We study cyclic proof systems for $μ\mathsf{PA}$, an extension of Peano arithmetic by positive inductive definitions that is arithmetically equivalent to the (impredicative) subsystem of second-order arithmetic $Π^1_2$-$\mathsf{CA}_0$ by Möllefeld. The main result of this paper is that cyclic and inductive $μ\mathsf{PA}$ have the same proof-theoretic strength. First, we translate cyclic proofs into an annotated variant based on Sprenger and Dam's systems for first-order $μ$-calculus, whose stronger validity condition allows for a simpler proof of soundness. We then formalise this argument within $Π^1_2$-$\mathsf{CA}_0$, leveraging Möllerfeld's conservativity properties. To this end, we build on prior work by Curzi and Das on the reverse mathematics of the Knaster-Tarski theorem. As a byproduct of our proof methods we show that, despite the stronger validity condition, annotated and "plain" cyclic proofs for $μ\mathsf{PA}$ prove the same theorems. This work represents a further step in the non-wellfounded proof-theoretic analysis of theories of arithmetic via impredicative fragments of second-order arithmetic, an approach initiated by Simpson's Cyclic Arithmetic, and continued by Das and Melgaard in the context of arithmetical inductive definitions.

We investigate the cyclic proof theory of extensions of Peano Arithmetic by (finitely iterated) inductive definitions. Such theories are essential to proof theoretic analyses of certain "impredicative" theories; moreover, our cyclic systems naturally subsume Simpson’s Cyclic Arithmetic. Our main result is that cyclic and inductive systems for arithmetical inductive definitions are equally powerful. We conduct a metamathematical argument, formalising the soundness of cyclic proofs within second-order arithmetic by a form of induction on closure ordinals, thence appealing to conservativity results. This approach is inspired by those of Simpson and Das for Cyclic Arithmetic, however we must further address a difficulty: the closure ordinals of our inductive definitions (around Church-Kleene) far exceed the proof theoretic ordinal of the appropriate metatheory (around Bachmann-Howard), so explicit induction on their notations is not possible. For this reason, we rather rely on formalisation of the theory of (recursive) ordinals within second-order arithmetic.