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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.
Keywords: boolean valued analysis, nonstandard analysis, forcing.
Language: English Download the full text  
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
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