The Barwise-Schlipf Theorem

TL;DR

This paper revises the proof of a key theorem relating nonstandard models of Peano arithmetic to second-order arithmetic subsystems, correcting an error and providing a valid proof for the challenging direction.

Contribution

It identifies a critical error in the original proof and supplies a correct proof for the nonstandard model characterization theorem.

Findings

Corrected the proof of the Barwise-Schlipf Theorem

Confirmed the equivalence between recursive saturation and certain second-order expansions

Clarified the conditions under which nonstandard models satisfy specific subsystems

Abstract

In 1975 Barwise and Schlipf published a landmark paper whose main theorem asserts that a nonstandard model $\mathcal{M}$ of PA (Peano arithmetic) is recursively saturated iff $\mathcal{M}$ has an expansion that satisfies the subsystem $\Delta_1^1$-${\sf CA}_0$ 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 additionally, we offer a correct proof of the problematic direction.