AN INTRODUCTION TO NONSTANDARD ANALYSIS: THEORY AND SOME RESULTS

In the 17th century Gottfried Wilhelm Leibniz was the first to postulate the existence of infinitesimal numbers, as part of its theory of calculus. According to Leibniz, infinitesimals are smaller than any positive real but different from zero, to which the usual algebric rules still apply, and it is upon them that he defined the familiar concepts of derivative and integral. Unfortunately, while extremely intuitive, the presence of infinitesimal numbers in the set of the reals was proven to be logically inconsistent by George Berkeley and modern calculus theory left them behind in favour of Augustin Cauchy and KarlWeierstrass’s formulation, based on Epsilon-Delta formulae and converging sequences of real numbers. It was only in 1966, with the pubblication of Non-Standard Analysis, that Abraham Robinson introduced the concept of a new model of R containing both unlimited and infinitesimal numbers, as a consequence of the Compactness Theorem, and presented a different theory of calculus, continuing Leibniz’s ideas. In the last 50 years, Robinson’s methodology has been successfully applied to a number of mathematical theories, including measure theory, probability, stochastic analysis and game theory, under the name of nonstandard analysis. Nonstandard analysis contribute to mathematics cannot be understated, both as a new research methodology and as a pedagogical instrument that introduces calculus in a more inuitive manner. The objective of this thesis is to introduce the reader to Robinsons’s work and its consequences. In particular, we will present a nonstandard foundation of R as complete metric space that includes infinitesimal numbers and introduce the means to, given any theory with at least one infinite model, find a nonstandard model of that theory. We will conclude with an example of the application of nonstandard analysis to a different field from that of the real numbers.