Zermelo-Fraenkel Set Theory

CHAPTER 4. ORDINALS 27
• 0 ∈ K,
• α ∈ K ⇒ α + 1 ∈ K,
• if γ is a limit and ∀ξ ∈γ (ξ ∈ K), then γ ∈ K.
Show: every ordinal is in K.
Hint. Induction.
60 ♣ We know that for every set A (i) there is a set B ⊂ A such that B ∉ A, and (ii)
there is a transitive set B ⊂ A such that B ∉ A.
Show: (iii) there is an ordinal β ⊂ A such that β ∉ A. Can you prove this without using
the Substitution Axiom?
4.2 Well-order Types
A well-ordering is a linear ordering that is well-founded. Classically, an ordinal is a wellorder
type. Theorem 4.10 shows that you can view ordinals as such.
Lemma 4.8 Suppose that (A, ≺) is a well-ordering. For every order-preserving function
h : A → A it holds that ∀a ∈ A(a ≼ h(a)).
In particular, (A, ≺) cannot be order-preservingly mapped into a proper initial.
Proof. Suppose that h : A → A is order-preserving (a ≺ b ⇒ h(a) ≺ h(b)) but for some
a ∈ A we have that h(a) ≺ a. Using that ≺ well-orders A, let b ∈ A be the least element
such that h(b) ≺ b. Then also h(h(b)) ≺ h(b). These properties of b ′ = h(b) contradict
minimality of b.
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Corollary 4.9 If (α,