chapter  7
39 Pages

Ordered Sets

This chapter presents the ground for the second of Georg Cantor's numbers, the ordinals, by looking in greater detail at ordered sets – an ordinal will be a special sort of ordered set. Much of the theory of ordered sets was developed by Cantor himself. To be sure that people have captured at least some of the essence of being 'the same' as ordered sets with this definition, it is customary to check that being order-isomorphic is an equivalence relation. A linearly ordered set need not have either a maximum or a minimum element. But examples of such sets have to be infinite. Ordinals are special sorts of ordered set. The chapter investigates arithmetic of ordered sets, to which ordinal arithmetic corresponds. To round off the excursion into the arithmetic of order, people should ask if there is any further nice connection between order sums and products.