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 | References | Similar Articles | Additional Information
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
