Transfinite induction principles and iterated reflection principles

Abstract

We study the relationships between partial transfinite induction principles obtained by restricting the induction formulae to a class $\Sigma_n$ or $\Pi_n$ in the arithmetical hierarchy and similarly restricted iterated reflection principles. We prove over the elementary arithmetic that the transfinite induction schema for $\Pi_n$ formulae along the ordinal $\omega^{\omega^\alpha}$ is equivalent to $\alpha+1$ times iterated uniform reflection principle for $\Pi_{n+2}$ formulae. This yields a complete characterisation of transfinite induction schemata in terms of reflection principles, closing the remaining gap in the overall picture.