Nonstandard set Theory

Abstract

Infinitely small and infinitely large quantities were systematically introduced into mathematics with the invention of calculus by Newton and Leibniz. The use of such quantities, however, was accompanied by logical contradictions, which mathematicians of the seventeenth and eighteenth centuries were unable to resolve. Although the method of infinitesimals generally yielded correct results, no one ever succeeded in formulating a precise, noncontradictory set of rules governing these objects; and infinitesimal quantities were gradually displaced (at least, in pure mathematics) by the familiar e-d calculus. A mathematically sound model of infinitely small and infinitely large objects became possible only after advances in mathematical logic in the twentieth century. Nonstandard Analysis, developed by A. Robinson in 1960, not only provided foundations for the calculus of infinitesimals in the classical spirit but also enabled mathematicians to use objects in ways that could not be attempted on the basis of vague, intuitive understanding alone. Since then, interesting applications were found in various branches of mathematics, mathematical physics, and economics. Robinson's exposition in [10] and its subsequent simplifications unfortunately involve the cumbersome apparatus of mathematical logic. Our aim here is to present methods of Nonstandard Analysis at a level of formalism customary in other branches of mathematics. We view nonstandard objects as ideal, imaginary elements adjoined to the universe of the standard mathematics and formulate a few simple and reasonably intuitive principles governing their behavior. We then show, on examples selected to illustrate a variety of nonstandard constructions, how nonstandard mathematics can be developed from these principles. The basic framework for Nonstandard Analysis is presented in ?? 1-3; this system was introduced in [3], where its relative consistency with respect to the Zermelo-Fraenkel set theory is shown. We examine the real line and some concepts of general topology from our point of view in ??4-5; these results can be found in Robinson [10] and Luxemburg [7], [8]. Section 6 is devoted to nonstandard measure theory; our approach is basically that of Loeb [6] (except that we construct Loeb's extension in Theorem 3 of ?6 directly, rather than using Caratheodory's Theorem), with some ideas coming from Anderson [1]. The final ?7, part I, contains a more formal description of the logical foundations, and parts II and III discuss the relationship between our approach and the classical one based on higher-order nonstandard models, as well as some other axiomatizations of Nonstandard Analysis.