The real numbers

At the beginning of the nineteenth century, it became clear that mathematical analysis (the study of functions and of series) lacked a satisfactory firm foundation. In 1821, Augustin-Louis Cauchy published his Cours d'Analyse, which contained the first rigorous account of mathematical analysis. Cauchy however took the properties of the real numbers for granted. In 1858, when Richard Dedekind was preparing a course of lectures on the elements of the differential calculus at the Polytechnic School in Zürich, he ‘felt more keenly than ever the lack of a really scientific foundation for arithmetic’, and discovered the construction of the real number system that is described in the Prologue. In fact, he only published his results in 1872. With hindsight, it has become clear that the properties of the real number system lie at the heart of all mathematical analysis, and that it is essential to obtain a full understanding of these properties in order to develop mathematical analysis.

In the Prologue, we have constructed Dedekind's model for the real numbers R and established some of its properties. It is however sensible to take the construction for granted, to write down the essential properties of R, and to use these properties to develop the theory of mathematical analysis. This we shall do.