An Introduction To The Theory Of Nonstandard Analysis
For over a century, the calculus has been understood via the limit process developed by Cauchy and Weierstrass. The delta-epsilon process of limits is always a new and usually confusing concept to the first-year calculus student. In my own experience, it was several years after being introduced to the delta-epsilon definition of the limit, that finally an "aha! Insight!" came and I understood the limit process. The delta-epsilon limit "was a 2 disaster for students to learn and teachers to teach," but for years there was no other mathematically sound concept by which one could understand analysis. It is an exciting development in mathematics, then, that Abraham Robinson of Yale University has succeeded in discovering a nonstandard way to logically warrant the infinitesimal. It is now possible to return the calculus to the Newtonian understanding that seems far easier to comprehend. The purpose of this thesis is to introduce the reader to some of these nonstandard techniques of analysis and show that they are equivalent to the more complex methods based upon the standard delta-epsilon limit. We will begin by reviewing the axiomatic development of the real number line and the standard techniques that Cauchy and Weierstrass developed that are based upon the limit. In order to warrant the infinitesimal, Robinson extended the reals into a hyperreal number system that is consistent with the infinitesimal. By reviewing the development of the reals, the reader will be better able to understand the similar development of the hyperreals. After defining hyperreal number systems, we will illustrate the nonstandard analysis associated with them by showing that the nonstandard definition of continuity, differentiation, and integration are equivalent to the standard delta-epsilon approach. Finally, we will end the development of the nonstandard analysis by proving, using only nonstandard techniques, the Fundamental Theorem of Integral Calculus (FTIC). While Robinson was able to extend the application of nonstandard techniques far beyond the major concepts of continuity, differentiation, and integration, these three will illustrate the essentials of the theory of nonstandard analysis, and serve as an ample introduction to the topic. The development and successful application to analysis of the hyperreals also prompted new interest in many complex epistemological questions. These questions illustrate the divergent concepts that lie at the foundations of mathematics and the distinctions between the Formalist and the Platonist understanding of mathematics. This discussion of the hyperreals will end by noting those insights into mathematical epistemology which are further illustrated by the nonstandard analysis.