. Jr., Metarecursively enumerable sets and their metadegrees, this Journal , vol. 33 (1968), pp. 389–411.Google Scholar
[4] Kleene, S. C., On notations for ordinal numbers, this Journal , vol. 3 (1938), pp. 150–155.Google Scholar
[5] Kleene, S. C., On the forms of the predicates in the theory of constructive ordinals (second paper), American Journal of Mathematics, vol. 77 (1955), pp. 405–428.Google Scholar
[6] Kreisel, G. and Sacks, G. E., Metarecursive sets, this Journal , vol. 30 (1965), pp. 318–338.Google Scholar
[7] Sacks, G. E., Post's problem, admissible ordinals, and regularity, Transactions of the American Mathematical Society, vol. 124 (1966), pp. 1–23.Google Scholar
[8] Sacks, G. E., Metarecursion theory, Sets, models, and recursion theory, Leicester Symposium Volume, North Holland Publ. Co., Amsterdam, 1965.Google Scholar
[9] Sacks, G. E., Metarecurswely enumerable sets and admissible ordinals, Bulletin of the American Mathematical Society, vol. 72 (1966), pp. 59–64.Google Scholar
[10] Spector, C., Recursive well-orderings, this Journal , vol. 20 (1955), pp. 151–163.Google Scholar