Tutorial

Abstract

Nonstandard Analysis, as is the case with Mathematics itself, can be applied to many domains; to illustrate this by means of examples is the object of this book. The reader may perhaps be surprised, and will surely be pleased to learn that very little need be known of the theoretical details of this modern theory of infinitesimals as introduced by A. Robinson [100] in order to apply it well. The purpose of this first chapter is to communicate the common background necessary to the rest of the book. It gathers the results that will be used in the sequel, the following chapters being all independent of each other, more or less. The ease with which it is possible to explain within twenty pages or so what mathematicians sought for in vain during the 19th century is a consequence of an idea of E. Nelson [91]: to resort to an adjective, the predicate standard, which is deliberately left undefined. This idea makes it possible to dissociate completely the logical foundation of the nonstandard method from its practical use.