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On the induction schema for decidable predicates

Published online by Cambridge University Press:  12 March 2014

Lev D. Beklemishev
Lev D. Beklemishev*
Affiliation:
Steklov Mathematical Institute, Gubkina 8, 117966 Moscow, Russia, E-mail: bekl@mi.ras.ru
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Abstract

We study the fragment of Peano arithmetic formalizing the induction principle for the class of decidable predicates, IΔ1. We show that IΔ1 is independent from the set of all true arithmetical Π2-sentences. Moreover, we establish the connections between this theory and some classes of oracle computable functions with restrictions on the allowed number of queries. We also obtain some conservation and independence results for parameter free and inference rule forms of Δ1-induction.

An open problem formulated by J. Paris (see [4, 5]) is whether IΔ1 proves the corresponding least element principle for decidable predicates, LΔ1 (or, equivalently, the Σ1-collection principle BΣ1). We reduce this question to a purely computation-theoretic one.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 68 , Issue 1 , March 2003 , pp. 17 - 34
Copyright
Copyright © Association for Symbolic Logic 2003

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References

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