On the Relationships between $ATR_0$ And $\widehat{ID}_{< \omega}$

Abstract

We show that the theory $ATR_0$ is equivalent to a second-order generalization of the theory $\widehat{ID}_{<\omega}$. As a result, $ATR_0$ is conservative over $\widehat{ID}_{<\omega}$ for arithmetic sentences, though proofs in $ATR_0$ can be much shorter than their $\widehat{ID}_{<\omega}$ counterparts.