1.4.1: Introduction

When Georg Cantor developed set theory in the 1870s, one of his aims was to make palatable the idea of an infinite collection—an actual infinity, as the medievals would say. A key part of this was his treatment of the size of different sets. If \(a\), \(b\) and \(c\) are all distinct, then the set \(\{a, b, c\}\) is intuitively larger than \(\{a, b\}\). But what about infinite sets? Are they all as large as each other? It turns out that they are not.

The first important idea here is that of an enumeration. We can list every finite set by listing all its elements. For some infinite sets, we can also list all their elements if we allow the list itself to be infinite. Such sets are called countable. Cantor’s surprising result, which we will fully understand by the end of this chapter, was that some infinite sets are not countable.

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