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Σ2 Induction and infinite injury priority argument, Part I: Maximal sets and the jump operator

Published online by Cambridge University Press:  12 March 2014

C. T. Chong  and
Yue Yang
C. T. Chong
Affiliation:
Department of Mathematics, Faculty of Science, National University of Singapore, Lower Kent Ridge Road, Singapore 119260, E-mail: chongct@math.nus.edu.sg
Yue Yang
Affiliation:
Department of Mathematics, Faculty of Science, National University of Singapore, Lower Kent Ridge Road, Singapore 119260, E-mail: matyangy@nus.edu.sg
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Extract

The study of recursion theory on models of fragments of Peano arithmetic has hitherto been concentrated on recursively enumerable (r.e.) sets and their degrees (with a few exceptions, such as that in [2] on minimal degrees). The reason for such a concerted effort is clear: priority arguments have occupied a central position in post Friedberg-Muchnik recursion theory, and after almost forty years of intensive development in the subject, they are still the essential tools on which investigations of r.e. sets and their degrees depend. There are two possible approaches to the study within fragments of arithmetic: To give a general analysis of strategies, and identify their proof-theoretic strengths (for example in [6] on infinite injury priority methods), or to consider specific theorems in recursion theory, and, if possible, pinpoint the exact levels of induction provably equivalent to the theorems. The work reported in this paper belongs to the second approach. More precisely, we single out two infinitary injury type constructions of r.e. sets—one concerning maximal sets and the other based on the notion of the jump operator—to be the topics of study.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 63 , Issue 3 , September 1998 , pp. 797 - 814
Copyright
Copyright © Association for Symbolic Logic 1998

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References

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