Some remarks about nonstandard methods in analysis. I

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.