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GENERIC STABILITY AND MODES OF CONVERGENCE

Part of: Model theory

Published online by Cambridge University Press:  26 August 2025

KARIM KHANAKI
KARIM KHANAKI*
Affiliation:
DEPARTMENT OF SCIENCE https://ror.org/053wftt74ARAK UNIVERSITY OF TECHNOLOGY ARAK 38181-15881 IRAN
*
E-mail: khanaki@arakut.ac.ir
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Abstract

We expand the study of generic stability in three different directions. Generic stability is best understood as a property of types in $NIP$ theories in classical logic. In this article, we make attempts to generalize our understanding to Keisler measures instead of types, arbitrary theories instead of $NIP$ theories, and continuous logic instead of classical logic. For this purpose, we study randomization of first-order structures/theories and modes of convergence of types/measures.

Keywords

generic stabilitytypesmeasuresBaire-1/2 convergence

MSC classification

Primary: 03C45: Classification theory, stability and related concepts

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Type
Article
Information
The Journal of Symbolic Logic , First View , pp. 1 - 25
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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References

Ben-Yaacov, I., Continuous and random vapnik-chervonenkis classes . Israel Journal of Mathematics , vol. 173 (2009), pp. 309–333.Google Scholar
Ben-Yaacov, I., Transfer of properties between measures and random types, unpublished research note, 2009. http://math.univ-lyon1.fr/begnac/articles/MsrPrps.pdf Google Scholar
Ben-Yaacov, I., On theories of random variables . Israel Journal of Mathematics , vol. 194 (2013), no. 2, pp. 957–1012.Google Scholar
Ben-Yaacov, I., Berenstein, A., Henson, C. W., and Usvyatsov, A., Model theory for metric structures , Model theory with Applications to Algebra and Analysis , vol. 2 (Chatzidakis, Z., Macpherson, D., Pillay, A., and Wilkie, A., editors), London Math Society Lecture Note Series, 350, Cambridge University Press, Cambridge, 2008.Google Scholar
Ben-Yaacov, I. and Keisler, H. J., Randomizations of models as metric structures . Confluentes Mathematici , vol. 1 (2009), pp. 197–223.Google Scholar
Conant, G. and Gannon, K., Remarks on generic stability in independent theories . Annals of Pure and Applied Logic , vol. 171 (2020), no. 2, Article no. 102736, 20 pp. MR 4033642Google Scholar
Conant, G., Gannon, K., and Hanson, J., Generic stability, randomizations, and NIP formulas, preprint, 2023, https://arxiv.org/abs/2308.01801 Google Scholar
Conant, G., Gannon, K., and Hanson, J., Keisler measures in the wild . Model Theory , vol. 2 (2023), no. 1, pp. 1–67.Google Scholar
Gannon, K., Sequential approximations for types and Keisler measures. Fundamenta Mathematicae , vol. 257 (2022), pp. 305–336. https://doi.org/10.4064/fm133-12-2021 Google Scholar
Gannon, K., A note on transfer maps and the Morley product in NIP theories, preprint, 2024. https://arxiv.org/abs/2402.04202v2 Google Scholar
Hrushovski, E. and Pillay, A., On $NIP$ and invariant measures. Journal of the European Mathematical Society , vol. 13 (2011), pp. 1005–1061.Google Scholar
Hrushovski, E., Pillay, A., and Simon, P., Generically stable and smooth measures in NIP theories . Transactions of the American Mathematical Society , vol. 365 (2013), no. 5, pp. 2341–2366.Google Scholar
Keisler, H. J., Randomizing a model . Advances in Mathematics , vol. 143 (1999), pp. 124–158.Google Scholar
Khanaki, K., Stability, NIP and NSOP; model theoretic properties of formulas via topological properties of function spaces . Mathematical Logic Quarterly , vol. 66 (2020), no. 2, pp. 136–149.Google Scholar
Khanaki, K., On classification of continuous first order theories, submitted, 2020.Google Scholar
Khanaki, K., Dependent measures in independent theories. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik : to appear (2026). https://doi.org/10.60866/CAM.253 Google Scholar
Khanaki, K., Dividing lines in unstable theories and subclasses of Baire 1 functions . Archive for Mathematical Logic , vol. 61 (2022), pp. 977–993. https://doi.org/10.1007/s00153-022-00816-8 Google Scholar
Khanaki, K., Remarks on convergence of Morley sequences . Journal of Symbolic Logic , vol. 89 (2023), no. 3, pp. 1339–1357. https://doi.org/10.1017/jsl.2023.18 Google Scholar
Khanaki, K., Glivenko-Cantelli classes and $NIP$ formulas. Archive for Mathematical Logic , vol. 63 (2024), pp. 1005–1031. https://doi.org/10.1007/s00153-024-00932-7 Google Scholar
Khanaki, K. and Pillay, A., Remarks on $NIP$ in a model. Mathematical Logic Quarterly , vol. 64 (2018), no. 6, pp. 429–434.Google Scholar
Pillay, A. and Tanović, P., Generic stability, regularity, and quasiminimality , Models, Logics, and Higher-Dimensional Categories , CRM Proceedings & Lecture Notes (B. Hart, T. G. Kucera, A. Pillay, P. J. Scott, and R. A. G. Steely, editors), 53, American Mathematical Society, Providence, 2011, pp. 189–211. MR 2867971Google Scholar
Poizat, B., A Course in Model Theory , Springer, New York, NY, 2000.Google Scholar
Simon, P., A Guide to NIP Theories . Cambridge University Press, Cambridge, 2015.Google Scholar
Simon, P., Invariant types in NIP theories . Journal of Mathematical Logic , vol. 15 (2015), no. 2, p. 1550006.Google Scholar
van den Dries, L., Tame Topology and O-Minimal Structures , London Mathematical Society Lecture Note Series, 248, Cambridge University Press, Cambridge, 1998.Google Scholar