On the induction schema for decidable predicates

Abstract

We study the fragment of Peano arithmetic formalizing the induction principle for the class of decidable predicates, $I\Delta₁$. We show that $I\Delta₁$ is independent from the set of all true arithmetical $\Pi₂$-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 $\Delta₁$-induction.

An open problem formulated by J. Paris (see \cite{CloKra,HP}) is whether $I\Delta₁$ proves the corresponding least element principle for decidable predicates, $L\Delta₁$ (or, equivalently, the $\Sigma₁$-collection principle $B\Sigma₁$). We reduce this question to a purely computation-theoretic one.