Zermelo-Fraenkel Set Theory

CHAPTER 6. CARDINALS 49
By Exercise 122.3, not every initial is regular. E.g., if α is a limit such that α < ω α , then
cf(ω α ) = cf(α) α < ω α . For instance, cf(ω ω ) = ω; by Exercise 122.3 and Lemma 6.34,
cf(ω ω1 ) = cf(ω 1 ) = ω 1 < ω ω1 .
The least regular initial is ω. All successor initials are regular:
Lemma 6.34 (AC) ω α+1 is regular.
Proof. Assume that ω α+1 has a cofinal subset B such that B < 1 ω α+1 . For every β ∈ B,
choose an injection : β → ω α . Then ω α+1 = ⋃ B 1
⋃β∈B β × {β} 1 ω α × B 1 ω α × ω α
= 1 ω α . □
Note that this proof shows, in particular, the familiar fact that a countable union of
countable sets is countable. But, even this needs AC.
The following lemma presents the close connection between ordinal and cardinal notions.
Lemma 6.35
1. cf(ℵ α ) = |cf(ω α )|, and hence
2. ℵ α is regular iff ω α is regular.
Proof. Using the AC-free definition, this can be shown without AC. Here is a proof using
AC: Assume that β = cf(ω α ) and that f : β → ω α is an order-preserving map onto a
cofinal subset of ω α . Then ω α = ⋃ ξ