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# Ordinal Cardinals

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### Ordinal Cardinals

1. 1. ORDINALS CARDINALS ORDINALS AND CARDINALS SEP Erik A. Andrejko University of Wisconsin - Madison Summer 2007 ERIK A. ANDREJKO ORDINALS AND CARDINALS
2. 2. ORDINALS CARDINALS NEUMANN ORDINALS VON FIGURE: John von Neumann ERIK A. ANDREJKO ORDINALS AND CARDINALS
3. 3. ORDINALS CARDINALS ORDINALS DEFINITION An ordinal is a set x that is transitive and well ordered by ∈. The class of ordinals is denoted ON. 0 ∈ ON zero / α ∈ ON =⇒ α ∪ {α} ∈ ON successor For any set X , X ⊆ ON =⇒ X ∈ ON limit ERIK A. ANDREJKO ORDINALS AND CARDINALS
4. 4. ORDINALS CARDINALS ORDINALS 0=0 / 0 ∪ {0} = {0} = 1 // / {0} ∪ {{0}} = {0, {0}} = 2 / / // . . . ω . . . ωω . . . ERIK A. ANDREJKO ORDINALS AND CARDINALS
5. 5. ORDINALS CARDINALS ORDER TYPES THEOREM If A, R is a well-ordering then there is a unique ordinal ξ such that ∼ A, R = ξ ∼ i.e. with A, R = ξ , ∈ . DEFINITION ξ is the order type of the well ordering A, R also denoted type(A, R) = ξ . ERIK A. ANDREJKO ORDINALS AND CARDINALS
6. 6. ORDINALS CARDINALS ORDINAL SUMS ... ... FIGURE: α + β e.g. 1+ω = ω = ω +1 ERIK A. ANDREJKO ORDINALS AND CARDINALS
7. 7. ORDINALS CARDINALS ORDINAL SUMS FACT For any ordinals α, β , γ α + (β + γ) = (α + β ) + γ, 1 α + 0 = α, 2 α + 1 = S(α), 3 α + S(β ) = S(α + β ), 4 If β is a limit ordinal 5 α + β = sup(α + ξ : ξ < β ). ERIK A. ANDREJKO ORDINALS AND CARDINALS
8. 8. ORDINALS CARDINALS ORDINAL PRODUCTS ... ... ... ... ... ... ... ... FIGURE: α · β ERIK A. ANDREJKO ORDINALS AND CARDINALS
9. 9. ORDINALS CARDINALS ORDINAL PRODUCTS ... ... FIGURE: α · β e.g. 2·ω = ω = ω ·2 = ω +ω ERIK A. ANDREJKO ORDINALS AND CARDINALS
10. 10. ORDINALS CARDINALS ORDINAL PRODUCTS FACT For ordinals α, β , γ α · (β · γ) = (α · β ) · γ, 1 α · 0 = 0 · α = 0, 2 α · 1 = 1 · α = α, 3 α · S(β ) = α · β + α, 4 For limit β 5 α · β = sup{α · ξ : ξ < β } α · (β + γ) = α · β + α · γ. 6 ERIK A. ANDREJKO ORDINALS AND CARDINALS
11. 11. ORDINALS CARDINALS ORDINAL WARNINGS WARNING The + operation is not commutative: α + β = β + α. (except on the natural numbers) WARNING The operation · is not commutative except on the natural numbers: 2 · ω = ω = ω · 2 = ω + ω. The right distributive law does not hold: (1 + 1) · ω = ω = 1 · ω + 1 · ω = ω + ω. ERIK A. ANDREJKO ORDINALS AND CARDINALS
12. 12. ORDINALS CARDINALS ORDINAL EXPONENTIATION For ordinals α, β deﬁne α β by α 0 = 1, 1 α β +1 = α β · α, 2 For limit β 3 α β = sup{α ξ : ξ < β } ERIK A. ANDREJKO ORDINALS AND CARDINALS
13. 13. ORDINALS CARDINALS CANTOR NORMAL FORM THEOREM (Cantor’s Normal Form Theorem) Every ordinal α > 0 can be written as α = ω β1 k1 + · · · + ω βn kn for ki ∈ ω {0}, α ≥ β1 > · · · > βn . Note that it is possible for α = β1 . The least such ordinal α is ε0 . i.e. ε0 = ω ε0 ERIK A. ANDREJKO ORDINALS AND CARDINALS
14. 14. ORDINALS CARDINALS SPECIAL ORDINALS DEFINITION 1 γ = ω. 0 γn+1 = ω γn . 2 ε0 = sup{γn : n < ω} 3 Then ω ε0 = ε0 ε0 is the least ordinal α such that ω α = α. DEFINITION ω1CK is the least non-computable ordinal. ERIK A. ANDREJKO ORDINALS AND CARDINALS
15. 15. ORDINALS CARDINALS CARDINALS DEFINITION Let A be a set that can be well-ordered. Then |A| is deﬁned to be the least ordinal α such that |A| ≈ α. Under AC every A can be well ordered and so |A| is deﬁned for all sets A. DEFINITION An ordinal α is called a cardinal if α = |α|. ERIK A. ANDREJKO ORDINALS AND CARDINALS
16. 16. ORDINALS CARDINALS CARDINALS DEFINITION Given an ordinal α, deﬁne α + to be the least cardinal > α. DEFINITION The cardinals ℵα = ωα are deﬁned as ℵ0 = ω0 = ω, 1 ℵα+1 = ωα+1 = (ωα )+ , 2 For limit γ, ℵγ = ωγ = sup{ωα : α < γ}. 3 ERIK A. ANDREJKO ORDINALS AND CARDINALS
17. 17. ORDINALS CARDINALS CARDINALS FACT 1. Each ωα is a cardinal, 2. Every inﬁnite cardinal is equal to ωα for some α. 3. α < β implies ωα < ωβ , ERIK A. ANDREJKO ORDINALS AND CARDINALS
18. 18. ORDINALS CARDINALS CARDINAL ARITHMETIC For cardinals κ and λ deﬁne the sum κ ⊕ λ = |κ × {0} ∪ λ × {1}| and the product κ ⊗ λ = |κ × λ | FACT ⊕ and ⊗ are commutative. FACT For cardinals κ, λ ≥ ω k ⊕ λ = κ ⊗ λ = max(κ, λ ), 1 |κ <ω | = κ. 2 ERIK A. ANDREJKO ORDINALS AND CARDINALS
19. 19. ORDINALS CARDINALS CARDINAL ARITHMETIC DEFINITION Using AC, for cardinals λ and κ deﬁne κ λ = | λ κ |. FACT For λ ≥ ω and 2 ≤ κ ≤ λ then κ ≈ λ 2 ≈ P(λ ) λ FACT (AC) For cardinals κ, λ , σ κ λ ⊕σ = κ λ ⊗ κ σ and (κ λ )σ = κ λ ⊗σ i.e. the normal rules for exponentiation apply. ERIK A. ANDREJKO ORDINALS AND CARDINALS
20. 20. ORDINALS CARDINALS HARTOG FUNCTION DEFINITION Given a set X deﬁne ℵ(X ), Hartog’s Aleph Function, ℵ(X ) = sup{α : ∃f ∈ α X f is 1 − 1} FACT (AC) ℵ(X ) = |X |+ ERIK A. ANDREJKO ORDINALS AND CARDINALS
21. 21. ORDINALS CARDINALS CARDINAL TYPES DEFINITION 1 ω is a limit cardinal if and only if α is a limit ordinal., α ωα is a successor cardinal if and only if α is a successor 2 ordinal. ERIK A. ANDREJKO ORDINALS AND CARDINALS
22. 22. ORDINALS CARDINALS CARDINAL TYPES DEFINITION Let f : α → β . Then f maps α coﬁnally if ran(f ) is unbounded in β . DEFINITION The coﬁnality of β , denoted cf(β ) is the least α such that there exists a map from α coﬁnally into β . DEFINITION A cardinal κ is regular if cf(κ) = κ, 1 A cardinal κ is singular if cf(κ) < κ. 2 FACT κ + is regular for any cardinal κ. ERIK A. ANDREJKO ORDINALS AND CARDINALS