Is the cartesian product of two cardinals a cardinal?

Question

Given two cardinals $\kappa$ and $\lambda$, is their cartesian product $\kappa \times \lambda$ also a cardinal?

Answer 1

Well, the Cartesian product is defined as a set of ordered pairs, so the Cartesian product of two cardinals isn't even an ordinal. It has a cardinality, of course, and that cardinality is determined by the cardinalities of $\lambda$ and $\kappa$, but it certainly isn't literally a cardinal.

Their ordinal product is an ordinal, but not a cardinal (if $\kappa > \lambda$, $\kappa\lambda$ will be an ordinal somewhere between $\kappa$ and $\kappa^+$). Their cardinal product is certainly a cardinal, by definition.

Answer 2

There may be some confusion going on. If you mean for $\kappa$ and $\lambda$ to be sets with a certain cardinality, and wonder if their cartesian product has a certain cardinality, then yes: the cartesian product is a set and all sets have a certain cardinality. If you are wondering what that cardinality is for infinite sets:

$|\kappa \times \lambda| = \max(|\kappa|,|\lambda|)$ unless one is empty, in which case $\kappa \times \lambda$ is empty too of course.

If $\kappa$ and $\lambda$ are cardinal numbers (cardinals), rather than sets, then $\kappa \times \lambda$ is usually understood to be their cardinal product (not cartesian product) which is another cardinal, and where again if either $\kappa$ or $\lambda$ is infinite, and neither is zero, then:

$\kappa \times \lambda = \max(\kappa,\lambda)$