| dc.description.abstract | There are two different approaches to nonstandard analysis: semantic
(model-theoretic) and syntactic (axiomatic). Both of these approaches
require some knowledge of mathematical logic. We present a method based
on an ultrapower construction which does not require any mathematical logic
prerequisites. On the one hand, it is a complementary course to a standard
calculus course. On the other hand, since it relies on a different intuitive
background, it provides an alternative approach. While in standard analysis
an intuition of being close is represented by the notion of limit, in nonstandard
analysis it finds its expression in the relation is infinitely close. As
a result, while standard courses focus on the ε − δ technique, we explore
an algebra of infinitesimals. In this paper, we offer a proof of the theorem
on the equivalency of limits and infinitesimals, showing that calculus can be
developed without the concept of limit. | en |
