Let and be cardinal numbers, and let and be disjoint sets such that and . (Here denotes the cardinality of a set , that is, the unique cardinal number equinumerous with .) Then we define cardinal addition, cardinal multiplication and cardinal exponentiation as follows.
(Here denotes the set of all functions from to .) These three operations are well-defined, that is, they do not depend on the choice of and . Also note that for multiplication and exponentiation and do not actually need to be disjoint.
We also define addition and multiplication for arbitrary numbers of cardinals. Suppose is an index set and is a cardinal for every . Then is defined to be the cardinality of the union , where the are pairwise disjoint and for each . Similarly, is defined to be the cardinality of the Cartesian product (http://planetmath.org/GeneralizedCartesianProduct) , where for each .
In the following, , , and are arbitrary cardinals, unless otherwise specified.
Some special cases involving and are as follows:
If at least one of and is infinite, then the following hold.
Also notable is that if and are cardinals with infinite and , then
Inequalities are also important in cardinal arithmetic. The most famous is Cantor’s theorem
If and , then
If, moreover, for all , then we have König’s theorem.
If for every in the index set , then
Thus it is possible to define exponentiation in terms of multiplication, and multiplication in terms of addition.
|Date of creation||2013-03-22 14:15:13|
|Last modified on||2013-03-22 14:15:13|
|Last modified by||yark (2760)|
|Defines||sum of cardinals|
|Defines||product of cardinals|