The Axiom of Choice:
Intuition and Paradox
Matthew Housley
housley@math.utah.edu
March 6th, 2009
Choice Functions
Given a finite collection of non-empty sets X = {Xα }α∈A , we can
choose an
[element from each set and define the choice function
c:A→
Xα such that for each α ∈ A, we have c(α) ∈ Xα .
α
The Axiom of Choice
Let X = {Xα }α∈A be a collection of non-empty [
sets, indexed by
the set A. Then there exists a function c : A →
Xα such that
α
for each α ∈ A, we have c(α) ∈ Xα .
Zermelo-Fraenkel-Choice (ZFC) Set Theory Axioms
1. (Extension Axiom) Two sets are identical if they have the same elements.
2. (Pair Axiom) If x, y are sets, then there is a set {x, y} whose elements are x
and y.
[
3. (Union Axiom) If x is a set (of sets), then there exists a set
x whose
elements are the elements of the elements of x.
4. (Power Set Axiom) If x is a set, then there exists a set P(x) whose elements are
the subsets of x.
5. (Selection Axiom) Let a be a set and let φ be a “statement” involving one free
variable x. Then there exists a set b whose elements are the elements x of a
such that φ(x) is true, i.e. b = {x ∈ a | φ(x) is true}
6. (Replacement Axiom) Let a be a set and ψ(x, y) be a statment involving two
free variables such that for any x, there exists at most one y that satisfies
ψ(x, y). Then, b := {y | ψ(x, y) holds for some x ∈ a} is a set.
7. (Empty Set Axiom) There exists a set ∅ which contains no elements.
8. (Axiom of Infinity) There exists a set a such that ∅ ∈ a and {x} ∈ a whenever
x ∈ a.
9. (Foundation Axiom) For any nonempty set x, there exists y ∈ x such that
x ∩ y = ∅.
10. The Axiom of Choice (AC).
Zermelo-Fraenkel-Choice (ZFC) Set Theory Axioms
1. (Extension Axiom) Two sets are identical if they have the same elements.
2. (Pair Axiom) If x, y are sets, then there is a set {x, y} whose elements are x
and y.
[
3. (Union Axiom) If x is a set (of sets), then there exists a set
x whose
elements are the elements of the elements of x.
4. (Power Set Axiom) If x is a set, then there exists a set P(x) whose elements are
the subsets of x.
5. (Selection Axiom) Let a be a set and let φ be a “statement” involving one free
variable x. Then there exists a set b whose elements are the elements x of a
such that φ(x) is true, i.e. b = {x ∈ a | φ(x) is true}
6. (Replacement Axiom) Let a be a set and ψ(x, y) be a statment involving two
free variables such that for any x, there exists at most one y that satisfies
ψ(x, y). Then, b := {y | ψ(x, y) holds for some x ∈ a} is a set.
7. (Empty Set Axiom) There exists a set ∅ which contains no elements.
8. (Axiom of Infinity) There exists a set a such that ∅ ∈ a and {x} ∈ a whenever
x ∈ a.
9. (Foundation Axiom) For any nonempty set x, there exists y ∈ x such that
x ∩ y = ∅.
10. The Axiom of Choice (AC).
(Cantor) The union of a countable family {Ai }∞
i=1 of countable
sets is countable.
Proof
Countability
6 ∅.
⇐⇒ Fi := {f : N → Ai | f bijective} =
of Ai
By AC, we may pick, for each Ai , an element fi ∈ Fi . We now see that
the map (i, j) 7−→ fi (j) is a bijection from N × N onto ∪Ai . And, N × N
is itself countable.
Cardinality
“Given any two sets A and B, there is
always an injective map either from A to
B or from B to A.”
Cardinality of products
“Given any infinite set X, the
product set X × X has the same
cardinality as X.”
Well Ordering Principle
Given any set X, there exists a comparison operation > satisfying
the following properties:
I
For any pair a 6= b ∈ X, either a > b or b > a.
I
Every subset of X has a least element.
Try this with the real numbers!
Well Ordering Principle
Given any set X, there exists a comparison operation > satisfying
the following properties:
I
For any pair a 6= b ∈ X, either a > b or b > a.
I
Every subset of X has a least element.
Try this with the real numbers!
Zorn’s Lemma
“A partially ordered set in which
every chain has an upper bound
contains a maximal element.”
Zorn’s lemma is used in the proof that every vector
space has a basis.
“The Axiom of Choice is obviously
true; the Well Ordering Principle
is obviously false; and who can tell
about Zorn’s Lemma?”
– Jerry Bona
AC Dependent Results
1. (Cantor, Borel, Russell) Every infinite set contains a countably
infinite subset.
2. (Cantor) A function f : R → R is continuous if and only if it
is sequentially continuous.
3. Tychonoff’s Theorem: The product of a collection of compact
topological spaces with the product topology is itself a
compact topological space. (equivalent to AC)
4. Every proper ideal of a ring with unity is contained in a
maximal ideal. (equivalent to AC)
To choose one sock from each of
infinitely many pairs of socks
requires the Axiom of Choice, but
for shoes the Axiom is not needed.
– Bertrand Russell
Shoes
Cantor’s diagonal argument:
.
.
.
.
.
.
.
2
1
1
2
1
0
1
1
0
1
1
2
0
2
0
1
1
2
2
0
2
1
0
2
2
2
2
2
0
1
1
1
2
1
1
2
0
0
1
0
2
1
x = .010000 · · ·
1. . .
1. . .
0. . .
0. . .
0. . .
1. . .
1. . .
Socks
Let X be the collection of all subsets of R. No one
has ever constructed a suitable choice function for
this collection.
Real Analysis or: How I Learned to Stop Worrying and
Love the Axiom of Choice
A brief overview: We want to generalize our notion of the length of
an interval to larger classes of subsets of R. Let A be a collection
of subsets of R with the following properties:
1. If A ∈ A, then R \ A ∈ A.
∞
[
2. If A1 , A2 , . . . ∈ A, then
Ai ∈ A.
i=1
A function µ : A → R
≥0
∪ ∞ is a measure if
1. µ(∅) = 0
∞
∞
[
X
2. µ
Ai =
µ(Ai ) if {A1 , A2 , . . .} is a disjoint sequence.
i=1
i=1
Nice Properties
I
We require that A contains all the intervals in R and that µ
gives the right length on these intervals.
I
We’ll assume that µ is translation invariant.
Theorem
There exists no translation invariant measure µ defined on 2R such
that 0 < µ([0, 1]) < ∞.
How do we prove this?
We need to construct a really bad set. . .
Proof
Define an equivalence relation ∼ on [0, 1] by:
x ∼ y ⇐⇒ x − y ∈ Q.
Each equivalence class of ∼ is non-empty.
By AC, we can construct a set E which intersects each equivalence
class at exactly one point.
Claim: µ(E) must be undefined for any set with the “nice”
properties we’ve stipulated.
Proof
Let r1 , r2 , . . . be an enumeration of Q ∩ [−1, 1], and for each
n ∈ N, let En := rn + E be the translation of the E by rn .
Then (a) En is pairwise disjoint, and
∞
[
En ⊆ [−1, 2].
(b) [0, 1] ⊆
n=1
Suppose that µ(E) is defined. Then µ(En ) is defined for each n
by translation-invariance of µ. Hence,
!
∞
∞
∞
[
X
X
3 = µ([−1, 2]) ≥ µ
En =
µ(En ) =
µ(E)
n=1
n=1
=⇒ µ(E) = 0.
n=1
Proof
But,
µ
∞
[
n=1
Contradiction!
!
En
=
∞
X
n=1
µ(E) ≥ µ([0, 1]) = 1 > 0.
I
We’ve just constructed a set that immeasurable under any
“nice” measure.
I
We’ve disjointly covered [0, 1] by countably many copies of
this nonmeasurable set. This hints at something very bad. . .
I
We’ve just constructed a set that immeasurable under any
“nice” measure.
I
We’ve disjointly covered [0, 1] by countably many copies of
this nonmeasurable set. This hints at something very bad. . .
The Banach-Tarski Paradox
It is possible to cut a solid sphere into
finitely many pieces and rearrange these
into two copies of the original sphere
using only translations and rotations.
Paradox?
This statement of the Banach-Tarski paradox is misleading:
I
The Lebesgue measure is the most general possible nice
measure. The “pieces” in the paradox are not Lebesgue
measureable. They have undefined volume.
I
We are moving around very nasty point sets, not geometric
objects.
I
Since we cannot talk about volume, the only relevent
comparison is cardinality. A single solid sphere and two solid
spheres have the same cardinality.
Paradox?
This statement of the Banach-Tarski paradox is misleading:
I
The Lebesgue measure is the most general possible nice
measure. The “pieces” in the paradox are not Lebesgue
measureable. They have undefined volume.
I
We are moving around very nasty point sets, not geometric
objects.
I
Since we cannot talk about volume, the only relevent
comparison is cardinality. A single solid sphere and two solid
spheres have the same cardinality.
Paradox?
This statement of the Banach-Tarski paradox is misleading:
I
The Lebesgue measure is the most general possible nice
measure. The “pieces” in the paradox are not Lebesgue
measureable. They have undefined volume.
I
We are moving around very nasty point sets, not geometric
objects.
I
Since we cannot talk about volume, the only relevent
comparison is cardinality. A single solid sphere and two solid
spheres have the same cardinality.
Alternate Set Theories
I
In 1970, Robert M. Solovay developed a set theory based on
the Zermelo-Fraenkel Axioms such that every set in 2R
becomes measurable.
I
This can be done by replacing the Axiom of Choice with the
Axiom of Determinancy which states that in every two-player
game where draws are not possible, one player has a winning
strategy.
Alternate Set Theories
I
In 1970, Robert M. Solovay developed a set theory based on
the Zermelo-Fraenkel Axioms such that every set in 2R
becomes measurable.
I
This can be done by replacing the Axiom of Choice with the
Axiom of Determinancy which states that in every two-player
game where draws are not possible, one player has a winning
strategy.
Alternate Set Theories
I
Countable Choice: choice functions exist for countable
collections of sets. This can be used to prove that any
countable collection of countable sets is countable. It allows
the proof of a number of results in analysis where countable
collections of sets appear.
I
Dependent Choice: The statement of this axiom in more
technical. It is weaker than AC but stronger than CC. It
allows construction of non-measurable sets and is equivalent
to the Baire Category Theorem for complete metric spaces.
Alternate Set Theories
I
Countable Choice: choice functions exist for countable
collections of sets. This can be used to prove that any
countable collection of countable sets is countable. It allows
the proof of a number of results in analysis where countable
collections of sets appear.
I
Dependent Choice: The statement of this axiom in more
technical. It is weaker than AC but stronger than CC. It
allows construction of non-measurable sets and is equivalent
to the Baire Category Theorem for complete metric spaces.
Dieudonné [1976]:
... there is an infinity of
different possible mathematics,
and for the time being no
definitive reason compels us to
choose one of them rather than
another.
Indeed, it has been said that
democracy is the worst form
of government except all
those other forms that have
been tried from time to time.
– Winston Churchill [1947]
Why ZFC?
I
AC is intuitive, simple, convenient and powerful.
I
ZFC is the most widely accepted set theory among
mathematicians.
I
Alternative set theories lead to paradoxes of their own, e.g.,
the measurability of all subsets of R.
Why ZFC?
I
AC is intuitive, simple, convenient and powerful.
I
ZFC is the most widely accepted set theory among
mathematicians.
I
Alternative set theories lead to paradoxes of their own, e.g.,
the measurability of all subsets of R.
Why ZFC?
I
AC is intuitive, simple, convenient and powerful.
I
ZFC is the most widely accepted set theory among
mathematicians.
I
Alternative set theories lead to paradoxes of their own, e.g.,
the measurability of all subsets of R.
Opinion
ZFC will remain the standard in the future, but it is possible that
alternate set theories will gain prominence in certain areas of
research.
Resources
Eric Schechter has built an excellent webpage on the Axiom of
Choice, currently located at:
http://www.math.vanderbilt.edu/~schectex/ccc/choice.html
A detailed but accessible paper on the Banach-Tarski Paradox by
Francis Edward Su is available at this address:
http://www.math.hmc.edu/~su/papers.dir/banachtarski.pdf
Special thanks to Kenneth Chu, who
provided an earlier version of this talk.