Non-Standard Analysis

1. As early as 1934 it was pointed out by Thoralf Skolem (see [17]) that there exist proper extensions of the natural number system which have, in some sense, ‘the same properties’ as the natural numbers. The title of Skolem’s paper indicates that the purpose of it was to show that no axiomatic system specified in a formal language, in Skolem’s case the lower predicate calculus, can characterize the natural numbers categorically. At that time, however, Skolem did not concern himself with the properties of the structures whose existence he had established. In due course these structures became known as non-standard models of arithmetic. For nearly thirty years since the appearance of Skolem’s paper non-standard models were not used or considered in any sense by the working mathematician. Robinson’s fundamental paper, which appeared in 1961 under the title ‘Non-standard Analysis’, (see [11]) changed this situation dramatically. In this paper Abraham Robinson was the first to point out that this highly abstract part of model theory could be applied fruitfully to a theory so far removed from it as the infinitesimal calculus. As a result Robinson obtained a firm foundation for the non-archimedian approach to the calculus based on a number system containing infinitely small and infinitely large numbers, in a manner almost identical to that suggested by Leibniz some three centuries ago, and which predominated the calculus until the middle of the nineteenth century when it was rejected as unsound and replaced by the ϵ, δ-method of Weierstrass