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The Continuum Hypothesis, the Axiom of
Choice, and Lebesgue Measurability
Chris Lambie-Hanson
CMU Graduate Student Seminar
11 October 2011
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Definitions
If S is a set, then a linear order on S is a binary relation ≤ such
that
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Definitions
If S is a set, then a linear order on S is a binary relation ≤ such
that
1 For all x ∈ S, x ≤ x.
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Definitions
If S is a set, then a linear order on S is a binary relation ≤ such
that
1 For all x ∈ S, x ≤ x.
2 For all x, y ∈ S, if x ≤ y and y ≤ x, then x = y.
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Definitions
If S is a set, then a linear order on S is a binary relation ≤ such
that
1 For all x ∈ S, x ≤ x.
2 For all x, y ∈ S, if x ≤ y and y ≤ x, then x = y.
3 For all x, y, z ∈ S, if x ≤ y and y ≤ z, then x ≤ z.
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Definitions
If S is a set, then a linear order on S is a binary relation ≤ such
that
1 For all x ∈ S, x ≤ x.
2 For all x, y ∈ S, if x ≤ y and y ≤ x, then x = y.
3 For all x, y, z ∈ S, if x ≤ y and y ≤ z, then x ≤ z.
4 For all x, y ∈ S, x ≤ y or y ≤ x.
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Definitions
If S is a set, then a linear order on S is a binary relation ≤ such
that
1 For all x ∈ S, x ≤ x.
2 For all x, y ∈ S, if x ≤ y and y ≤ x, then x = y.
3 For all x, y, z ∈ S, if x ≤ y and y ≤ z, then x ≤ z.
4 For all x, y ∈ S, x ≤ y or y ≤ x.
A linear order ≤ on a set S is a well-order if, for every nonempty
X ⊆ S, there is a ≤-least element in X, i.e. there is x ∈ X such
that, for all y ∈ X, x ≤ y.
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The Axiom of Choice
The Axiom of Choice is the assertion: For every family of
nonempty sets F, there is a function g such that dom(g) = F
and, for every X ∈ F, g(X) ∈ X.
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The Axiom of Choice
The Axiom of Choice is the assertion: For every family of
nonempty sets F, there is a function g such that dom(g) = F
and, for every X ∈ F, g(X) ∈ X.
The Axiom of Choice is equivalent to the Well-ordering
Theorem, which asserts that every set can be well-ordered.
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Ordinals, Informally
An ordinal number describes the order type of a well-ordering.
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Ordinals, Informally
An ordinal number describes the order type of a well-ordering.
• A natural number n is an ordinal describing a well-ordering
with n elements.
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Ordinals, Informally
An ordinal number describes the order type of a well-ordering.
• A natural number n is an ordinal describing a well-ordering
with n elements.
• The ordinal describing the order type of the natural numbers
is ω.
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Ordinals, Informally
An ordinal number describes the order type of a well-ordering.
• A natural number n is an ordinal describing a well-ordering
with n elements.
• The ordinal describing the order type of the natural numbers
is ω.
• The ordinal describing the order type of the natural numbers
plus one element larger than all of the natural numbers is
ω + 1.
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Ordinals, Informally
An ordinal number describes the order type of a well-ordering.
• A natural number n is an ordinal describing a well-ordering
with n elements.
• The ordinal describing the order type of the natural numbers
is ω.
• The ordinal describing the order type of the natural numbers
plus one element larger than all of the natural numbers is
ω + 1.
An ordinal α is actually a set of ordinals, well-ordered by ∈, of
order-type α.
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Ordinals, Informally
An ordinal number describes the order type of a well-ordering.
• A natural number n is an ordinal describing a well-ordering
with n elements.
• The ordinal describing the order type of the natural numbers
is ω.
• The ordinal describing the order type of the natural numbers
plus one element larger than all of the natural numbers is
ω + 1.
An ordinal α is actually a set of ordinals, well-ordered by ∈, of
order-type α.
If α = β + 1 = β ∪ {β}, then α is called a successor ordinal.
Otherwise, it is called a limit ordinal.
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Cardinals
Two sets A and B have the same cardinality if there is a bijection
f : A → B.
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Cardinals
Two sets A and B have the same cardinality if there is a bijection
f : A → B.
The cardinality of A, written |A|, can be thought of as the
equivalence class of all sets for which bijections with A exist.
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Cardinals
Two sets A and B have the same cardinality if there is a bijection
f : A → B.
The cardinality of A, written |A|, can be thought of as the
equivalence class of all sets for which bijections with A exist.
For two sets A and B, we say |A| |B| if there is an injective
function f : A → B.
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Cardinals
Two sets A and B have the same cardinality if there is a bijection
f : A → B.
The cardinality of A, written |A|, can be thought of as the
equivalence class of all sets for which bijections with A exist.
For two sets A and B, we say |A| |B| if there is an injective
function f : A → B.
Under the Axiom of Choice, a cardinality is often identified with
the smallest ordinal of that cardinality. Thus, under AC,
cardinalities are well-ordered by .
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Cardinals
Two sets A and B have the same cardinality if there is a bijection
f : A → B.
The cardinality of A, written |A|, can be thought of as the
equivalence class of all sets for which bijections with A exist.
For two sets A and B, we say |A| |B| if there is an injective
function f : A → B.
Under the Axiom of Choice, a cardinality is often identified with
the smallest ordinal of that cardinality. Thus, under AC,
cardinalities are well-ordered by .
If α is an ordinal, the α-th infinite cardinal is written ℵα. The
smallest ordinal of cardinality ℵα is written ωα.
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Cardinals
A set is called countable if it is either finite or has the same
cardinality as the set of natural numbers.
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Cardinals
A set is called countable if it is either finite or has the same
cardinality as the set of natural numbers.
Every natural number n is a cardinal number.
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Cardinals
A set is called countable if it is either finite or has the same
cardinality as the set of natural numbers.
Every natural number n is a cardinal number.
ω0 = ω, and |ω| = ℵ0.
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Cardinals
A set is called countable if it is either finite or has the same
cardinality as the set of natural numbers.
Every natural number n is a cardinal number.
ω0 = ω, and |ω| = ℵ0.
ω1 is the set of all countable ordinals, and |ω1| = ℵ1.
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Cardinals
A set is called countable if it is either finite or has the same
cardinality as the set of natural numbers.
Every natural number n is a cardinal number.
ω0 = ω, and |ω| = ℵ0.
ω1 is the set of all countable ordinals, and |ω1| = ℵ1.
ω2 is the set of all ordinals of cardinality ≤ ℵ1, and |ω2| = ℵ2.
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Cantor’s Theorem
If A is a set, then the power set of A, written P(A), is the set of
all subsets of A.
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Cantor’s Theorem
If A is a set, then the power set of A, written P(A), is the set of
all subsets of A.
Theorem
For every set A, |A| ≺ |P(A)|.
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Cantor’s Theorem
If A is a set, then the power set of A, written P(A), is the set of
all subsets of A.
Theorem
For every set A, |A| ≺ |P(A)|.
It is clear that |A| |P(A)|, so it suffices to prove that there is no
bijection f : A → P(A).
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Hypergame
We say a finite game is a two-player game with alternating moves
which is guaranteed to end in a finite number of moves.
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Hypergame
We say a finite game is a two-player game with alternating moves
which is guaranteed to end in a finite number of moves.
Hypergame is a two-player game with alternating moves played as
follows:
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Hypergame
We say a finite game is a two-player game with alternating moves
which is guaranteed to end in a finite number of moves.
Hypergame is a two-player game with alternating moves played as
follows:
1 Player 1 names a finite game.
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Hypergame
We say a finite game is a two-player game with alternating moves
which is guaranteed to end in a finite number of moves.
Hypergame is a two-player game with alternating moves played as
follows:
1 Player 1 names a finite game.
2 Player 2 makes the first move in that finite game.
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Hypergame
We say a finite game is a two-player game with alternating moves
which is guaranteed to end in a finite number of moves.
Hypergame is a two-player game with alternating moves played as
follows:
1 Player 1 names a finite game.
2 Player 2 makes the first move in that finite game.
3 The players continue playing that finite game until it ends.
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Hypergame
Example:
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Hypergame
Example:
I: “Let’s play chess!”
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Hypergame
Example:
I: “Let’s play chess!”
II:f3
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Hypergame
Example:
I: “Let’s play chess!”
II:f3
I: e6
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Hypergame
Example:
I: “Let’s play chess!”
II:f3
I: e6
II:g4
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Hypergame
Example:
I: “Let’s play chess!”
II:f3
I: e6
II:g4
I: Qh4#
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Hypergame
Is Hypergame a finite game?
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Hypergame
Is Hypergame a finite game?
Clearly, yes, but...
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Hypergame
Is Hypergame a finite game?
Clearly, yes, but...
I: “Let’s play hypergame!”
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Hypergame
Is Hypergame a finite game?
Clearly, yes, but...
I: “Let’s play hypergame!”
II:“Let’s play hypergame!”
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Hypergame
Is Hypergame a finite game?
Clearly, yes, but...
I: “Let’s play hypergame!”
II:“Let’s play hypergame!”
I: “Let’s play hypergame!”
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Hypergame
Is Hypergame a finite game?
Clearly, yes, but...
I: “Let’s play hypergame!”
II:“Let’s play hypergame!”
I: “Let’s play hypergame!”
II:“Let’s play hypergame!”
. . .
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Proof of Cantor’s Theorem
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Proof of Cantor’s Theorem
Suppose for sake of contradiction that there is an infinite set A and
a bijection f : A → P(A).
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Proof of Cantor’s Theorem
Suppose for sake of contradiction that there is an infinite set A and
a bijection f : A → P(A).
We say that (a0, a1, a2, . . .) is a path in A if:
1 a0 ∈ A
2 For all n, an+1 ∈ f (an)
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Proof of Cantor’s Theorem
Suppose for sake of contradiction that there is an infinite set A and
a bijection f : A → P(A).
We say that (a0, a1, a2, . . .) is a path in A if:
1 a0 ∈ A
2 For all n, an+1 ∈ f (an)
Let X = {a ∈ A | there is no infinite path in A starting with a}.
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Proof of Cantor’s Theorem
Suppose for sake of contradiction that there is an infinite set A and
a bijection f : A → P(A).
We say that (a0, a1, a2, . . .) is a path in A if:
1 a0 ∈ A
2 For all n, an+1 ∈ f (an)
Let X = {a ∈ A | there is no infinite path in A starting with a}.
Since f is a bijection, there is x ∈ A such that f (x) = X.
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Proof of Cantor’s Theorem
Suppose for sake of contradiction that there is an infinite set A and
a bijection f : A → P(A).
We say that (a0, a1, a2, . . .) is a path in A if:
1 a0 ∈ A
2 For all n, an+1 ∈ f (an)
Let X = {a ∈ A | there is no infinite path in A starting with a}.
Since f is a bijection, there is x ∈ A such that f (x) = X.
There can be no infinite path in A starting with x, as the second
element of such a path would have to be in X. Thus, x ∈ f (x).
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Proof of Cantor’s Theorem
Suppose for sake of contradiction that there is an infinite set A and
a bijection f : A → P(A).
We say that (a0, a1, a2, . . .) is a path in A if:
1 a0 ∈ A
2 For all n, an+1 ∈ f (an)
Let X = {a ∈ A | there is no infinite path in A starting with a}.
Since f is a bijection, there is x ∈ A such that f (x) = X.
There can be no infinite path in A starting with x, as the second
element of such a path would have to be in X. Thus, x ∈ f (x).
But then (x, x, x, . . .) is an infinite path in A starting with x.
Contradiction.
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Real numbers
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Real numbers
Definition
A Polish space is a separable, completely metrizable topological
space.
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Real numbers
Definition
A Polish space is a separable, completely metrizable topological
space.
Fact
There is a Borel isomorphism between any two uncountable Polish
spaces.
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Real numbers
Definition
A Polish space is a separable, completely metrizable topological
space.
Fact
There is a Borel isomorphism between any two uncountable Polish
spaces.
Examples
R, [0, 1], P(ω), ω2 = {countable infinite sequences of 0s and 1s},
ωω = {countable infinite sequences of natural numbers}
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Real numbers
Definition
A Polish space is a separable, completely metrizable topological
space.
Fact
There is a Borel isomorphism between any two uncountable Polish
spaces.
Examples
R, [0, 1], P(ω), ω2 = {countable infinite sequences of 0s and 1s},
ωω = {countable infinite sequences of natural numbers}
The cardinality of each of these examples is c = 2ℵ0 .
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The Continuum Hypothesis
CH: 2ℵ0 = ℵ1.
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The Continuum Hypothesis
CH: 2ℵ0 = ℵ1.
Alternative formulations:
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The Continuum Hypothesis
CH: 2ℵ0 = ℵ1.
Alternative formulations:
• There is a surjection f : ω1 → R.
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The Continuum Hypothesis
CH: 2ℵ0 = ℵ1.
Alternative formulations:
• There is a surjection f : ω1 → R.
• There is an injection g : R → ω1.
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The Continuum Hypothesis
CH: 2ℵ0 = ℵ1.
Alternative formulations:
• There is a surjection f : ω1 → R.
• There is an injection g : R → ω1.
• Every uncountable subset of R has cardinality 2ℵ0 .
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The Continuum Hypothesis
CH: 2ℵ0 = ℵ1.
Alternative formulations:
• There is a surjection f : ω1 → R.
• There is an injection g : R → ω1.
• Every uncountable subset of R has cardinality 2ℵ0 .
Hilbert’s 1st problem.
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Cantor-Bendixson Theorem
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Cantor-Bendixson Theorem
Definition
If X is a topological space, then a set S ⊆ X is a perfect set if it
is closed and has no isolated points.
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Cantor-Bendixson Theorem
Definition
If X is a topological space, then a set S ⊆ X is a perfect set if it
is closed and has no isolated points.
Fact
If S is a nonempty perfect subset of a Polish space, then |S| = 2ℵ0 .
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Cantor-Bendixson Theorem
Definition
If X is a topological space, then a set S ⊆ X is a perfect set if it
is closed and has no isolated points.
Fact
If S is a nonempty perfect subset of a Polish space, then |S| = 2ℵ0 .
Definition
A subset A of a Polish space has the perfect set property if
either A is countable or A contains a nonempty perfect subset.
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Cantor-Bendixson Theorem
Definition
If X is a topological space, then a set S ⊆ X is a perfect set if it
is closed and has no isolated points.
Fact
If S is a nonempty perfect subset of a Polish space, then |S| = 2ℵ0 .
Definition
A subset A of a Polish space has the perfect set property if
either A is countable or A contains a nonempty perfect subset.
Theorem
If X is a Polish space, then every closed set C ⊆ X can be written
uniquely as the disjoint union of a perfect set and a countable set.
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Cantor-Bendixson Theorem
Definition
If X is a topological space, then a set S ⊆ X is a perfect set if it
is closed and has no isolated points.
Fact
If S is a nonempty perfect subset of a Polish space, then |S| = 2ℵ0 .
Definition
A subset A of a Polish space has the perfect set property if
either A is countable or A contains a nonempty perfect subset.
Theorem
If X is a Polish space, then every closed set C ⊆ X can be written
uniquely as the disjoint union of a perfect set and a countable set.
Corollary
For every uncountable closed set C ⊆ R, |C| = 2ℵ0 .
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G˝
odel’s Constructible Universe
The constructible universe, denoted L, was introduced by Kurt
G˝
odel in 1938. It is a model of ZFC.
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G˝
odel’s Constructible Universe
The constructible universe, denoted L, was introduced by Kurt
G˝
odel in 1938. It is a model of ZFC.
L is, in a sense, the smallest model of ZFC containing all of the
ordinals.
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G˝
odel’s Constructible Universe
The constructible universe, denoted L, was introduced by Kurt
G˝
odel in 1938. It is a model of ZFC.
L is, in a sense, the smallest model of ZFC containing all of the
ordinals.
CH is true in L.
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G˝
odel’s Constructible Universe
The constructible universe, denoted L, was introduced by Kurt
G˝
odel in 1938. It is a model of ZFC.
L is, in a sense, the smallest model of ZFC containing all of the
ordinals.
CH is true in L.
Shelah: “L looks like the head of a gay chapter of the Ku Klux
Klan - a case worthy of study, but probably not representative.”
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Forcing
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Forcing
In 1963, Paul Cohen introduced the technique of forcing, which
allows for the construction of new models of ZFC.
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Forcing
In 1963, Paul Cohen introduced the technique of forcing, which
allows for the construction of new models of ZFC.
”Adding sets, but very gently.”
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Forcing
In 1963, Paul Cohen introduced the technique of forcing, which
allows for the construction of new models of ZFC.
”Adding sets, but very gently.”
Cohen used forcing to construct a model of ZFC in which CH is
false by adding ℵ2-many new subsets of ω.
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Forcing
In 1963, Paul Cohen introduced the technique of forcing, which
allows for the construction of new models of ZFC.
”Adding sets, but very gently.”
Cohen used forcing to construct a model of ZFC in which CH is
false by adding ℵ2-many new subsets of ω.
This “settled” Hilbert’s first problem.
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The Axiom of Symmetry
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The Axiom of Symmetry
Let I = [0, 1], and let Iω denote the set of countable subsets of
[0, 1].
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The Axiom of Symmetry
Let I = [0, 1], and let Iω denote the set of countable subsets of
[0, 1].
AS: For every f : I → Iω, there are x, y ∈ I such that x ∈ f (y) and
y ∈ f (x).
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The Axiom of Symmetry
Let I = [0, 1], and let Iω denote the set of countable subsets of
[0, 1].
AS: For every f : I → Iω, there are x, y ∈ I such that x ∈ f (y) and
y ∈ f (x).
Lemma
If CH is true, then AS is false.
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The Axiom of Symmetry
Let I = [0, 1], and let Iω denote the set of countable subsets of
[0, 1].
AS: For every f : I → Iω, there are x, y ∈ I such that x ∈ f (y) and
y ∈ f (x).
Lemma
If CH is true, then AS is false.
Proof.
Suppose CH is true and fix an enumeration of I: xα | α < ω1 .
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The Axiom of Symmetry
Let I = [0, 1], and let Iω denote the set of countable subsets of
[0, 1].
AS: For every f : I → Iω, there are x, y ∈ I such that x ∈ f (y) and
y ∈ f (x).
Lemma
If CH is true, then AS is false.
Proof.
Suppose CH is true and fix an enumeration of I: xα | α < ω1 .
Define f : I → Iω by f (xα) = {xβ | β < α}.
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The Axiom of Symmetry
Let I = [0, 1], and let Iω denote the set of countable subsets of
[0, 1].
AS: For every f : I → Iω, there are x, y ∈ I such that x ∈ f (y) and
y ∈ f (x).
Lemma
If CH is true, then AS is false.
Proof.
Suppose CH is true and fix an enumeration of I: xα | α < ω1 .
Define f : I → Iω by f (xα) = {xβ | β < α}.
Then, for any α < β < ω1, xα ∈ f (xβ), so f is a counterexample
to AS.
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The Axiom of Symmetry
AS is actually equivalent to the negation of CH and was
introduced by Chris Freiling as an argument against CH.
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The Axiom of Symmetry
AS is actually equivalent to the negation of CH and was
introduced by Chris Freiling as an argument against CH.
Freiling’s Dartboard Argument:
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The Axiom of Symmetry
AS is actually equivalent to the negation of CH and was
introduced by Chris Freiling as an argument against CH.
Freiling’s Dartboard Argument: Fix an f : I → Iω.
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The Axiom of Symmetry
AS is actually equivalent to the negation of CH and was
introduced by Chris Freiling as an argument against CH.
Freiling’s Dartboard Argument: Fix an f : I → Iω. Throw two
darts at the interval [0, 1].
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The Axiom of Symmetry
AS is actually equivalent to the negation of CH and was
introduced by Chris Freiling as an argument against CH.
Freiling’s Dartboard Argument: Fix an f : I → Iω. Throw two
darts at the interval [0, 1]. If the first dart is thrown and lands at
point x, then, since f (x) is countable and thus has measure 0, the
probability that the second dart lands in f (x) is 0.
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The Axiom of Symmetry
AS is actually equivalent to the negation of CH and was
introduced by Chris Freiling as an argument against CH.
Freiling’s Dartboard Argument: Fix an f : I → Iω. Throw two
darts at the interval [0, 1]. If the first dart is thrown and lands at
point x, then, since f (x) is countable and thus has measure 0, the
probability that the second dart lands in f (x) is 0. Since this is
true no matter where the first dart lands, it should be true even
before the first dart is thrown.
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The Axiom of Symmetry
AS is actually equivalent to the negation of CH and was
introduced by Chris Freiling as an argument against CH.
Freiling’s Dartboard Argument: Fix an f : I → Iω. Throw two
darts at the interval [0, 1]. If the first dart is thrown and lands at
point x, then, since f (x) is countable and thus has measure 0, the
probability that the second dart lands in f (x) is 0. Since this is
true no matter where the first dart lands, it should be true even
before the first dart is thrown. By symmetry of the order in which
the darts are thrown, if y denotes the point where the second dart
lands, the probability that the first dart lands in f (y) is 0.
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The Axiom of Symmetry
AS is actually equivalent to the negation of CH and was
introduced by Chris Freiling as an argument against CH.
Freiling’s Dartboard Argument: Fix an f : I → Iω. Throw two
darts at the interval [0, 1]. If the first dart is thrown and lands at
point x, then, since f (x) is countable and thus has measure 0, the
probability that the second dart lands in f (x) is 0. Since this is
true no matter where the first dart lands, it should be true even
before the first dart is thrown. By symmetry of the order in which
the darts are thrown, if y denotes the point where the second dart
lands, the probability that the first dart lands in f (y) is 0. So,
almost surely, the two darts will land at two points witnessing that
AS holds for f .
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Problems with Freiling’s Arguments
Two main objections can be raised.
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Problems with Freiling’s Arguments
Two main objections can be raised.
One objection involves Lebesgue measurability:
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Problems with Freiling’s Arguments
Two main objections can be raised.
One objection involves Lebesgue measurability: Freiling’s argument
seems to appeal to an intuition that {(x, y) | x ∈ f (y)} and
{(x, y) | y ∈ f (x)} both have measure 0.
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Problems with Freiling’s Arguments
Two main objections can be raised.
One objection involves Lebesgue measurability: Freiling’s argument
seems to appeal to an intuition that {(x, y) | x ∈ f (y)} and
{(x, y) | y ∈ f (x)} both have measure 0. This intuition seems to
require that there is a very well-behaved measure on the reals,
which is precluded by the Axiom of Choice.
Slide 98
Slide 98 text
Problems with Freiling’s Arguments
Two main objections can be raised.
One objection involves Lebesgue measurability: Freiling’s argument
seems to appeal to an intuition that {(x, y) | x ∈ f (y)} and
{(x, y) | y ∈ f (x)} both have measure 0. This intuition seems to
require that there is a very well-behaved measure on the reals,
which is precluded by the Axiom of Choice. In fact, if f is a
counterexample to AS, then both {(x, y) | x ∈ f (y)} and
{(x, y) | y ∈ f (x)} are non-measurable.
Slide 99
Slide 99 text
Problems with Freiling’s Arguments
Two main objections can be raised.
One objection involves Lebesgue measurability: Freiling’s argument
seems to appeal to an intuition that {(x, y) | x ∈ f (y)} and
{(x, y) | y ∈ f (x)} both have measure 0. This intuition seems to
require that there is a very well-behaved measure on the reals,
which is precluded by the Axiom of Choice. In fact, if f is a
counterexample to AS, then both {(x, y) | x ∈ f (y)} and
{(x, y) | y ∈ f (x)} are non-measurable.
The other objection to Freiling’s argument also features the Axiom
of Choice in a prominent role.
Slide 100
Slide 100 text
Problems with Freiling’s Arguments
Two main objections can be raised.
One objection involves Lebesgue measurability: Freiling’s argument
seems to appeal to an intuition that {(x, y) | x ∈ f (y)} and
{(x, y) | y ∈ f (x)} both have measure 0. This intuition seems to
require that there is a very well-behaved measure on the reals,
which is precluded by the Axiom of Choice. In fact, if f is a
counterexample to AS, then both {(x, y) | x ∈ f (y)} and
{(x, y) | y ∈ f (x)} are non-measurable.
The other objection to Freiling’s argument also features the Axiom
of Choice in a prominent role. These two objections suggest that
AS might really be more about Choice than about CH.
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Slide 101 text
Solovay’s Model
Soon after Cohen introduced it, Solovay used the technique of
forcing to construct a model S of ZF + DC in which:
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Slide 102 text
Solovay’s Model
Soon after Cohen introduced it, Solovay used the technique of
forcing to construct a model S of ZF + DC in which:
1 All sets of real numbers are Lebesgue measurable.
Slide 103
Slide 103 text
Solovay’s Model
Soon after Cohen introduced it, Solovay used the technique of
forcing to construct a model S of ZF + DC in which:
1 All sets of real numbers are Lebesgue measurable.
2 All sets of real numbers have the perfect set property.
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Slide 104 text
Solovay’s Model
Soon after Cohen introduced it, Solovay used the technique of
forcing to construct a model S of ZF + DC in which:
1 All sets of real numbers are Lebesgue measurable.
2 All sets of real numbers have the perfect set property.
The existence of an injective function f : ω1 → R is enough to
construct a non-measurable set, so, in S, ℵ1 and 2ℵ0 are not even
comparable.
Slide 105
Slide 105 text
Solovay’s Model
Soon after Cohen introduced it, Solovay used the technique of
forcing to construct a model S of ZF + DC in which:
1 All sets of real numbers are Lebesgue measurable.
2 All sets of real numbers have the perfect set property.
The existence of an injective function f : ω1 → R is enough to
construct a non-measurable set, so, in S, ℵ1 and 2ℵ0 are not even
comparable.
However, 2 implies that, in S, every set of real numbers is either
countable or of cardinality 2ℵ0 , so a form of CH is true in S.
Slide 106
Slide 106 text
Solovay’s Model
Soon after Cohen introduced it, Solovay used the technique of
forcing to construct a model S of ZF + DC in which:
1 All sets of real numbers are Lebesgue measurable.
2 All sets of real numbers have the perfect set property.
The existence of an injective function f : ω1 → R is enough to
construct a non-measurable set, so, in S, ℵ1 and 2ℵ0 are not even
comparable.
However, 2 implies that, in S, every set of real numbers is either
countable or of cardinality 2ℵ0 , so a form of CH is true in S.
1 implies that AS is true in S.
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Closing Remarks