The Barwise-Schlipf theorem

Abstract

In 1975 Barwise and Schlipf published a landmark paper whose main theorem asserts that a nonstandard model M \mathcal {M} of P A \mathsf {PA} (Peano arithmetic) is recursively saturated iff M \mathcal {M} has an expansion that satisfies the subsystem Δ 1 1 \Delta _{1}^{1} - C A 0 \mathsf {CA}_{0} of second order arithmetic. In this paper we identify a crucial error in the Barwise–Schlipf proof of the right-to-left direction of the theorem, and we offer a correct proof of the problematic direction.