cardinal arithmetic

cardinal arithmetic

Definitions

Let

κ

and

λ

be

cardinal numbers

,
and let

A

and

B

be

disjoint sets

such that

|

A

|

=

κ

and

|

B

|

=

λ

.
(Here

|

X

|

denotes the

cardinality

of a set

X

,
that is, the unique cardinal number equinumerous with

X

.)
Then we define

cardinal addition

,

cardinal multiplication

and

cardinal exponentiation

as follows.

κ

+

λ

=

|

A

∪

B

|

.

κ

⁢

λ

=

|

A

×

B

|

.

κ

λ

=

|

A

B

|

.

(Here

A

B

denotes the set of all

functions

from

B

to

A

.)
These three

operations

are

well-defined

, that is,
they do not depend on the choice of

A

and

B

.
Also note that for

multiplication

and

exponentiation

A

and

B

do not actually need to be

disjoint

.

We also define

addition

and multiplication for arbitrary numbers of cardinals.
Suppose

I

is an

index set

and

κ

i

is a cardinal for every

i

∈

I

.
Then

∑

i

∈

I

κ

i

is defined to be
the cardinality of the union

⋃

i

∈

I

A

i

,
where the

A

i

are

pairwise disjoint

and

|

A

i

|

=

κ

i

for each

i

∈

I

.
Similarly,

∏

i

∈

I

κ

i

is defined to be the cardinality of the

Cartesian product

(

http://planetmath.org/GeneralizedCartesianProduct

)

∏

i

∈

I

B

i

, where

|

B

i

|

=

κ

i

for each

i

∈

I

.

Properties

In the following,

κ

,

λ

,

μ

and

ν

are arbitrary cardinals,
unless otherwise specified.

Cardinal arithmetic obeys many of the same algebraic laws as real

arithmetic

.
In particular, the following

properties

hold.

κ

+

λ

=

λ

+

κ

.

(

κ

+

λ

)

+

μ

=

κ

+

(

λ

+

μ

)

.

κ

⁢

λ

=

λ

⁢

κ

.

(

κ

⁢

λ

)

⁢

μ

=

κ

⁢

(

λ

⁢

μ

)

.

κ

⁢

(

λ

+

μ

)

=

κ

⁢

λ

+

κ

⁢

μ

.

κ

λ

⁢

κ

μ

=

κ

λ

+

μ

.

(

κ

λ

)

μ

=

κ

λ

⁢

μ

.

κ

μ

⁢

λ

μ

=

(

κ

⁢

λ

)

μ

.

Some special cases involving

0

and

1

are as follows:

κ

+

0

=

κ

.

0

⁢

κ

=

0

.

κ

0

=

1

.

0

κ

=

0

,

for

⁢

κ

>

0

.

1

⁢

κ

=

κ

.

κ

1

=

κ

.

1

κ

=

1

.

If at least one of

κ

and

λ

is

infinite

, then the following hold.

κ

+

λ

=

max

⁡

(

κ

,

λ

)

.

κ

⁢

λ

=

max

⁡

(

κ

,

λ

)

,

provided

⁢

κ

≠

0

≠

λ

.

Also notable is that if

κ

and

λ

are cardinals
with

λ

infinite and

2

≤

κ

≤

2

λ

,
then

κ

λ

=

2

λ

.

Inequalities are also important in cardinal arithmetic.
The most famous is Cantor’s

theorem

κ

<

2

κ

.

If

μ

≤

κ

and

ν

≤

λ

, then

μ

+

ν

≤

κ

+

λ

.

μ

⁢

ν

≤

κ

⁢

λ

.

μ

ν

≤

κ

λ

,

unless

⁢

μ

=

ν

=

κ

=

0

<

λ

.

Similar

inequalities hold for infinite sums and

products

.
Let

I

be an index set,
and suppose that

κ

i

and

λ

i

are cardinals for every

i

∈

I

.
If

κ

i

≤

λ

i

for every

i

∈

I

, then

∑

i

∈

I

κ

i

≤

∑

i

∈

I

λ

i

.

∏

i

∈

I

κ

i

≤

∏

i

∈

I

λ

i

.

If, moreover,

κ

i

<

λ

i

for all

i

∈

I

,
then we have König’s theorem.

∑

i

∈

I

κ

i

<

∏

i

∈

I

λ

i

.

If

κ

i

=

κ

for every

i

in the index set

I

, then

∑

i

∈

I

κ

i

=

κ

⁢

|

I

|

.

∏

i

∈

I

κ

i

=

κ

|

I

|

.

Thus it is possible to define exponentiation in terms of multiplication,
and multiplication in terms of addition.

Title

cardinal arithmetic

Canonical

name

CardinalArithmetic

Date of creation

2013-03-22 14:15:13

Last modified on

2013-03-22 14:15:13

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

38

Author

yark (2760)

Entry type

Topic

Classification

msc 03E10

Related topic

OrdinalArithmetic

Related topic

CardinalNumber

Related topic

CardinalExponentiationUnderGCH

Related topic

CardinalityOfTheContinuum

Defines

cardinal addition

Defines

cardinal multiplication

Defines

cardinal exponentiation

Defines

sum of cardinals

Defines

product of cardinals

Defines

addition

Defines

multiplication

Defines

exponentiation

Defines

sum

Defines

product

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