Zermelo-Fraenkel Set Theory

CHAPTER 4. ORDINALS 37
Theorem 4.32 (Cantor Normal Form) Fix θ > 1. Every ordinal α can be written in
exactly one way as α = θ β1 · γ 1 + · · · + θ β k
· γ k with β 1 > · · · > β k and γ 1 , . . . , γ k < θ.
The Cantor Normal Form Theorem for θ = 10 justifies our use of decimal notations
for natural numbers. For θ = ω it offers “notations” (in terms of natural numbers) for
all ordinals below ε 0 := ⋂ {α | ω α = α} = ⋃ {ω, ω ω , ω ωω , . . .}. This is used in Gentzen’s
consistency proof for Peano arithmetic.
Definition 4.33 α is called critical for the operation F : OR 2 → OR if β, γ < α ⇒
F (β, γ) < α.
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Exercises
94 ♣ Show: α · ω is the least ordinal > α that is critical for +.
95 ♣ For α ω, the following are equivalent:
1. α is critical for +; 2. β < α ⇒ β + α = α; 3. ∃ξ (α = ω ξ ).
96 ♣ Assume α ω.
Show: α ω is the least ordinal > α that is critical for multiplication.
97 ♣ Show that, for α ω, the following are equivalent:
1. α is critical for multiplication; 2. β < α ⇒ β · α = α; 3. ∃ξ (α = ω ωξ ).
98 ♣ Show: every initial is critical for addition, multiplication and exponentiation.
4.8 Well-ordering of OR × OR
Definition 4.34 Define the relation < on OR × OR by:
(α, β) < (α ′ , β ′ ) :≡
max(α, β) < max(α ′ , β ′ ), or: max(α, β) = max(α ′ , β ′ ) ∧ [α < α ′ ∨ (α = α ′ ∧ β < β ′ )].
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99 ♣ Exercise Show:
1. < well-orders OR × OR,
2. every product γ × γ is an initial segment
(if (α, β) < (α ′ , β ′ ) ∈ γ × γ, then (α, β) ∈ γ × γ),
3. the product ω × ω is well-ordered in type ω,
4. every product ω α × ω α (α > 0) is well-ordered in type ω α .
Hint. 4 says that, if Γ : OR × OR → OR is the unique order-preserving map, then
Γ(ω α , ω α ) = ω α . Use induction w.r.t. α. If equality doesn’t hold, then we must have
Γ(ω α , ω α ) > ω α . Then (β, γ) ∈ ω α × ω α exists such that Γ(β, γ) = ω α , etc.
Part 4 of this exercise has the important
Corollary 4.35 ω α × ω α = 1 ω α .
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