Summary

This chapter is concerned with relative sizes of sets, through the idea of functions between them. The distinctions between finite and infinite sets, and between countable sets and uncountable sets, are made. Properties of countable sets are derived, and the sets ℤ and ℚ are shown to be countable. ℝ is shown to be uncountable, and properties of sets equinumerous with ℝ are derived. Two sets are said to have the same cardinal number if there is a bijection between them. Properties of the cardinal numbers ℵ and ℵ are derived.

The reader is presumed to be familiar with the algebra of sets and with the notions of injection, surjection and bijection. Apart from one reference to Theorem 1.29, this chapter is independent of Chapter 1, although knowledge of the basic properties of integers, rational numbers and real numbers is required.

Finite and countable sets

How can we measure the size of a set? Perhaps the crudest criterion is that of finiteness. A set is either finite or infinite, and the former is ‘smaller’ than the latter. For finite sets there is an obvious further measure of size, namely, the number of elements in the set, and using this criterion it is easy to judge when one finite set is ‘larger’ than another. For infinite sets the question is not so easy however. This book is largely about the mathematical ideas necessary for sensible discussion of the nature and behaviour of infinite sets.