Sets, Structures, Numbers

In this chapter we shall introduce most of the background needed to develop the foundations of mathematical analysis. We start with sets and algebraic structures. Then the real numbers are defined axiomatically. This in turn allows one to define natural numbers, integers, rational numbers, and irrational numbers as well as to derive fundamental properties of these numbers. Next, we study representations of real numbers. Then we turn our attention to mappings and the numerosity of sets. In particular, it is shown that there are many more real numbers than rational numbers. Furthermore, we introduce linear spaces. The chapter is concluded by defining the complex numbers.