By W. Weiss
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Then λ = |P(κ)| = |κ 2| = |(κ×κ) 2| = |κ (κ 2)| = |κ λ| ≥ |cf (λ) λ| > λ. Cantor’s Theorem guarantees that for each ordinal α there is a set, P(α), which has cardinality greater than α. However, it does not imply, for example, that ω + = |P(ω)|. This statement is called the Continuum Hypothesis, and is equivalent to the third question in the introduction. 64 CHAPTER 7. CARDINALITY The aleph function ℵ : ON → ON is defined as follows: ℵ(0) = ω ℵ(α) = sup {ℵ(β)+ : β ∈ α}. We write ℵα for ℵ(α). We also sometimes write ωα for ℵ(α).
This leads to the contradiction ON ∈ ON. Theorem 11. (Trichotomy of Ordinals) (∀α ∈ ON)(∀β ∈ ON)(α ∈ β ∨ β ∈ α ∨ α = β). Proof. The reader may check that a proof of this theorem can be obtained by replacing “N” with “ON” in the proof of Theorem 7. Because of this theorem, when α and β are ordinals, we often write α < β for α ∈ β. Since N ⊆ ON, it is natural to wonder whether N = ON. In fact, we know that “N = ON” can be neither proved nor disproved from the axioms that we have stated (provided, of course, that those axioms are actually consistent).
Theorem 22. (∀X)(∃ <) [ X, < is a well ordered set]. Proof. We begin by using Theorem 5 to obtain a choice function f : P(X) \ {∅} → X such that for each nonempty A ⊆ X we have f (A) ∈ A. By recursion on ON we define g : ON → X ∪ {X} as: g(β) = f (X \ {g(α) : α < β}), if X \ {g(α) : α < β} = ∅; X, otherwise. 1) Now replace each x ∈ X ∩ran(g) by the unique ordinal β such that g(β) = x. The Axiom of Replacement gives the resulting set S ⊆ ON, where S = {β ∈ ON : g(β) ∈ X}. By Theorem 10 there is a δ ∈ ON \ S.