chapter  8
61 Pages

Ordinal Numbers

In this chapter, the authors look at a construction developed by John von Neumann which gives a particular set out of each order-type of well-ordered sets in a systematic way, without any need for axioms beyond those in Ernst Zermelo-Abraham Fraenkel. They reconstruct Georg Cantor's arithmetic on his ordinals using the von Neumann ordinals, which from on they shall just refer to as the ordinals. Cantor's arithmetic on ordinals gives an insight into what infinite ordinals look like. The authors show how Cantor's two sorts of infinite number, ordinals and cardinals, relate to each other. They investigate the properties of ordinal addition and multiplication, to build up a full arithmetic of ordinals. The authors explain why they defined order product using the anti-lexicographic order, rather than the lexicographic order. Cantor's original work, and many modern texts, present some of the results in terms of limits of sequences of ordinals, rather than unions of sets of ordinals.