Some Sets, More Numbers

Abstract

In the last chapter we raised, in a fairly informal way, a number of problems one has to examine before going on to study functions and the calculus. In particular, we kept stumbling over the use of sets; we need some clarification of our idea of numbers and of the different sorts of endlessness (infinity); and we will acquire a more precise approach to relations and functions using set notation. However, as we dispel some of the mists generated in the last chapter, we will bring to light whole new areas of study, in particular, the specification of axioms and their use in classifying sets, which will absorb us later. We will have cause to be considerably more formal in this chapter, in places, formality being a device which enables mathematicians to be simultaneously brief and incomprehensible; we shall also lean far more heavily on symbolic notation, again for the sake of brevity.