In Chapter 5, we defined certain sets to represent collections of numbers. Despite being sets themselves, the elements of those sets were called numbers. We continue this association with sets as numbers but for a different purpose. While before we defined ω, , , , and to represent N, Z, Q, R, and C, the definitions of this chapter are intended to be a means by which all sets can be classified according to a particular criterion. Specifically, in the later part of the chapter, we will define sets for the purpose of identifying the size of a given set, and we begin the chapter by defining sets that are used to identify whether two well-ordered sets have the same order type (Definition 4.5.24). A crucial tool in this pursuit is the following generalization of Theorem 5.5.1 to well-ordered infinite sets.
THEOREM 6.1.1 [Transfinite Induction 1]
Let (A, ) be a well-ordered set. If B ⊆ A and (A, x) ⊆ B implies x ∈ B for all x ∈ A, then