The Axiom of Choice:Intuition and Paradox

Matthew Housleyhousley@math.utah.edu

March 6th, 2009

Choice Functions

Given a finite collection of non-empty sets X = {Xα}α∈A, we canchoose an element from each set and define the choice functionc : A→

⋃α

Xα such that for each α ∈ A, we have c(α) ∈ Xα.

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→⋃α

Xα such that

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 xand 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 arethe subsets of x.

5. (Selection Axiom) Let a be a set and let φ be a “statement” involving one freevariable x. Then there exists a set b whose elements are the elements x of asuch 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 twofree variables such that for any x, there exists at most one y that satisfiesψ(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 wheneverx ∈ a.

9. (Foundation Axiom) For any nonempty set x, there exists y ∈ x such thatx ∩ 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 xand 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 arethe subsets of x.

5. (Selection Axiom) Let a be a set and let φ be a “statement” involving one freevariable x. Then there exists a set b whose elements are the elements x of asuch 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 twofree variables such that for any x, there exists at most one y that satisfiesψ(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 wheneverx ∈ a.

9. (Foundation Axiom) For any nonempty set x, there exists y ∈ x such thatx ∩ y = ∅.

10. The Axiom of Choice (AC).

(Cantor) The union of a countable family {Ai}∞i=1 of countablesets is countable.

Proof

Countabilityof Ai

⇐⇒ Fi := {f : N→ Ai | f bijective} 6= ∅.

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×Nis 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, theproduct set X ×X has the same

cardinality as X.”

Well Ordering Principle

Given any set X, there exists a comparison operation > satisfyingthe 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 > satisfyingthe 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 whichevery chain has an upper boundcontains a maximal element.”

Zorn’s lemma is used in the proof that every vectorspace 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 countablyinfinite subset.

2. (Cantor) A function f : R→ R is continuous if and only if itis sequentially continuous.

3. Tychonoff’s Theorem: The product of a collection of compacttopological spaces with the product topology is itself acompact topological space. (equivalent to AC)

4. Every proper ideal of a ring with unity is contained in amaximal 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 0 1 0 2 1. . .

. 1 0 1 0 1 0 1. . .

. 1 1 1 2 1 0 0. . .

. 2 1 2 2 1 1 0. . .

. 1 2 2 2 2 0 0. . .

. 0 0 0 2 1 2 1. . .

. 1 2 2 2 1 1 1. . .

x = .010000 · · ·

Socks

Let X be the collection of all subsets of R. No onehas ever constructed a suitable choice function forthis collection.

Real Analysis or: How I Learned to Stop Worrying andLove the Axiom of Choice

A brief overview: We want to generalize our notion of the length ofan interval to larger classes of subsets of R. Let A be a collectionof subsets of R with the following properties:

1. If A ∈ A, then R \A ∈ A.

2. If A1, A2, . . . ∈ A, then∞⋃i=1

Ai ∈ A.

A function µ : A → R≥0 ∪∞ is a measure if

1. µ(∅) = 0

2. µ( ∞⋃i=1

Ai

)=∞∑i=1

µ(Ai) if {A1, A2, . . .} is a disjoint sequence.

Nice Properties

I We require that A contains all the intervals in R and that µgives the right length on these intervals.

I We’ll assume that µ is translation invariant.

TheoremThere exists no translation invariant measure µ defined on 2R suchthat 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 equivalenceclass 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 eachn ∈ N, let En := rn + E be the translation of the E by rn.

Then (a) En is pairwise disjoint, and

(b) [0, 1] ⊆∞⋃n=1

En ⊆ [−1, 2].

Suppose that µ(E) is defined. Then µ(En) is defined for each nby translation-invariance of µ. Hence,

3 = µ([−1, 2]) ≥ µ

( ∞⋃n=1

En

)=∞∑n=1

µ(En) =∞∑n=1

µ(E)

=⇒ µ(E) = 0.

Proof

But,

µ

( ∞⋃n=1

En

)=∞∑n=1

µ(E) ≥ µ([0, 1]) = 1 > 0.

Contradiction!

I We’ve just constructed a set that immeasurable under any“nice” measure.

I We’ve disjointly covered [0, 1] by countably many copies ofthis 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 ofthis 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 nicemeasure. The “pieces” in the paradox are not Lebesguemeasureable. They have undefined volume.

I We are moving around very nasty point sets, not geometricobjects.

I Since we cannot talk about volume, the only releventcomparison is cardinality. A single solid sphere and two solidspheres have the same cardinality.

Paradox?

This statement of the Banach-Tarski paradox is misleading:

I The Lebesgue measure is the most general possible nicemeasure. The “pieces” in the paradox are not Lebesguemeasureable. They have undefined volume.

I We are moving around very nasty point sets, not geometricobjects.

I Since we cannot talk about volume, the only releventcomparison is cardinality. A single solid sphere and two solidspheres have the same cardinality.

Paradox?

This statement of the Banach-Tarski paradox is misleading:

I The Lebesgue measure is the most general possible nicemeasure. The “pieces” in the paradox are not Lebesguemeasureable. They have undefined volume.

I We are moving around very nasty point sets, not geometricobjects.

I Since we cannot talk about volume, the only releventcomparison is cardinality. A single solid sphere and two solidspheres have the same cardinality.

Alternate Set Theories

I In 1970, Robert M. Solovay developed a set theory based onthe Zermelo-Fraenkel Axioms such that every set in 2R

becomes measurable.

I This can be done by replacing the Axiom of Choice with theAxiom of Determinancy which states that in every two-playergame where draws are not possible, one player has a winningstrategy.

Alternate Set Theories

I In 1970, Robert M. Solovay developed a set theory based onthe Zermelo-Fraenkel Axioms such that every set in 2R

becomes measurable.

I This can be done by replacing the Axiom of Choice with theAxiom of Determinancy which states that in every two-playergame where draws are not possible, one player has a winningstrategy.

Alternate Set Theories

I Countable Choice: choice functions exist for countablecollections of sets. This can be used to prove that anycountable collection of countable sets is countable. It allowsthe proof of a number of results in analysis where countablecollections of sets appear.

I Dependent Choice: The statement of this axiom in moretechnical. It is weaker than AC but stronger than CC. Itallows construction of non-measurable sets and is equivalentto the Baire Category Theorem for complete metric spaces.

Alternate Set Theories

I Countable Choice: choice functions exist for countablecollections of sets. This can be used to prove that anycountable collection of countable sets is countable. It allowsthe proof of a number of results in analysis where countablecollections of sets appear.

I Dependent Choice: The statement of this axiom in moretechnical. It is weaker than AC but stronger than CC. Itallows construction of non-measurable sets and is equivalentto the Baire Category Theorem for complete metric spaces.

Dieudonne [1976]:... there is an infinity ofdifferent possible mathematics,and for the time being nodefinitive reason compels us tochoose one of them rather thananother.

Indeed, it has been said thatdemocracy is the worst formof government except allthose other forms that havebeen 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 amongmathematicians.

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 amongmathematicians.

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 amongmathematicians.

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 thatalternate set theories will gain prominence in certain areas ofresearch.

Resources

Eric Schechter has built an excellent webpage on the Axiom ofChoice, currently located at:

http://www.math.vanderbilt.edu/~schectex/ccc/choice.html

A detailed but accessible paper on the Banach-Tarski Paradox byFrancis 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.