 PA\(^2\) is a very strong theory. Via the method of interpretations (see below), one can show that it is proof-theoretically as strong as the Zermelo-Fraenkel set theory ZFC without the power-set axiom, call it ZFC–Pow (whereas the standard, first-order PA is similarly proof-theoretically equivalent to ZFC without the axiom of infinity, ZFC–Inf). (Cf. the section on the axioms of ZFC in the entry on  set theory.)