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The Barwise-Schlipf theorem


Authors: Ali Enayat and James H. Schmerl
Journal: Proc. Amer. Math. Soc. 149 (2021), 413-416
MSC (2010): Primary 03C50, 03C62, 03H15
DOI: https://doi.org/10.1090/proc/15216
Published electronically: October 20, 2020
MathSciNet review: 4172616
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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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Additional Information

Ali Enayat
Affiliation: University of Gothenburg, Gothenburg, Sweden
MR Author ID: 63375
ORCID: 0000-0003-0372-3354
Email: ali.enayat@gu.se

James H. Schmerl
Affiliation: University of Connecticut, Storrs, Connecticut 06269
MR Author ID: 156275
ORCID: 0000-0003-0545-8339
Email: james.schmerl@uconn.edu

Received by editor(s): November 10, 2019
Received by editor(s) in revised form: May 24, 2020
Published electronically: October 20, 2020
Additional Notes: The authors are grateful to Roman Kossak, Mateusz Łełyk, and an anonymous referee for their help in improving the paper’s exposition.
Communicated by: Heike Mildenberger
Article copyright: © Copyright 2020 American Mathematical Society