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Proceedings of the American Mathematical Society

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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
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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
Email: ali.enayat@gu.se

James H. Schmerl
Affiliation: University of Connecticut, Storrs, Connecticut 06269
Email: james.schmerl@uconn.edu

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