Introduction

The reader might wish to skip this appendix, referring back to it when necessary. The most elementary properties of the axiom of choice, Zorn's lemma, ordinals, and cardinals are outlined as briefly as possible on an intuitive level. Quasi-ordered and partially ordered sets, equivalence relations, and functions are discussed.

One aim of this book is to make the reader aware that the material presented here might hold in a broader framework. For this reason some definitions and the isomorphism theorems are reviewed in the context of nonassociative rings, although the latter are not really used in this book. Another reason is that hopefully this chapter may say something new also to those readers who do not need a review.

The goal here is to supply self-contained proofs of the facts about cardinal numbers needed later in the book, and the one such result which makes this possible is that for any infinite set X, the size or cardinality of X × X equals that of X, i.e. |X × X| = |X| (A-1.11). This implies that if “∼” is an equivalence relation on an infinite set X whose equivalence classes are finite, that then the cardinality of the set of equivalence classes X/∼ is the same as that of X, that is |X/∼| = |X| (A-1.12). We will already use the latter fact in Chapter 2 to show that for infinite rank free modules the rank is well defined.