axiom of choice

axiom of choice

The

Zermelo-Fraenkel axioms

for

set theory

are often accepted as a basis for an

axiomatic set theory

.
On the other hand, the

axiom of choice

is somewhat controversial, and it is currently segregated from the ZF system of set theory axioms. When the axiom of choice is combined with the ZF axioms, the whole

axiom system

is
called “ZFC” (for “Zermelo-Fraenkel with Choice”).

Axiom

(Axiom of choice).

Let

C

be a set of

nonempty sets

. Then there exists a

function

f

:

C

→

⋃

S

∈

C

S

such that

f

⁢

(

x

)

∈

x

for all

x

∈

C

.

The function

f

is sometimes called a

choice function

on

C

.

For

finite sets

C

, a choice function can be constructed without appealing to the axiom of choice. In particular, if

C

=

∅

, then the choice function is clear: it is the

empty set

! It is only for

infinite

(and usually

uncountable

) sets

C

that the

existence

of a choice function becomes an issue. Here one can see why it is not considered “obvious” and always taken for an axiom by everyone:
one really cannot imagine any process which makes uncountably many selections without also imagining some rule for making the selections. Given such a rule, the axiom of choice is not needed. Thus, objects that are proved to exist using the axiom of choice cannot generally be described by any kind of systematic rule, for if they could it would not be necessary to their construction.

Let us consider a couple of examples. Imagine that there are infinitely many pairs of shoes (each consisting of one left shoe and one right shoe). Let

𝒫

denote the set of all pairs of shoes. In this scenario, it can be verified that the function

f

:

𝒫

→

⋃

𝒫

defined by

f

⁢

(

P

)

=

the left shoe of

⁢

P

is a choice function. Similarly, imagine that there are infinitely many pairs of socks. Let

𝒮

denote the set of all pairs of socks. In this scenario, one cannot assume that a function

g

:

𝒮

→

⋃

𝒮

exists without appealing to the axiom of choice (or something

equivalent

to it). Note that this scenario cannot be resolved in the same manner as the previous scenario because most people do not a ”left sock” and a ”right sock”.

Strange objects that can be constructed using the axiom of choice include non-measurable sets (leading to the

http://planetmath.org/node/7057

Hausdorff

and

http://planetmath.org/node/4464

Banach-Tarski paradoxa), and Hamel bases for any

vector space

. A

Hamel
basis

may not seem strange, but try to imagine a set

S

of

continuous functions

such that every continuous function can be expressed uniquely as a

linear combination

of

finitely many

elements of

S

. Since in fact the existence of a basis for every vector space is equivalent to the axiom of
choice, it is almost guaranteed that no such set

S

can ever be described. It is for this reason that some mathematicians dislike the axiom of choice.

On the other hand, many very useful facts can be proven using the axiom of choice. For example, the fact that every vector space has a basis, every

ring with identity

element

1

≠

0

has a

maximal ideal

and many other algebraic

theorems

which are difficult or impossible to prove without using the axiom of choice.

In pure set theory, the axiom of choice is only relevant where most people’s intuition more or less breaks down, when dealing with hierarchies of uncountable

infinities

.

The relevance of the axiom of choice to various of mathematics has led to a detailed study of its truth. It turns out that if the Zermelo-Fraenkel axioms are

consistent

, then they remain consistent upon adding the axiom of choice. But they also remain consistent upon adding the

negation

of the axiom of choice (see

[

G

]

and

[

C

]

).

Some mathematicians have suggested an axiom that would result in all subsets of the

real numbers

being

measurable

; this would of course imply the negation of the axiom of choice.

There are many alternative formulations of the axiom of choice, although it is not always

trivial

to prove . These include:

•

The

generalized Cartesian product

of a non-empty family of sets is non-empty iff every set is non-empty. Due to this equivalence the axiom of choice is sometimes called the

multiplicative axiom

(

http://planetmath.org/node/11514

Proof).

•

Every

relation

R

is a union of functions with the same domain as

R

(

http://planetmath.org/node/8763

Proof).

•

Every ring with identity element

1

≠

0

has a maximal ideal (

http://planetmath.org/node/4713

Proof).

•

Every

surjection

is a

retraction

(

http://planetmath.org/TypesOfMorphisms

) (

http://planetmath.org/node/11517

Proof).

•

Every tree has a branch.

•

Every vector space has a basis (for AC implying existence of vector bases, see

http://planetmath.org/node/3494

here, and its

converse

, proved by Andreas Blass in 1984,

http://www.math.lsa.umich.edu/ ablass/bases-AC.pdf

here).

•

Given any two sets, there exists an

injection

from one to another (

http://planetmath.org/node/11645

Proof).

•

Hausdorff’s

maximum principle

(

http://planetmath.org/node/3493

Proof).

•

http://planetmath.org/node/5598

König’s theorem.

•

Kuratowski’s lemma (

http://planetmath.org/node/11512

Proof).

•

Tukey’s lemma (

http://planetmath.org/node/4672

Proof).

•

Tychonoff

’s theorem (for AC implying Tychonoff, see

http://planetmath.org/node/9797

here, and its converse

http://planetmath.org/node/11540

here).

•

Zermelo’s

postulate

(

http://planetmath.org/node/8342

Proof).

•

Zermelo’s well-ordering theorem (

http://planetmath.org/node/3493

Proof).

•

Zorn’s lemma (

http://planetmath.org/node/3358

Proof).

Figure

1

shows how these equivalences are proven on PlanetMath.

Figure 1:

Structure

of the equivalence proofs on PlanetMath. The
abbreviations are explained in table

1

. An arrow

A

→

B

means that “

A

implies

B

” is proven on
PlanetMath.

AC

Axiom of Choice

Hamel

Every vector space has a basis

Hausdorff

Hausdorff’s maximum principle

König

König’s theorem

Krull

Every ring with identity element

1

≠

0

has a maximal ideal

Tukey

Tukey’s lemma

Well

Zermelo’s well-ordering theorem (

http://planetmath.org/ZermelosWellOrderingTheorem

)

Zorn

Zorn’s lemma

Table 1:

Abbreviations

References

C

P. J. Cohen

, The independence of the

continuum
hypothesis

. I, II,

Proc. Natl. Acad. Sci.

USA 50,
1143–1148 (1963); 51, 105–110 (1964).

G

K. Gödel

, The

consistency

of the axiom of
choice and of the generalized continuum-hypothesis,

Proc. Natl. Acad. Sci.

USA 24, 556–557 (1938).

Title

axiom of choice

Canonical name

AxiomOfChoice

Date of creation

2013-03-22 11:47:42

Last modified on

2013-03-22 11:47:42

Owner

GrafZahl (9234)

Last modified by

GrafZahl (9234)

Numerical id

43

Author

GrafZahl (9234)

Entry type

Axiom

Classification

msc 03E30

Classification

msc 03E25

Synonym

multiplicative axiom

Related topic

MaximalityPrinciple

Related topic

ZermeloFraenkelAxioms

Related topic

ContinuumHypothesis

Related topic

GeneralizedContinuumHypothesis

Related topic

HaudorffsMaximumPrinciple

Related topic

ZornsLemmaAndTheWellOrderingTheoremEquivalenceOfHaudorffsMaximumPrinciple

Related topic

EveryVectorSpaceHasABasis

Related topic

TukeysLemma

Related topic

ZermelosPostulat

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