Hartogs number

Hartogs number

A set

A

is said to be

embeddable

in another set

B

if there is a one-to-one function

f

:

A

→

B

. For example, every subset of a set is embeddable in the set. In particular,

∅

is embeddable in every set. Clearly, given any set, there is an

ordinal

embeddable in it. On the other hand, is any set embeddable in an ordinal? If so, then the set is

well-orderable

(as it is

equipollent

to one), which is

equivalent

to the

well-ordering principle

. In other words, in ZF, AC is equivalent to saying that every set is embeddable in an ordinal. Without AC, how much can one deduce? In 1915, Hartogs proved the following:

Theorem 1

.

Given any set

A

, there is an ordinal

α

not embeddable in

A

.

Proof.

Let

α

be the class of all ordinals embeddable in

A

. We want to show that

α

is in fact an ordinal, not embeddable in

A

. We have the following steps:

•

α

is a set.

Let

ℬ

be the subset of

P

⁢

(

A

)

, the

powerset

of

A

, consisting of all well-orderable subsets of

A

. For each element of

B

∈

ℬ

, let

W

⁢

(

B

)

be the

collection

of

well-orderings

on

B

. Each element of

W

⁢

(

B

)

is a subset of

B

×

B

, so that

W

⁢

(

B

)

∈

P

⁢

(

B

×

B

)

. For any

R

∈

W

⁢

(

B

)

, the well-ordered set

(

B

,

R

)

is

order isomorphic

to exactly one

element

β

of

α

.

Conversely

, every

β

∈

α

, by definition, is embeddable in

A

, so equipollent to a subset

B

of

A

. We may well-order

B

via

β

, and this well-ordering

R

∈

W

⁢

(

B

)

. Therefore, there is a

surjection

from

W

:=

{

(

B

,

R

)

∣

B

∈

ℬ

,

R

∈

W

⁢

(

B

)

}

onto

α

. Since

W

is a set, so is its range (by the replacement axiom), which is just

α

.

•

α

is an ordinal.

Since

α

is a set consisting of ordinals,

α

is well-ordered. Now, suppose

γ

∈

β

∈

α

. Since

β

is an ordinal,

γ

⊆

β

. If

ϕ

:

β

→

(

B

,

R

)

is an order isomorphism, then

ϕ

restricted to

γ

is an order isomorphism onto

ϕ

⁢

(

γ

)

⊆

A

, whose well-ordering is that

induced

by

R

. Therefore,

γ

∈

α

, so

α

is a

transitive set

. This shows that

α

is an ordinal.

•

α

is not embeddable in

A

.

Otherwise,

α

∈

α

, contradicting the fact that an ordinal can never be a member of itself.

∎

The proof is done within ZF, without the aid of the

axiom of choice

.

Since the class of ordinals

On

is well-ordered, so is the

subclass

C

of all ordinals not embeddable in

A

. The least element in

C

is called the

Hartogs number

of

A

, and is denoted by

h

⁢

(

A

)

.

In fact, the

α

constructed above is the Hartogs number of

A

, for if

δ

is another ordinal distinct from

α

that is not embeddable in

A

, then

δ

∉

α

, so

α

∈

δ

by

trichotomy

.

Remark

. For every set

A

, its Hartogs number

h

⁢

(

A

)

is a

cardinal number

: it is first of all an ordinal, so

|

h

⁢

(

A

)

|

≤

h

⁢

(

A

)

, where

≤

is the

ordering

on the ordinals, and if

|

h

⁢

(

A

)

|

<

h

⁢

(

A

)

(meaning

|

h

⁢

(

A

)

|

∈

h

⁢

(

A

)

), then

|

h

⁢

(

A

)

|

is embeddable in

A

. Since

h

⁢

(

A

)

is equipollent to

|

h

⁢

(

A

)

|

,

h

⁢

(

A

)

is embeddable in

A

, contradicting the definition of

h

⁢

(

A

)

. Hence

h

⁢

(

A

)

=

|

h

⁢

(

A

)

|

. From the discussion so far, we see that

h

can be thought of as a class function from the class

V

of all sets onto the class

Cn

of all cardinal numbers. In

addition

, assuming AC, every set is well-orderable, so that

h

⁢

(

A

)

is the least cardinal greater

|

A

|

, for every set

A

.

Title

Hartogs number

Canonical name

HartogsNumber

Date of creation

2013-03-22 18:49:48

Last modified on

2013-03-22 18:49:48

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03E10

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