Well-Orders and Ordinals

Abstract

This chapter develops the classical theory of well-orders and ordinals in a naive setting. Ordinals are defined as order types of well-orders, not as von Neumann ordinals. We cover the basic ordinal operations of sum and product, transfinite induction and recursion, uniqueness of isomorphisms and ranks, unique representation of well-orders by initial sets of ordinals, the comparability theorem for well-orders, the division algorithm, remainder ordinals, ordinal exponentiation, and the Cantor Normal Form.