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An α-finite injury method of the unbounded type1

Published online by Cambridge University Press:  12 March 2014

C. T. Chong
C. T. Chong*
Affiliation:
Yale University, New Haven, Connecticut 06520 University of Singapore, Republic of Singapore
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Extract

Let α be an admissible ordinal. In this paper we study the structure of the upper semilattice of α-recursively enumerable degrees. Various results about the structure which are of fundamental importance had been obtained during the past two years (Sacks-Simpson [7], Lerman [4], Shore [9]). In particular, the method of finite priority argument of Friedberg and Muchnik was successfully generalized in [7] to an α-finite priority argument to give a solution of Post's problem for all admissible ordinals. We refer the reader to [7] for background material, and we also follow closely the notations used there.

Whereas [7] and [4] study priority arguments in which the number of injuries inflicted on a proper initial segment of requirements can be effectively bounded (Lemma 2.3 of [7]), we tackle here priority arguments in which no such bounds exist. To this end, we focus our attention on the fine structure of Lα , much in the fashion of Jensen [2], and show that we can still use a priority argument on an indexing set of requirements just short enough to give us the necessary bounds we seek.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 41 , Issue 1 , March 1976 , pp. 1 - 17
Copyright
Copyright © Association for Symbolic Logic 1976

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Footnotes

1

The results in this paper form a portion of the author's Ph.D. dissertation (Yale University, 1973) written under the guidance of Professor M. Lerman. We would like to thank him for the suggestions and encouragement that we received during the course of this work.

References

BIBLIOGRAPHY

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