ORDINALS
CARDINALS
ORDINALS AND CARDINALS
SEP
Erik A. Andrejko
University of Wisconsin - Madison
Summer 2007
ERIK A. ANDREJKO ORDINALS AND CARDINALS

ORDINALS
CARDINALS
NEUMANN ORDINALS
VON
FIGURE: John von Neumann
ERIK A. ANDREJKO ORDINALS AND CARDINALS

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

ORDINALS
CARDINALS
ORDINALS
0=0
/
0 ∪ {0} = {0} = 1
// /
{0} ∪ {{0}} = {0, {0}} = 2
/ / //
.
.
.
ω
.
.
.
ωω
.
.
.
ERIK A. ANDREJKO ORDINALS AND CARDINALS

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

ORDINALS
CARDINALS
ORDINAL SUMS
... ...
FIGURE: α + β
e.g.
1+ω = ω = ω +1
ERIK A. ANDREJKO ORDINALS AND CARDINALS

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

ORDINALS
CARDINALS
ORDINAL PRODUCTS
...
...
...
...
...
...
...
...
FIGURE: α · β
ERIK A. ANDREJKO ORDINALS AND CARDINALS

ORDINALS
CARDINALS
ORDINAL PRODUCTS
...
...
FIGURE: α · β
e.g.
2·ω = ω = ω ·2 = ω +ω
ERIK A. ANDREJKO ORDINALS AND CARDINALS

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

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

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

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

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

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

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

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

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

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

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

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

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