Address: Vatutina st. 53, Vladikavkaz,
362025, RNO-A, Russia
Phone: (8672)23-00-54
E-mail: rio@smath.ru
DOI: 10.23671/VNC.2019.21.44619
Some Remarks About Nonstandard Methods in Analysis. I
Gordon, E. I.
Vladikavkaz Mathematical Journal 2019. Vol. 21. Issue 4.
Abstract: This and forthcoming articles discuss two of the most known nonstandard methods of analysis-the Robinson's infinitesimal analysis and the Boolean valued analysis, the history of their origination, common features, differences, applications and prospects. This article contains a review of infinitesimal analysis and the original method of forcing. The presentation is intended for a reader who is familiar only with the most basic concepts of mathematical logic-the language of first-order predicate logic and its interpretations. It is also desirable to have some idea of the formal proofs and the Zermelo--Fraenkel axiomatics of the set theory. In presenting the infinitesimal analysis, special attention is paid to formalizing the sentences of ordinary mathematics in a first-order language for a superstructure. The presentation of the forcing method is preceded by a brief review of C.~Godel's result on the compatibility of the Axiom of Choice and the Continuum Hypothesis with Zermelo--Fraenkel's axiomatics. The forthcoming article is devoted to Boolean valued models and to the Boolean valued analysis, with particular attention to the history of its origination.
For citation: Gordon, E. I. Some Remarks about Nonstandard Methods in Analysis. I, Vladikavkaz Math. J., 2019, vol. 21, no. 4, pp. 25-41. DOI 10.23671/VNC.2019.21.44619
1. Kusraev, A. G. and Kutateladze, S. S. Nonstandard Methods of Analysis,
Dordrecht, Kluwer Academic Publishers, 1995, 398 p.
2. Maltsev, A. I. A General Method for Obtaining Local Theorems in Group Theory, Uchen. Zap. Ivanovsk. Gos. Ped. Inst. Uch. Zap. Fiz.-Mat. Fak., 1941, vol. 1, no. 1, pp. 3-9 (in Russian).
3. Robinson, A. Nonstandard Analysis, Revised Edition, Princeton, Princeton University Press, 1996, 312 p.
4. Albeverio, S., Fenstad, J. E. and et al. Nonstandard Methods in Stochastic Analysis and
Mathematical Physics, Academic Press, Orlando, etc., 1990, 514 p.
5. Scott, D. A Proof of the Independence of the Continuum Hypothesis, Mathematical System Theory, 1967, vol. 1, no. 2, pp. 89-111. DOI: 10.1007/BF01705520.
6. Scott, D. Lectures of Boolean Valued Models of Set Theory, Amer. Math. Soc., Summer Institute on Axiomatic Set Theory, UCLA, 1967.
7. Scott, D. and Solovay, R. Boolean Valued Models of Set Theory, Axiomatic Set Theory, Proc. Sympos.
Pure Math., vol. 13, Amer. Math. Soc., Providence, R.I.
8. Saks, G. Measure-Theoretical Uniformity in Recursion Theory and Set Theory, Trans. Amer. Math. Soc., 1969, vol. 142, no. 2, pp. 381-420. DOI: 10.1090/S0002-9947-1969-0253895-6.
9. Takeuti, G. Two Applications of Logic to Mathematics, Tokio and Princeton, Iwanami and Princeton University Press, 1978, p. 148.
10. Takeuti, G. A Transfer Principle in Harmonic Analysis, Journal of Symbolic Logic, 1979, vol. 44, no. 3, pp. 417-440.
11. Kusraev, A. G. and Kutateladze, S. S. Introduction to Boolean Valued Analysis, Moscow, Nauka, 2005, 526 p. (in Russian).
12. Kusraev, A. G. and Kutateladze, S. S. Boolean Valued Analysis: Selected Topics,
Vladikavkaz, Southern Mathematical Institute, VSC RAS & RNO-A, 2014, 400 p.
13. Gordon, E. I. Real Numbers in Boolean Valued Models of Set Theory, and \(K\)-Spaces, Soviet Mathematics Doklady, 1977, vol. 18, pp. 1481-1484.
14. Loeb, P. A. and Wolff, M. Ph. Nonstandard Analysis for the Working Mathematician, Dordrecht, Kluwer Academic Publishers, 2000, 325 p.
15. Davis, M. Applied Nonstandard Analysis, New York, etc., John Wiley & Sons, 1977.
16. Manin, Yu. I. A Course in Mathematical Logic for Mathematicians. Second Edition, New York, etc., Springer, 2010, p. 403.
17. Gordon E. I. Nonstandard Methods in Commutative Harmonic Analysis,
Amer. Math. Soc., Providence, R.I., 1977, 180 p.
18. McDuff, D. On Theory of \(II_1\) Factors, Russian Mathematical Surveys, 1970, vol. 25, no. 1, pp. 29-51 (in Russian).
19. Solovay, R. A Model of Set-Theory in Which Every Set of Reals is Lebesgue Measurable, Annals of Mathematics, 1970, vol. 92, no. 1, pp. 1-56. DOI: 10.2307/1970696.
20. Godel, K. Consistency of Continuum Hypothesis, Princeton University Press, Princeton, 1940.
21. Jech, T. J. Lectures in Set Theory with Particular Emphasis on the Method of Forcing, Lecture Notes in Mathematics, vol. 217, Berlin, Springer-Verlag, 1971. DOI: 10.1007/BFb0061131.
22. Cohen, P. J. Set Theory and the Continuum Hypothesis, New York, etc., Benjamin, 1966.
23. Vladimirov, D. A. Boolean Algebras in Analysis, Dordrecht, Springer Science & Business Media, 2002.
24. Solovay, R. and Tennenbaum, S. Iterated Cohen Extensions and Souslin's Problem, Annals of Mathematics, 1971, vol. 94, no. 2, pp. 201-245. DOI: 10.2307/2272650.
25. Solovаy, R. On the Cardinality of \(\sum_1^2\)-Sets of Reals, Foundations of Mathematics, Symposium Papers Commemorating the Sixtieth Birthday of Kurt Godel, Berlin, Springer Verlag, 1969, pp.58-73. DOI: 10.1007/978-3-642-86745-3_7.
26. Shelah, S. Can you Take Solovay's Inaccessible Away? Israel Journal of Mathematics, 1984, vol. 48, pp. 1-47. DOI: 10.1007/BF02760522.
27. Gordon, E. I. On the Extension of Haar Measure in \(\sigma\)-Compact Groups, Moscow, Lenin Moscow State Pedagogical Institute, VINITI, no. 1243-81, 1980 (in Russian).
The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses.
Сайт использует файлы cookie, необходимые для корректной работы сайта, и сервисы Яндекс-метрики, используемые для анализа статистики посещаемости, которые не содержат сведений, на основании которых можно идентифицировать личность пользователя. Продолжение пользования сайтом является согласием на применение данных технологий.
