The Real Number System

Abstract

From what we have seen in the preceding chapters, around 1870 there were several indications that the notion of set might prove of fundamental importance for algebra, function theory, and even geometry. In the next chapter we shall consider another line of development, consolidated around that time, which led mathematicians working on real functions to pay attention to point-sets. In the present chapter we consider more elementary questions in analysis that also stimulated the emergence of a theory of sets, and which firmly established the conception of pure mathematics as the science of number. This conception was crucial for Weierstrass, Dedekind and Cantor, the central (German) figures in the rigorization of the real number system. Each one of them presented a rigorous definition of the real numbers, together with basic notions and results on the topology of ℝ.