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The infinite injury priority method1

Published online by Cambridge University Press:  12 March 2014

Robert I. Soare*
Affiliation:
University of Illinois, Chicago Circle, Illinois 60680 University of Chicago, Chicago, Illinois 60637

Extract

One of the most important and distinctive tools in recursion theory has been the priority method whereby a recursively enumerable (r.e.) set A is constructed by stages to satisfy a sequence of conditions {Rn }n∈ω called requirements. If n < m, requirement Rn is given priority over Rm and action taken for Rm at some stage s may at a later stage t > s be undone for the sake of Rn thereby injuring Rm at stage t. The first priority method was invented by Friedberg [2] and Muchnik [11] to solve Post's problem and is characterized by the fact that each requirement is injured at most finitely often.

Shoenfield [20, Lemma 1], and then independently Sacks [17] and Yates [25] invented a much more powerful method in which a requirement may be injured infinitely often, and the method was applied and refined by Sacks [15], [16], [17], [18], [19] and Yates [25], [26] to obtain many deep results on r.e. sets and their degrees. In spite of numerous simplifications and variations this infinite injury method has never been as well understood as the finite injury method because of its apparently greater complexity.

The purpose of this paper is to reduce the Sacks method to two easily understood lemmas whose proofs are very similar to the finite injury case. Using these lemmas we can derive all the results of Sacks on r.e. degrees, and some by Yates and Robinson as well, in a manner accessible to the nonspecialist. The heart of the method is an ingenious observation of Lachlan [7] which is combined with a further simplification of our own.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1976

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Footnotes

1

This research was supported by NSF grant 19958A#3. There have been several illuminating explanations of the infinite injury method including Sacks [16], Yates [24], Martin [8], Shoen-field [21] and Lachlan [5], [6], and [7]. We gratefully acknowledge our debt to all of these sources, particuarly the last. In addition this paper reflects conversations with C. G. Jockusch, Jr., A. H. Lachlan, M. Lermari, A. Manaster, D. A. Martin and R. A. Shore. This paper was presented on April 12, 1975 at a Special Session for Recursion Theory during the American Mathematical Society meeting in St. Louis.

References

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