Introduction

This introduction presents an overview of the key concepts discussed in the subsequent chapters of this book. The book provides constructions of the real numbers in terms of rational numbers and shows how these ultimately stem from set constructions involving natural numbers. It also shows how the natural numbers themselves can be constructed from very simple sorts of set, like the empty set. The book introduces axioms for set theory which underpin all the constructions and describes axioms were introduced after Georg Cantor had produced his theory of infinite sets. It looks at some of Cantor's theory of infinite cardinal numbers, including remarkable results about the sizes of the real line, the real plane and the set of irrational numbers, all stemming from the construction of the reals. The book also look and Cantor's theory of ordinal numbers, a variety of richly structured sets extending ideas about natural numbers into the realm of infinite sets.