Model approach to nonstandard analysis in the context of axiomatic set theory. (English. Russian original) Zbl 1151.26020

Math. Notes 79, No. 1, 122-128 (2006); translation from Mat. Zametki 79, No. 1, 134-141 (2006).
Author’s abstract: The model approach to nonstandard analysis is developed on the basis of Zermelo-Fraenkel axiomatic set theory with atoms. The traditional consideration of the standard superstructure \(V\) as the primary object of nonstandard analysis is justified. Set-theoretic axioms for the nonstandard system \(*V\) are obtained.


26E35 Nonstandard analysis
03H05 Nonstandard models in mathematics
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[1] M. Davis, Applied Nonstandard Analysis, Wiley, New York, 1977; Russian translation: Mir, Moscow, 1980. · Zbl 0359.02060
[2] S. Albeverio, J. Fenstad, R. Hoegh-Krohn, and T. Lindstrom, Nonstandard Methods in Stochastic Analysis and Mathematical Physics, Academic Press, New York, 1986; Russian translation: Mir, Moscow, 1990.
[3] Nonstandard Analysis for the Working Mathematician (P. Loeb, M. P. H. Wolff, Eds.), Kluwer, Dordrecht, 2000.
[4] E. Nelson, ”Internal set theory: a new approach to nonstandard analysis,” Bull. Amer. Math. Soc., 83 (1977), no. 6, 1165–1198. · Zbl 0373.02040
[5] E. I. Gordon, A. G. Kusraev, and S. S. Kutateladze, Infinitesimal Analysis, pt. 1 [in Russian], Izd. IM, Novosibirsk, 2001. · Zbl 1013.46004
[6] V. F. Yakovlev, ”Enlarged universe for systems with minimal structures,” Dokl. Akad. Nauk SSSR [Soviet Math. Dokl.], 319 (1991), no. 1, 103–105.
[7] T. Jech, Lectures in Set Theory, Springer-Verlag, Berlin, 1971. · Zbl 0236.02048
[8] C. C. Chang and H. J. Keisler, Model Theory, North Holland, Amsterdam, 1973.
[9] M. Makkai, ”Admissible sets and infinitary logic,” in: Handbook of Mathematical Logic (J. Barwise, Ed.), North-Holland, Amsterdam, 1977, pp. 233–282.
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