In this chapter, I provide a very quick tour though some of the parts of mathematics that are relevant to the discussion of infinities of the large and small. The tour begins with set theory, which – according to many – provides foundations for the rest of mathematics. It then takes in the natural numbers – ordinal and cardinal – and Cantor's theory of the infinite ordinals and cardinals. Next, we look at the theory of the real numbers, and the standard analysis of limits and continuity. Finally, we have a brief look at some alternatives to the standard account of limits and continuity, based on the notion of infinitesimals, and a very quick glance at the area of finite mathematics. The aim throughout is merely to introduce some of the mathematical tools and results that might be considered in a serious discussion of the infinite and the infinitesimal.

SET THEORY

As noted above, there are many people who suppose that set theory can provide secure foundations for all of mathematics. During the 1980s, some people touted Category theory as a competitor; but enthusiasm for this proposal seems to be on the wane.

There are different versions of set theory. We shall look at what is probably the best-known, and best-loved, version of set theory, Zermelo-Frankel set theory (ZF for short). We shall note some alternatives to, and some extensions of, ZF.