Download Axiomatic Set Theory Alexandru Baltag

Axiomatic Set Theory
Alexandru Baltag
(ILLC, University of Amsterdam)
LECTURE NOTES 5: AC, the Well-Ordering Theorem and
Operations on Ordinals
1
Various Formulations of the Axiom of Choice
Axiom of Choice (AC) (first formulation):
If F is a set of pairwise-disjoint, non-empty sets, then there
exists a set M (called a “choice set”) that consists of exactly
one element from each member of F.
(AC’) (second formulation):
If F is a set of non-empty sets, then there exists a function
S
f : F → F (called a “choice function”) such that f (X) ∈ X for
all X ∈ F.
(AC”) (third formulation):
If F is a set of non-empty sets, then the Cartesian product
ΠF = ΠX∈F X is non-empty.
Proposition (in ZF): (AC) ⇔ (AC 0 ) ⇔ (AC 00 ).
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The Well-Ordering Principle
Another equivalent formulation is Zermelo’s Well-Ordering
Principle (WO), also known as the Well-Ordering Theorem:
(WO) Well-Ordering Theorem (in ZF C):
Every set can be well-ordered.
Proposition (in ZF , see Theorem 2.7.3 in Devlin):
(AC) ⇔ (W O).
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Zorn’s Lemma
Another equivalent formulation is “Zorn’s lemma”.
An element a of a poset (P, ≤) is maximal (in P ) iff there is no
element b ∈ P s.t. a < b.
A chain in a poset (P, ≤) is a subset of P that is totally ordered (by ≤).
An element a of a poset (P, ≤) is an upper bound of a subset B ⊆ P
iff b ≤ a for all b ∈ B.
“Zorn’s Lemma” (ZL):
If every chain in a poset (P, ≤) has an upper bound (in P ),
then P has a maximal element.
Proposition (in ZF , see Theorem 2.7.5 in Devlin): (AC) ⇔ (ZL).
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Ordinals in ZF C
DEF: A set X is transitive iff
∀x∀y(y ∈ x ∧ x ∈ X ⇒ y ∈ X)
Lemma 3.1.1. (Devlin): A set X is an ordinal iff it is transitive and
totally ordered by ∈.
Consequences (provable in ZF , see Lemmas 3.1.2 and 3.1.3 in
Devlin):
If α is an ordinal, then its immediate successor α + 1 := α ∪ {α} is an
ordinal.
S
If A is a set of ordinals, then A is an ordinal.
5
Proving existence of ordinals in ZF
The above results allow us to prove in ZF the existence of all finite
ordinals, and then (using the Axiom of Infinity and the Replacement
Axiom) the existence of ω.
Next, one can iterate the successor function to generate all ordinal
ω + n.
To show existence of ω + ω, let
F : ω → V, F (n) = ω + n
By Replacement, f [ω] = {f (n)|n ∈ ω} is a set (of ordinals).
S
By the above result, f [ω] = ω + ω is an ordinal.
Similarly, one can show existence of ω + ω + ω etc.
6
Addition of Ordinals
To define ordinal sum α + β, we take the (unique ordinal isomorphic
to the) disjoint union of the two ordinals well-ordered it by putting first
the elements of α (in their usual order as ordinals) and then the
elements of β (in their usual order).
Formally: take the set
A = (α × {0}) ∪ (β × {1})
with the anti-lexicographic well-ordering
(µ, i) <A (τ, j) ⇔ (i < j) ∨ (i = j ∧ µ < τ ),
then define
α + β := Ord(A, <A )
where Ord(A, <A ) is the unique ordinal isomorphic to (A, <A ).
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Properties of Ordinal Addition
Proposition: Ordinal addition is associative:
α + (β + γ) = (α + β) + γ.
BUT it is NOT commutative!
n+ω =ω
but
ω+n>ω
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Monotonic Laws for Ordinal Addition
The First Monotonic Law for Addition:
Ordinal addition is strictly increasing in the second argument:
α<β ⇒ γ+α<γ+β
The Second Monotonic Law for Addition:
Ordinal addition is monotonic in the first argument:
α≤β ⇒ α+γ ≤β+γ
BUT ordinal addition is NOT strictly increasing in the first argument:
1 < 2, but 1 + ω = ω = 2 + ω.
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Ordinal Subtraction
Theorem:
If α ≥ β are ordinals, then there exists a unique ordinal γ (called the
difference of the two ordinals, and denoted by α − β) such that
α = β + γ.
CONSEQUENCE:
if α ≥ β, then α = β + (α − β).
EXAMPLES:
ω−n=ω
ω2 − ω = ω2
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Monotonic Laws for Ordinal Subtraction
Difference is strictly increasing in the first argument:
α>β ≥γ ⇒ α−γ >β−γ
Difference is anti-monotonic in the second argument:
γ ≥α≥β ⇒ γ−α≤γ−β
BUT it is NOT strictly decreasing in the second argument:
ω > 2 > 1 but ω − 2 = ω = ω − 1.
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Ordinal Multiplication
To define the ordinal multiplication α · β of two ordinals α and β, we
take (the unique ordinal isomorphic to) the Cartesian Product
α×β
well-ordered with the anti-lexicographic order
(µ, η) < (µ0 , η 0 ) ⇔ (η < η 0 ) ∨ (η = η 0 ∧ µ < µ0 ),
(for µ ∈ α, η ∈ β).
As we’ll see, ordinal multiplication can also be seen as iterated ordinal
addition.
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Ordinal Addition of A Sequence of Ordinals
We generalize the above ordinal sum to well-ordered sequences
(αη )η<λ of ordinals. (i.e. functions f : λ → On, with f (η) = αη ).
The ordinal sum
X
αη
η<λ
is defined by taking the disjoint union
[
(αη × {η})
A=
η<λ
together with the anti-lexicographic well-ordering
0
0
0
0
0
(µ, η) < (µ , η ) ⇔ (η < η ) ∨ (η = η ∧ µ < µ ),
X
and putting
αη := Ord(A, <A ).
η<λ
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Ordinal Multiplication
Then it is easy to see that ordinal multiplication is iterated
addition:
X
α·β =
α
η<β
(Formally, this is the ordinal sum of the sequence given by the function
f : β → On, given by f (η) = α, for every η ∈ β ).
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Properties of Ordinal Multiplication
Ordinal Multiplication is associative
α · (β · γ) = (α · β) · γ
and distributes to the right over addition:
α · (β + γ) = α · β + α · γ
But it is NOT commutative
n · ω = ω, but ω · n > ω
and it does NOT distribute to the left:
(1 + 1) · ω = 2 · ω = ω
but
1 · ω + 1 · ω = ω + ω > ω.
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Monotonic Laws for Ordinal Multiplication
The First Monotonic Law for Multiplication:
Ordinal multiplication with a non-zero first argument is strictly
increasing in the second argument:
α < β ∧ γ 6= 0 ⇒ γ · α < γ · β
The Second Monotonic Law for Multiplication:
Ordinal multiplication is monotonic in the first argument:
α≤β ⇒ α·γ ≤β·γ
BUT ordinal multiplication (even with a non-zero second argument) is
NOT strictly increasing in the first argument:
1 < 2, but 1 · ω = ω = 2 · ω.
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Distributivity to the Right over Subtraction
Ordinal multiplication is distributive to the right with respect
to ordinal subtraction:
α · (β − γ) = α · β − α · γ
But it does NOT distribute to the left:
(2 − 1) · ω = 1 · ω = ω
while
2·ω−1·ω =ω−ω =0
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Division with Remainder
Theorem (Division with Remainder):
If α is an ordinal and β > 0 is a non-zero ordinal, then there exist
uniquely determined ordinals γ and δ, such that
α = β · γ + δ, and δ < β.
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Limit of a Sequence of Ordinals
For a given limit ordinal λ, let (αη )η<λ be a λ-sequence of ordinals (i.e.
a function f : λ → On, with f (η) = αη ).
For an ordinal α, we write
α = lim αη
η<λ
iff ∀β < α ∃η < λ ∀ζ (η < ζ < λ ⇒ β < αζ ≤ α) .
It is easy to see that, if α exists, then it is unique.
If it exists, we call α the limit of the sequence (αη )η<λ .
Lemma 3.4.1 (Devlin): For every limit ordinal λ, every increasing
λ-sequence (αη )η<λ has a (unique) limit, given by
[
lim αη =
αη .
η<λ
η<λ
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Properties of limits
lim (α + βη ) = α + lim βη
η<λ
η<λ
But NOT the other way around: e.g.
lim (n + ω) = lim ω = ω, BUT
lim n + ω = ω + ω > ω.
n<ω
n<ω
n<ω
Similarly:
lim (α · βη ) = α · lim βη
η<λ
η<λ
But NOT the other way around: e.g.
2
lim (n · ω) = lim ω = ω, BUT
lim n · ω = ω · ω = ω > ω.
n<ω
n<ω
n<ω
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Ordinal Exponentiation
We can define ordinal exponention by recursion on ordinals:
α0 = 1
αβ+1 = αβ · α
αβ = lim αγ , if β is a limit ordinal.
γ<β
(More precisely, for each given ordinal α, we use the above recursive
clauses to define a class function Fα : On → On, Fα (β) = αβ .)
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Properties of Ordinal Exponentiation
Lemma 3.5.1 (Devlin):
αβ · αγ = αβ+γ
β γ
α
= αβ·γ
But left-distributivity (of exponentiation over multiplication) FAILS:
(ω · 2)2 6= ω 2 · 22
Other properties:
1α = 1
α1 = α
0α = 0, for α > 0
nω = ω for any 1 < n < ω
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Monotonic Laws for Ordinal Exponentiation
The First Monotonic Law for Exponentiation:
Ordinal exponentiation with a non-zero first argument is strictly
increasing in the second argument:
α < β ∧ γ 6= 0 ⇒ γ α < γ β
The Second Monotonic Law for Exponentiation:
Ordinal exponentiation is monotonic in the first argument:
α ≤ β ⇒ αγ ≤ β γ
BUT ordinal exponentiation (even with a non-zero second argument) is
NOT strictly increasing in the first argument:
2 < 3, but 2ω = ω = 3ω .
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