variations on axiom of choice

variations on axiom of choice

The

axiom of choice

states that every set

C

of non-empty sets has a

choice function

. There are a number of ways to modify the statement so as to produce something that is

similar

but perhaps weaker version of the axiom. For example, we can play with the size (

cardinality

) of the set

C

, the sizes of the

elements

in

C

, as well as the choice function itself. Below are some of the variations of AC:

1.

Varying the size of

C

: let

λ

be a

cardinal

. Then AC

(

λ

,

∞

)

is the statement that every set

C

of non-empty sets, where

|

C

|

=

λ

, has a choice function. If

λ

=

ℵ

0

, then we have the

axiom of countable choice

.

2.

Varying sizes of members of

C

: let

λ

be a cardinal. Then AC

(

∞

,

κ

)

is the statement that every set

C

of non-empty sets of cardinality

κ

has a choice function. Another variation is called the

axiom of choice for finite sets

AC

(

∞

,

<

ℵ

0

)

: every set

C

of non-empty

finite sets

has a choice function.

3.

Varying choice function: The most popular is what is known as the

axiom of multiple choice

(AMC), which states that every set

C

of non-empty sets, there is a

multivalued function

from

C

to

⋃

C

such that

f

⁢

(

A

)

is finite non-empty and

f

⁢

(

A

)

⊆

A

.

4.

Varying any

combination

of the above three, for example AC

(

λ

,

κ

)

is the statement that every

collection

C

of size

λ

of non-empty sets of size

κ

has a choice function.

It’s easy to see that all of the variations are provable in ZFC. In

addition

, some of them are provable in ZF, for example, AC

(

n

,

∞

)

for any finite cardinal, and AC

(

∞

,

1

)

.

Conversely

, it can be shown that AMC implies AC in ZF.

Other

implications

include: AC

(

∞

,

κ

)

for all

κ

implies the

axiom of dependent choices

, another weaker version of AC. For finite cardinals, we have the following: AC

(

∞

,

m

⁢

n

)

implies AC

(

∞

,

n

)

, where

m

,

n

are finite cardinals.

Proof.

Let

C

be a set of non-empty sets of cardinality

n

. For each

a

∈

C

, define

a

*

:=

a

×

m

:=

{

(

x

,

i

)

∣

x

∈

a

,

and

⁢

i

∈

m

}

. Then the set

D

=

{

a

*

∣

a

∈

C

}

is a set of non-empty sets of cardinality

m

⁢

n

, hence has a choice function

f

by

assumption

. Then

p

∘

f

∘

g

is a choice function for

C

, where

p

:

⋃

D

→

⋃

C

is the projection given by

p

⁢

(

x

,

i

)

=

x

, and

g

:

C

→

D

is the

function

given by

g

⁢

(

a

)

=

a

*

.
∎

For more implications, see the references below.

References

1

H. Herrlich,

Axiom of Choice

, Springer, (2006).

2

T. J. Jech,

The Axiom of Choice

, North-Holland Pub. Co., Amsterdam, (1973).

Title

variations on axiom of choice

Canonical

name

VariationsOnAxiomOfChoice

Date of creation

2013-03-22 18:47:09

Last modified on

2013-03-22 18:47:09

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03E30

Classification

msc 03E25

Synonym

AMC

Synonym

MC

Defines

axiom of choice for finite sets

Defines

axiom of multiple choice

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