Zermelo-Fraenkel Set Theory

CHAPTER 4. ORDINALS 35
4.6 Initial Numbers
See the beginning of Section 4.1 (p. 23): note that all ordinals of the initial segement of
OR that is sketched there are countable. This section starts with proving that uncountable
ordinals exist.
Definition 4.23
1. A 1 B :≡ there exists an injection : A → B,
2. A = 1 B :≡ there exists a bijection : A → B,
3. A < 1 B :≡ A 1 B ∧ ¬A = 1 B. □
Definition 4.24 A is countable iff A 1 ω; A is countably infinite iff A is countable and
infinite. (Cf. Definition 3.15 p. 15.)
□
Definition 4.25 (Hartogs’ operation.) Γ(A) := {α | α 1 A}.
□
Lemma 4.26 Γ(A) is the least ordinal α such that ¬α 1 A.
In particular, if β ∈ OR, then Γ(β) is the least ordinal α such that β < 1 α.
Proof. Γ(A) is easily seen to be a transitive class of ordinals. To see that it is an ordinal, it
therefore suffices to show that it is a set. But this follows (using the Axioms of Powerset,
Separation, and Substitution) from the fact, that Γ(A) = {type(X, R) | X ⊂ A ∧ R
well-orders X}. Finally, if Γ(A) 1 A, then Γ(A) ∈ Γ(A); hence, ¬Γ(A) 1 A. And if
α < Γ(A), then α 1 A.
□
Without Axiom of Choice, it is unprovable that A < 1 Γ(A) and Γ(A) 1 ℘(A). In this
connection, see Exercise 89.
Definition 4.27 An initial number is an ordinal α ω such that ∀ξ < α(ξ < 1 α).
The equivalence = 1 partitions OR in number classes. The classes of natural numbers
are singletons. Then follows the (uncountable) class of ω: the countably infinite ordinals
which are ω and < Γ(ω). Γ(ω) is the first uncountable ordinal in a long series of ordinals
of the same power, etc.
Lemma 4.28
1. ω is the least initial.
2. If ω α, then Γ(α) is the least initial > α.
3. If α β < Γ(α), then β = 1 α.
4. Every infinite ordinal is = 1 to an initial.
Definition 4.29 Using recursion on OR, define the series ω α as follows:
1. ω 0 = ω,
2. ω α+1 = Γ(ω α ),
3. for limits γ: ω γ = ⋃ ξ