Nonstandard Analysis -- Jacob Barandes, November 25, 2003

After the invention of calculus, great minds like Euler used an intuitive approach to infinitesimal and infinite quantities to derive a variety of important results. However, the imprecise nature of these ideas led later mathematicians into making a number of clearly erroneous claims. After much work was expended by the mathematical community in an attempt to put infinitesimals and infinites on a rigorous foundation, by the 1800s it appeared that the effort was futile and analysis was consequently reformulated using the precise but somehow less intuitive notion of limits. In the 1960s, Abraham Robinson resurrected the infinitesimal calculus, or nonstandard analysis, by discovering a novel way of making the subject rigorous and well-defined. His method, however, required digging all the way down to the system of fundamental logic on which all mathematics rests. In this talk, I will present a modern treatment of his approach, which rests on deep principles like the axiom of choice and the well-ordering theorem, and show how the techniques of nonstandard analysis can recast previous definitions of continuity, derivatives, and integration, as well as greatly simplify proofs and even be used as a new way to teach calculus.