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.
- Jon Barwise and John Schlipf, On recursively saturated models of arithmetic, Model theory and algebra (A memorial tribute to Abraham Robinson), Springer, Berlin, 1975, pp. 42–55. Lecture Notes in Math., Vol. 498. MR 0409172
- Matt Kaufmann and James H. Schmerl, Saturation and simple extensions of models of Peano arithmetic, Ann. Pure Appl. Logic 27 (1984), no. 2, 109–136. MR 763736, DOI https://doi.org/10.1016/0168-0072%2884%2990008-3
- Matt Kaufmann and James H. Schmerl, Remarks on weak notions of saturation in models of Peano arithmetic, J. Symbolic Logic 52 (1987), no. 1, 129–148. MR 877860, DOI https://doi.org/10.2307/2273867
- Roman Murawski, On expandability of models of Peano arithmetic. I, Studia Logica 35 (1976), no. 4, 409–419. MR 441723, DOI https://doi.org/10.1007/BF02123406
- Stephen G. Simpson, Subsystems of second order arithmetic, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1999. MR 1723993
- C. Smoryński, Recursively saturated nonstandard models of arithmetic, J. Symbolic Logic 46 (1981), no. 2, 259–286. MR 613281, DOI https://doi.org/10.2307/2273620
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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