Product of cardinals

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

• Product of infinitely many cardinals generalizes this to products of infinite length, where the cardinals individually may be finite or infinite.
• Restricted product of infinitely many cardinals