cardinal arithmetic


Let κ and λ be cardinal numbersMathworldPlanetmath, 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 additionMathworldPlanetmath, cardinal multiplication and cardinal exponentiation as follows.

κ+λ =|AB|.
κλ =|A×B|.
κλ =|AB|.

(Here AB denotes the set of all functions from B to A.) These three operationsMathworldPlanetmath 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 setMathworldPlanetmathPlanetmath and κi is a cardinal for every iI. Then iIκi is defined to be the cardinality of the union iIAi, where the Ai are pairwise disjoint and |Ai|=κi for each iI. Similarly, iIκi is defined to be the cardinality of the Cartesian product ( iIBi, where |Bi|=κi for each iI.


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 arithmeticPlanetmathPlanetmath. In particular, the following properties hold.

κ+λ =λ+κ.
(κ+λ)+μ =κ+(λ+μ).
κλ =λκ.
(κλ)μ =κ(λμ).
κ(λ+μ) =κλ+κμ.
κλκμ =κλ+μ.
(κλ)μ =κλμ.
κμλμ =(κλ)μ.

Some special cases involving 0 and 1 are as follows:

κ+0 =κ.
0κ =0.
κ0 =1.
0κ =0, for κ>0.
1κ =κ.
κ1 =κ.
1κ =1.

If at least one of κ and λ is infiniteMathworldPlanetmath, then the following hold.

κ+λ =max(κ,λ).
κλ =max(κ,λ), provided κ0λ.

Also notable is that if κ and λ are cardinals with λ infinite and 2κ2λ, then

κλ =2λ.

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

κ <2κ.

If μκ and νλ, then

μ+ν κ+λ.
μν κλ.
μν κλ, unless μ=ν=κ=0<λ.

Similar inequalities hold for infinite sums and productsMathworldPlanetmath. Let I be an index set, and suppose that κi and λi are cardinals for every iI. If κiλi for every iI, then

iIκi iIλi.
iIκi iIλi.

If, moreover, κi<λi for all iI, then we have König’s theorem.

iIκi <iIλi.

If κi=κ for every i in the index set I, then

iIκi =κ|I|.
iIκi =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