The Barwise-Schlipf theorem

Enayat, Ali; Schmerl, James

doi:10.1090/proc/15216

The Barwise-Schlipf theorem
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by Ali Enayat and James H. Schmerl

Proc. Amer. Math. Soc. 149 (2021), 413-416

DOI: https://doi.org/10.1090/proc/15216

Published electronically: October 20, 2020

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Abstract:

In 1975 Barwise and Schlipf published a landmark paper whose main theorem asserts that a nonstandard model $\mathcal {M}$ of $\mathsf {PA}$ (Peano arithmetic) is recursively saturated iff $\mathcal {M}$ has an expansion that satisfies the subsystem $\Delta _{1}^{1}$-$\mathsf {CA}_{0}$ 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.

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