Induction Rules, Reflection Principles, and Provably Recursive Functions (1995)
| Venue: | University of Utrecht |
| Citations: | 8 - 3 self |
BibTeX
@INPROCEEDINGS{Beklemishev95inductionrules,,
author = {Lev D. Beklemishev},
title = {Induction Rules, Reflection Principles, and Provably Recursive Functions},
booktitle = {University of Utrecht},
year = {1995}
}
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Abstract
A well-known result of D. Leivant states that, over basic Kalmar elementary arithmetic EA, the induction schema for \Sigma n formulas is equivalent to the uniform reflection principle for \Sigma n+1 formulas. We show that fragments of arithmetic axiomatized by various forms of induction rules admit a precise axiomatization in terms of reflection principles as well. Thus, the closure of EA under the induction rule for \Sigma n (or \Pi n+1 ) formulas is equivalent to ! times iterated \Sigma n reflection principle. Moreover, for k ! !, k times iterated \Sigma n reflection principle over EA precisely corresponds to the extension of EA by k nested applications of \Sigma n induction rule. The above relationship holds in greater generality than just stated. In fact, we give general formulas characterizing in terms of iterated reflection principles the extension of any given theory (containing EA) by k nested applications of \Sigma n or \Pi n induction rules. In particular, the closure of a ...
