Non-standard Analysis

Abstract

Non-standard analysis is an area of mathematics that provides a natural framework for the discussion of infinite economies. It is more suitable in many ways than Lebesgue measure theory as a source of models for large but finite economies since the sets of traders in such models are infinite sets which can be manipulated as though they were finite sets. The number system used to describe non-standard economies is an extension of the real numbers R; it is denoted by ^∗R. The set ^∗R contains ‘infinite natural numbers’ and their multiplicative inverses, which are positive infinitesimals. It was with the development in 1960 of such a number system that Abraham Robinson (1974) solved an age-old problem by making rigorous the use of infinitesimals in mathematical analysis. Robinson gave a model-theoretic approach to his theory that is relevant to any infinite mathematical structure; that approach starts by listing the basic properties of the new number system. Before taking up this approach, however, it will be helpful to consider a simple nonstandard extension of the real numbers system that is constructed from sequences of real numbers.