Ordinal Cardinals
description
Transcript of Ordinal Cardinals
ORDINALSCARDINALS
ORDINALS AND CARDINALSSEP
Erik A. Andrejko
University of Wisconsin - Madison
Summer 2007
ERIK A. ANDREJKO ORDINALS AND CARDINALS
ORDINALSCARDINALS
VON NEUMANN ORDINALS
FIGURE: John von Neumann
ERIK A. ANDREJKO ORDINALS AND CARDINALS
ORDINALSCARDINALS
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
ORDINALSCARDINALS
ORDINALS
/0 = 0
/0∪ { /0} = { /0} = 1
{ /0}∪ {{ /0}} = { /0, { /0}} = 2...
ω
...
ωω
...
ERIK A. ANDREJKO ORDINALS AND CARDINALS
ORDINALSCARDINALS
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 denotedtype(A,R) = ξ .
ERIK A. ANDREJKO ORDINALS AND CARDINALS
ORDINALSCARDINALS
ORDINAL SUMS
... ...
FIGURE: α +β
e.g.1+ω = ω 6= ω +1
ERIK A. ANDREJKO ORDINALS AND CARDINALS
ORDINALSCARDINALS
ORDINAL SUMS
FACT
For any ordinals α,β ,γ
1 α +(β + γ) = (α +β )+ γ ,2 α +0 = α ,3 α +1 = S(α),4 α +S(β ) = S(α +β ),5 If β is a limit ordinal
α +β = sup(α +ξ : ξ < β ).
ERIK A. ANDREJKO ORDINALS AND CARDINALS
ORDINALSCARDINALS
ORDINAL PRODUCTS
...
...... ... ... ... ... ...
FIGURE: α ·β
ERIK A. ANDREJKO ORDINALS AND CARDINALS
ORDINALSCARDINALS
ORDINAL PRODUCTS
...
...
FIGURE: α ·β
e.g.2 ·ω = ω 6= ω ·2 = ω +ω
ERIK A. ANDREJKO ORDINALS AND CARDINALS
ORDINALSCARDINALS
ORDINAL PRODUCTS
FACT
For ordinals α,β ,γ
1 α · (β · γ) = (α ·β ) · γ ,2 α ·0 = 0 ·α = 0,3 α ·1 = 1 ·α = α ,4 α ·S(β ) = α ·β +α ,5 For limit β
α ·β = sup{α ·ξ : ξ < β }
6 α · (β + γ) = α ·β +α · γ .
ERIK A. ANDREJKO ORDINALS AND CARDINALS
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ORDINAL WARNINGS
WARNING
The + operation is not commutative: α +β 6= β +α . (except on thenatural numbers)
WARNING
The operation · is not commutative except on the natural numbers:
2 ·ω = ω 6= ω ·2 = ω +ω.
The right distributive law does not hold:
(1+1) ·ω = ω 6= 1 ·ω +1 ·ω = ω +ω.
ERIK A. ANDREJKO ORDINALS AND CARDINALS
ORDINALSCARDINALS
ORDINAL EXPONENTIATION
For ordinals α,β define αβ by1 α0 = 1,2 αβ+1 = αβ ·α ,3 For limit β
αβ = sup{αξ : ξ < β }
ERIK A. ANDREJKO ORDINALS AND CARDINALS
ORDINALSCARDINALS
CANTOR NORMAL FORM
THEOREM
(Cantor’s Normal Form Theorem) Every ordinal α > 0 can be writtenas
α = ωβ1k1 + · · ·+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 = ωε0
ERIK A. ANDREJKO ORDINALS AND CARDINALS
ORDINALSCARDINALS
SPECIAL ORDINALS
DEFINITION
1 γ0 = ω .2 γn+1 = ωγn .3 ε0 = sup{γn : n < ω}
Then ωε0 = ε0
ε0 is the least ordinal α such that ωα = α .
DEFINITION
ω1CK is the least non-computable ordinal.
ERIK A. ANDREJKO ORDINALS AND CARDINALS
ORDINALSCARDINALS
CARDINALS
DEFINITION
Let A be a set that can be well-ordered. Then |A| is defined to be theleast ordinal α such that |A|≈ α .
Under AC every A can be well ordered and so |A| is defined for all setsA.
DEFINITION
An ordinal α is called a cardinal if α = |α |.
ERIK A. ANDREJKO ORDINALS AND CARDINALS
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CARDINALS
DEFINITION
Given an ordinal α , define α+ to be the least cardinal > α .
DEFINITION
The cardinals ℵα = ωα are defined as1 ℵ0 = ω0 = ω ,2 ℵα+1 = ωα+1 = (ωα)+,3 For limit γ , ℵγ = ωγ = sup{ωα : α < γ}.
ERIK A. ANDREJKO ORDINALS AND CARDINALS
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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 ,
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ORDINALSCARDINALS
CARDINAL ARITHMETIC
For cardinals κ and λ define the sum
κ⊕λ = |κ× {0}∪λ × {1}|
and the productκ⊗λ = |κ×λ |
FACT
⊕ and ⊗ are commutative.
FACT
For cardinals κ,λ ≥ ω
1 k ⊕λ = κ⊗λ = max(κ,λ ),2 |κ<ω | = κ .
ERIK A. ANDREJKO ORDINALS AND CARDINALS
ORDINALSCARDINALS
CARDINAL ARITHMETIC
DEFINITION
Using AC, for cardinals λ and κ define κλ = |λκ |.
FACT
For λ ≥ ω and 2≤ κ ≤ λ then
λκ ≈ λ 2 ≈ P(λ )
FACT
(AC) For cardinals κ,λ ,σ
κλ⊕σ = κ
λ ⊗κσ and (κλ )σ = κ
λ⊗σ
i.e. the normal rules for exponentiation apply.ERIK A. ANDREJKO ORDINALS AND CARDINALS
ORDINALSCARDINALS
HARTOG FUNCTION
DEFINITION
Given a set X define ℵ(X), Hartog’s Aleph Function,
ℵ(X) = sup{α : ∃f ∈ αX f is 1−1}
FACT
(AC) ℵ(X) = |X |+
ERIK A. ANDREJKO ORDINALS AND CARDINALS
ORDINALSCARDINALS
CARDINAL TYPES
DEFINITION
1 ωα is a limit cardinal if and only if α is a limit ordinal.,2 ωα is a successor cardinal if and only if α is a successor
ordinal.
ERIK A. ANDREJKO ORDINALS AND CARDINALS
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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 existsa map from α cofinally into β .
DEFINITION
1 A cardinal κ is regular if cf(κ) = κ ,2 A cardinal κ is singular if cf(κ) < κ .
FACT
κ+ is regular for any cardinal κ .
ERIK A. ANDREJKO ORDINALS AND CARDINALS