axiom of dependent choices

axiom of dependent choices

The

axiom of dependent choices

(DC), or the

principle of dependent choices

, is the following statement:

given a set

A

and a

binary relation

R

≠

∅

on

A

such that

ran

⁡

(

R

)

⊆

dom

⁡

(

R

)

, then there is a

sequence

(

a

n

)

n

∈

ℕ

in

A

such that

a

n

⁢

R

⁢

a

n

+

1

.

Here,

ℕ

is the set of all

natural numbers

.

The

relation

between DC, AC (

axiom of choice

), and CC (

axiom of countable choice

) are the following:

Proposition 1

.

ZF+AC implies ZF+DC.

We prove this by using one of the

equivalents

of AC: Zorn’s lemma. For this proof, we define

seg

⁡

(

n

)

:=

{

m

∈

ℕ

∣

m

≤

n

}

, the

initial segment

of

ℕ

with the

greatest element

n

. Before starting the proof, we need a fact about initial segments:

Lemma 1

.

The union of initial segments of

N

is either

N

or an initial segment.

Proof.

Let

S

be a set of initial segments of

ℕ

. If

s

:=

⋃

S

≠

ℕ

, then

B

:=

ℕ

-

S

≠

∅

, so

B

has a least

element

r

. As a result, none of

i

∈

seg

⁡

(

r

-

1

)

is in

B

, and

seg

⁡

(

r

-

1

)

⊆

s

. If some

m

≥

r

is in

s

, then there is an initial segment

seg

⁡

(

n

)

∈

S

with

m

∈

seg

⁡

(

n

)

, so that

r

∈

seg

⁡

(

m

)

⊆

seg

⁡

(

n

)

⊆

s

, contradicting

r

∈

B

. ∎

Remark

. The fact that

B

in the proof above has a least element is a direct result of ZF, so the

well-ordering principle

(and hence AC) is not needed.

We are now ready for the proof of

proposition

1.

Proof.

Let

R

be a non-empty binary relation on a set

A

(of course non-empty). We want to find a

function

f

:

ℕ

→

dom

⁡

(

R

)

such that

f

⁢

(

n

)

⁢

R

⁢

f

⁢

(

n

+

1

)

.

Let

P

be the set of all

partial functions

f

:

ℕ

⇒

dom

⁡

(

R

)

such that

dom

⁡

(

f

)

is either an initial segment of

ℕ

, or

ℕ

itself, such that

f

⁢

(

n

)

⁢

R

⁢

f

⁢

(

n

+

1

)

, whenever

n

,

n

+

1

∈

dom

⁡

(

f

)

. Since

R

≠

∅

, some

(

a

,

b

)

∈

R

. Additionally,

b

∈

ran

⁡

(

R

)

⊆

dom

⁡

(

R

)

. Define function

g

:

{

1

,

2

}

→

dom

⁡

(

R

)

by

g

⁢

(

1

)

=

a

and

g

⁢

(

2

)

=

b

. Then

g

⁢

(

1

)

⁢

R

⁢

g

⁢

(

2

)

, so that

g

∈

P

, or

P

is non-empty.

Partial order

P

by

inclusion

so it is a poset. Let

C

be a chain in

P

, then

h

:=

⋃

C

is a partial function from

ℕ

to

A

. Since

dom

⁡

(

h

)

is the union of initial segments or

ℕ

,

dom

⁡

(

h

)

itself is either an initial segment or

ℕ

by Lemma 1.

Now, suppose

m

,

m

+

1

∈

dom

⁡

(

h

)

, then

m

+

1

∈

dom

⁡

(

s

)

for some

s

∈

C

, so

m

∈

dom

⁡

(

s

)

as well. Therefore

s

⁢

(

m

)

⁢

R

⁢

s

⁢

(

m

+

1

)

. Since

h

⁢

(

i

)

=

s

⁢

(

i

)

for any

i

∈

dom

⁡

(

s

)

, we see that

h

⁢

(

m

)

⁢

R

⁢

h

⁢

(

m

+

1

)

. This shows that

h

∈

P

, or that

C

has an

upper bound

in

P

.

By Zorn’s lemma,

P

has a

maximal element

f

. We claim that

f

is a

total function

. If not, then

dom

⁡

(

f

)

=

{

1

,

…

,

n

}

for some

n

. Since

f

⁢

(

n

)

∈

dom

⁡

(

R

)

, there is some

d

∈

ran

⁡

(

R

)

such that

f

⁢

(

n

)

⁢

R

⁢

d

. Define a partial function

g

:

ℕ

⇒

dom

⁡

(

R

)

such that

dom

⁡

(

g

)

=

{

1

,

…

,

n

+

1

}

, and

g

⁢

(

i

)

=

f

⁢

(

i

)

for all

i

=

1

,

…

,

n

, and

g

⁢

(

n

+

1

)

=

b

. So

g

≠

f

extends

f

, contradicting the maximality of

f

. Hence,

f

is a total function, and we are done. ∎

Proposition 2

.

ZF+DC implies ZF+CC.

Proof.

Let

C

be a

countable set

of non-empty sets. We assume that

C

is

countably infinite

, for the finite case can be proved using ZF alone, and is left for the reader.

Since there is a

bijection

ϕ

:

C

→

ℕ

, index each element in

C

by its image in

ℕ

, so that

C

=

{

A

i

∣

i

∈

ℕ

}

. Let

A

:=

⋃

C

. We want to find a function

f

:

C

→

A

such that

f

⁢

(

A

i

)

∈

A

i

for every

i

∈

ℕ

.

Define a binary relation

R

on

A

as follows:

a

⁢

R

⁢

b

iff there is an

i

∈

ℕ

such that

a

∈

A

i

and

b

∈

A

i

+

1

. Since each

A

i

≠

∅

,

R

≠

∅

. Furthermore, if

b

∈

ran

⁡

(

R

)

, then

b

∈

A

i

+

1

for some

i

∈

ℕ

. Pick any

c

∈

A

i

+

2

(since

A

i

+

2

≠

∅

), so that

b

⁢

R

⁢

c

, and therefore

b

∈

dom

⁡

(

R

)

. This shows that

ran

⁡

(

R

)

⊆

dom

⁡

(

R

)

.

By DC, there is a function

g

:

ℕ

→

dom

⁡

(

R

)

such that

g

⁢

(

i

)

⁢

R

⁢

g

⁢

(

i

+

1

)

for every

i

∈

ℕ

. Now,

g

⁢

(

1

)

∈

A

j

for some

j

∈

ℕ

. Define a function

h

:

ℕ

→

A

as follows, for each

i

∈

seg

⁡

(

j

-

1

)

, pick

a

i

∈

A

i

and set

h

⁢

(

i

)

:=

a

i

(this can be done by

induction

), and for

i

≥

j

, set

h

⁢

(

i

)

:=

g

⁢

(

j

-

i

+

1

)

(

arithmetic

of finite

cardinals

is possible in ZF). Then

h

⁢

(

i

)

∈

A

i

for all

i

∈

ℕ

.

Finally, define

f

:

C

→

A

as follows: for each

A

i

∈

C

, set

f

⁢

(

A

i

)

:=

h

⁢

(

i

)

. Then

f

has the desired property

f

⁢

(

A

i

)

∈

A

i

, and the proof is

complete

. ∎

However, the

converses

of both of these

implications

are false. Jensen proved the independence of DC from ZF+CC, and Mostowski and Jech proved the independence of AC from ZF+DC. In fact, it was shown that the weaker version of AC, which states that every set with

cardinality

at most

ℵ

1

has a

choice function

, is

independent

from ZF+DC.

Remark

. DC is related to

Baire spaces

in point-set topology. It can be shown that DC is equivalent to each of the following statements in ZF:

•

Any complete

pseudometric space

is Baire under the

topology

induced by the pseudometric.

•

Any

product

of

compact

Hausdorff spaces

is Baire under the product topology.

References

1

T. Jech,

Interdependence of weakened forms of the axiom of choice

, Comment. Math. Univ. Carolinae 7, pp. 359-371, (1966).

2

R. B. Jensen

Independence of the axiom of dependent choices from the countable axiom of choice

(abstract), Jour.

Symbolic Logic

31, 294, (1966).

3

A. Levy,

Basic Set Theory

, Dover Publications Inc., (2002).

4

A. Mostowski

On the principle of dependent choices

, Fund. Math. 35, pp 127-130 (1948).

Title

axiom of dependent choices

Canonical name

AxiomOfDependentChoices

Date of creation

2013-03-22 18:46:39

Last modified on

2013-03-22 18:46:39

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03E25

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