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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


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


ORDINALSCARDINALS

ORDINALS

/0 = 0

/0∪ { /0} = { /0} = 1

{ /0}∪ {{ /0}} = { /0, { /0}} = 2...

ω

...

ωω

...


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 .


ORDINALSCARDINALS

ORDINAL SUMS

... ...

FIGURE: α +β

e.g.1+ω = ω 6= ω +1


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 ).


ORDINALSCARDINALS

ORDINAL PRODUCTS

...

...... ... ... ... ... ...

FIGURE: α ·β


ORDINALSCARDINALS

ORDINAL PRODUCTS

...

...

FIGURE: α ·β

e.g.2 ·ω = ω 6= ω ·2 = ω +ω


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 .


ORDINALSCARDINALS

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.


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 }


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


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.


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 |.


ORDINALSCARDINALS

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}.


ORDINALSCARDINALS

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 ,


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 .


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 |+


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.


ORDINALSCARDINALS

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 .


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