Product of cardinals

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Definition

Product of two cardinals

The product of two cardinals \alpha,\beta is defined as the cardinality of any set obtained as the Cartesian product of a set of cardinality \alpha with a set of cardinality \beta. The product may be denoted \alpha\beta.

Product of finitely many cardinals

Suppose \alpha_1,\alpha_2,\alpha_3,\dots,\alpha_n are cardinals (possibly equal, possibly distinct). The product cardinal, denoted \alpha_1\alpha \dots \alpha_n is the cardinality of any set that is expressed as the Cartesian product of sets A_1,A_2,\dots,A_n where the cardinality of A_i is \alpha_i for each i \in \{ 1,2,\dots,n\}.

We can also define this iteratively by multiplying the cardinals two at a time. Cardinal multiplication is commutative and associative, so the manner of parenthesization and order of multiplication do not matter.

Computation of product

We list the following simple rules to compute products of cardinals:

Hypothesis on cardinals being multiplied Conclusion about product Does it rely on the axiom of choice?
one or more of the cardinals being multiplied is equal to zero the product is zero No
all the cardinals being multiplied are finite and nonzero the product is the product in the sense of multiplication of natural numbers. Note that the product is also a finite cardinal No
one or more of the cardinals being multiplied is infinite, and none of the cardinals being multiplied is zero the product is the maximum of the infinite cardinals being multiplied Yes

Note that if the cardinals are arising as orders of groups, we do not need to worry about any of them being zero.

Computation of factors from product

Hypothesis on product of cardinals Conclusion about cardinals being multiplied Does it rely on the axiom of choice?
the product is zero one or more of the cardinals being multiplied is zero Yes
the product is finite and nonzero all the cardinals being multiplied are finite and nonzero, and the multiplication is the usual multiplication of natural numbers. In particular, this forces each of the cardinals individually to be factors of the claimed product. No
the product is infinite all the cardinals being multiplied are less than or equal to the claimed product, and at least one of the cardinals being multiplied is equal to the claimed product. Yes

Computation of one factor given product and other factor

We consider the case of a product of two cardinals.

Hypothesis on product cardinal Hypothesis on one of the cardinals being multiplied Conclusion about the other cardinal
zero zero could be anything
zero nonzero zero
finite and nonzero finite and nonzero quotient of the product by the first cardinal, uniquely determined
infinite finite or infinite, and strictly smaller than the product cardinal infinite, equal to the product cardinal
infinite infinite, equal to the product cardinal nonzero and less than or equal to the product cardinal

Related notions