ordering on cardinalities

ordering on cardinalities

When there is a one-to-one function from a set

A

to a set

B

, we say that

A

is

embeddable

in

B

, and write

A

≤

B

. Thus

≤

is a (class)

binary relation

on the class

V

of all sets. This

relation

is clearly

reflexive

and

transitive

. If

A

≤

B

and

B

≤

A

, then, by Schröder-Bernstein

theorem

,

A

is

bijective

to

B

,

A

∼

B

. However, clearly

A

≠

B

in general. Therefore

≤

fails to be a

partial order

. However, since

A

∼

B

iff they have the same

cardinality

,

|

A

|

=

|

B

|

, and since

cardinals

are by

definition

sets, the class of all cardinals becomes a

partially ordered set

with partial order

≤

. We record this result as a theorem:

Theorem 1

.

In ZF, the relation

≤

is a partial order on the cardinals.

With the

addition

of the

axiom of choice

, one can show that

≤

is a

linear order

on the cardinals. In fact, the statement “

≤

is a linear order on the cardinals” is

equivalent

to the axiom of choice.

Theorem 2

.

In ZF, the following are equivalent:

1.

the axiom of choice

2.

≤

is a linear order on the cardinals

Proof.

Restating the second statement, we have that for any two sets

A

,

B

, there is an

injection

from one to the other. The plan is to use Zorn’s lemma to prove the second statement, and use the second statement to prove the

well-ordering principle

(WOP).

Zorn implies Statement 2:

Suppose there are no injections from

A

to

B

. We need to find an injection from

B

to

A

. We may assume that

B

≠

∅

, for otherwise

∅

is an injection from

B

to

A

. Let

P

be the

collection

of all partial injective functions from

B

to

A

.

P

, as a collection of relations between

B

and

A

, is a set.

P

≠

∅

, since any

function

from a

singleton

subset of

B

into

A

is in

P

. Order

P

by

set inclusion

, so

P

becomes a poset. Suppose

F

is a chain of

partial functions

in

P

, let us look at

f

:=

⋃

F

. Suppose

(

a

,

b

)

,

(

a

,

c

)

∈

f

. Then

(

a

,

b

)

∈

p

and

(

a

,

c

)

∈

q

for some

p

,

q

∈

F

. Since

F

is a chain, one is a subset of the other, so say,

p

⊆

q

. Then

(

a

,

b

)

∈

q

, and since

q

is a partial function,

b

=

c

. This shows that

f

is a partial function. Next, suppose

(

a

,

c

)

,

(

b

,

c

)

∈

f

. By the same

argument

used to show that

f

is a function, we see that

a

=

b

, so that

f

is injective. Therefore

f

∈

P

. Thus, by Zorn’s lemma,

P

has a

maximal element

g

. We want to show that

g

is defined on all of

B

. Now,

g

can not be

surjective

, or else

g

is a bijection from

dom

⁡

(

g

)

onto

A

. Then

g

-

1

:

A

→

B

is an injection, contrary to the

assumption

. Therefore, we may pick an

element

a

∈

A

-

ran

⁡

(

g

)

. Now, if

dom

⁡

(

g

)

≠

B

, we may pick an element

b

∈

B

-

dom

⁡

(

g

)

. Then the partial function

g

*

:

dom

⁡

(

g

)

∪

{

b

}

→

A

given by

g

*

⁢

(

x

)

=

{

g

⁢

(

x

)

if

⁢

x

∈

dom

⁡

(

g

)

,

a

if

⁢

x

=

b

.

Since

g

*

is injective by construction,

g

*

∈

P

. Since

g

*

properly extends

g

, we have reached a

contradiction

, as

g

is maximal in

P

. Therefore the domain of

g

is all of

B

, and is our desired injective function from

B

to

A

.

Statement 2 implies WOP:

Let

A

be a set and let

h

⁢

(

A

)

be its

Hartogs number

. Since

h

⁢

(

A

)

is not embeddable in

A

, by statement 2,

A

is embeddable in

h

⁢

(

A

)

. Let

f

be the injection from

A

to

h

⁢

(

A

)

is injective. Since

h

⁢

(

A

)

is an

ordinal

, it is

well-ordered

. Therefore, as

f

⁢

(

A

)

is well-ordered, and because

A

∼

f

⁢

(

A

)

,

A

itself is

well-orderable

via the well-ordering on

f

⁢

(

A

)

.

Since Zorn’s lemma and the well-ordering principles are both equivalent to AC in ZF, the theorem is proved.
∎

Title

ordering on cardinalities

Canonical

name

OrderingOnCardinalities

Date of creation

2013-03-22 18:50:23

Last modified on

2013-03-22 18:50:23

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

4

Author

CWoo (3771)

Entry type

Theorem

Classification

msc 03E10

Classification

msc 03E25

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