We show that the theory ATR 0 is equivalent to a second-order generalization of the theory $\widehat{ID}_{ . As a result, ATR 0 is conservative over $\widehat{ID}_{ for arithmetic sentences, though proofs in ATR 0 can be much shorter than their $\widehat{ID}_{ counterparts.

The paper relativizes the method of ordinal analysis developed for Kripke–Platek set theory to theories which have the power set axiom. We show that it is possible to use this technique to extract information about Power Kripke–Platek set theory, KP.As an application it is shown that whenever KP+AC proves a ΠP2 statement then it holds true in the segment Vτ of the von Neumann hierarchy, where τ stands for the Bachmann–Howard ordinal.