 Section  3.4  discusses a more recent mathematical theory of infinitely large and infinitesimal numbers that provides an alternate setting for calculus. This theory of “non-standard analysis” has not become as central a part of the mathematics curriculum as real analysis and set theory. It may thus be unfamiliar to most readers, and it is harder to find accessible introductions elsewhere. Although non-standard analysis is not as central a part of the cultural understanding of infinity in mathematics as cardinals are, we include it both because it is a topic of growing interest in mathematics research, and because it can help make mathematically rigorous sense of both many intuitive thoughts about infinity and some of the early work on calculus in the 17th and 18th centuries.