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Maximal sets and fragments of Peano arithmetic

Published online by Cambridge University Press:  22 January 2016

C.T. Chong
C.T. Chong*
Affiliation:
Department of Mathematics, National University of Singapore, Kent Ridge, Singapore 0511, Republic of Singapore
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This work is inspired by the recent paper of Mytilinaios and Slaman [9] on the infinite injury priority method. It may be considered to fall within the general program of the study of reverse recursion theory: What axioms of Peano arithmetic are required or sufficient to prove theorems in recursion theory? Previous contributions to this program, especially with respect to the finite and infinite injury priority methods, can be found in the works of Groszek and Mytilinaios [4], Groszek and Slaman [5], Mytilinaios [8], Slaman and Woodin [10]. Results of [4] and [9], for example, together pinpoint the existence of an incomplete, nonlow r.e. degree to be provable only within some fragment of Peano arithmetic at least as strong as P- + IΣ2. Indeed an abstract principle on infinite strategies, such as that on the construction of an incomplete high r.e. degree, was introduced in [4] and shown to be equivalent to Σ2 induction over the base theory P- + IΣ0 of Peano arithmetic.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 115 , September 1989 , pp. 165 - 183
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1989

References

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