Real Numbers

In this chapter, we describe the system of real numbers, deducing some of their essential properties from the axioms for a complete ordered field. Before doing so, we take a quick look at the ideas and notations of sets, relations, and functions, sketch the construction of the integers and the rational numbers (starting from the natural numbers), and indicate the need for a field larger than the rational numbers. At the end of the chapter, we sketch the proof of the existence and (essential) uniqueness of a complete ordered field.