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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

$\Sigma_n$-bounding and $\Delta_n$-induction


Author: Theodore A. Slaman
Journal: Proc. Amer. Math. Soc. 132 (2004), 2449-2456
MSC (2000): Primary 03F30, 03H15
Published electronically: March 25, 2004
MathSciNet review: 2052424
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Abstract | References | Similar Articles | Additional Information

Abstract: Working in the base theory of $\mathrm{PA}^- + \mathrm{I}\Sigma_0 +\exp$, we show that for all $n\in\omega$, the bounding principle for $\Sigma_n$-formulas ( $\mathrm{B}\Sigma_n$) is equivalent to the induction principle for $\Delta_n$-formulas ( $\mathrm{I}\Delta_n$). This partially answers a question of J. Paris.


References [Enhancements On Off] (What's this?)

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Additional Information

Theodore A. Slaman
Affiliation: Department of Mathematics, University of California Berkeley, Berkeley, California 94720-3840
Email: slaman@math.berkeley.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07294-6
PII: S 0002-9939(04)07294-6
Keywords: $\Sigma_n$-bounding, $\Delta_n$-induction
Received by editor(s): November 20, 2002
Received by editor(s) in revised form: February 20, 2003
Published electronically: March 25, 2004
Additional Notes: During the preparation of this paper, the author was partially supported by the Alexander von Humboldt Foundation and by the National Science Foundation Grant DMS-9988644. The author is grateful to Jan Krajíček for reading a preliminary version of this paper and suggesting improvements to it.
Communicated by: Carl G. Jockusch, Jr.
Article copyright: © Copyright 2004 American Mathematical Society