alternative definition of cardinality

alternative definition of cardinality

The

concept

of

cardinality

comes from the notion of equinumerosity of sets. To define the cardinality

|

A

|

of a set

A

, one desirable property is that

A

is equinumerous to

B

precisely when

|

A

|

=

|

B

|

. The first attempt, due to Frege and Russel, is to define a

relation

∼

on the class

V

of sets so that

A

∼

B

iff there is a

bijection

from

A

to

B

. This relation is an

equivalence relation

on

V

. Then we can define

|

A

|

as the

equivalence class

containing the set

A

. However,

|

A

|

is not a set, so we can’t do much with

|

A

|

in ZF.

The second attempt, due to Von Neumann, defines

|

A

|

to be the smallest

ordinal

card

⁡

(

A

)

equinumerous to

A

. Now,

card

⁡

(

A

)

exists if

A

is

well-orderable

. But in general, we do not know if

A

is well-orderable unless the

well-ordering principle

is applied, which is just another form of the

axiom of choice

. Thus, this definition depends on AC, and, in everyday mathematical usage (which assumes ZFC),

|

A

|

:=

card

⁡

(

A

)

suffices.

The third way, due to Scott, of looking at

|

A

|

, without AC, is to modify the first attempt somewhat, so that

|

A

|

is a set. Recall that the rank of a set

A

is the least ordinal

α

such that

A

⊆

V

α

in the

cumulative hierarchy

. A set having a rank is said to be

grounded

. By the

axiom of foundation

, every set is grounded. For any set

A

, let

R

⁢

(

A

)

:=

{

ρ

⁢

(

B

)

∣

B

∼

A

}

. Then

R

⁢

(

A

)

, as a class of ordinals, has a least

element

r

⁢

(

A

)

. So

r

⁢

(

A

)

≤

ρ

⁢

(

A

)

. Next, we define (borrowing the terminology used in the first reference below)

kard

⁡

(

A

)

:=

{

B

∣

B

∼

A

⁢

and

⁢

ρ

⁢

(

B

)

=

r

⁢

(

A

)

}

,

and set

|

A

|

:=

kard

⁡

(

A

)

. Since every element in

kard

⁡

(

A

)

is a subset of

V

r

⁢

(

A

)

,

kard

⁡

(

A

)

⊆

V

r

⁢

(

A

)

+

, so that

|

A

|

is a set. This method is known as Scott’s trick. It can also be used in defining other

isomorphism

types on sets. It is easy to see that

|

A

|

=

|

B

|

iff

A

∼

B

. However, with this definition,

kard

⁡

(

n

)

≠

n

in general, where

n

is a

natural number

.

Nevertheless, it is known that every

finite set

is well-orderable, and so we come to the fourth definition of the cardinality of a set: given a set

A

:

|

A

|

:=

{

card

⁡

(

A

)

⁢

if

⁢

A

⁢

is well-orderable

,

kard

⁡

(

A

)

⁢

otherwise

.

The one big advantage of this definition is clear: it does not require AC, and with AC, it is identical to the second definition above. At the same time, it also resolves the conflict with our intuitive notion about cardinality: the cardinality of a finite set is the number of elements in the set. However, the one big disadvantage in this definition is that we do not have

A

∼

|

A

|

in general (of course,

A

is

infinite

). There is no way, without AC, to find a definition of

|

A

|

, such that

A

∼

B

iff

|

A

|

=

|

B

|

, and

A

∼

|

A

|

at the same time.

References

1

H. Enderton,

Elements of

Set Theory

, Academic Press, Orlando, FL (1977).

2

T. J. Jech,

Set Theory

, 3rd Ed., Springer, New York, (2002).

3

A. Levy,

Basic Set Theory

, Dover Publications Inc., (2002).

Title

alternative definition of cardinality

Canonical name

AlternativeDefinitionOfCardinality

Date of creation

2013-03-22 18:50:11

Last modified on

2013-03-22 18:50:11

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03E10

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