 In the late nineteenth century, it was a widespread idea that pure mathematics is nothing but an elaborate form of arithmetic. Thus it was usual to talk about the “arithmetisation” of mathematics, and how it had brought about the highest standards of rigor. With Dedekind and Hilbert, this viewpoint led to the idea of grounding all of pure mathematics in set theory. The most difficult steps in bringing forth this viewpoint had been the establishment of a theory of the real numbers, and a set-theoretic reduction of the natural numbers. Both problems had been solved by the work of Cantor and Dedekind. But precisely when mathematicians were celebrating that “full rigor” had been finally attained, serious problems emerged for the foundations of set theory. First Cantor, and then Russell, discovered the paradoxes in set theory.