More on Cardinals and Ordinals

By now we have made sufficient progress to introduce ordinal and cardinal numbers, the two most fundamental concepts in set theory. In this chapter, we shall make a detailed study of ordinal and cardinal numbers. Let ON be a class of sets such that ON consists of well-ordered sets satisfying the following two properties: (i) Well-ordered sets in ON are pairwise non-isomorphic. (ii) Every well-ordered set is order isomorphic to a well-ordered set in ON.