nLab product-regular cardinal

Idea

The version of regular cardinal for products/Cartesian products instead of sums/unions/disjoint unions.

Definition

A cardinal $\kappa$ is product-regular if, for all families of sets $(X_i)_{i \in I}$, if $\vert I \vert \lt \kappa$ and for all elements $i \in I$ $\vert X_i \vert \lt \kappa$, then the indexed product

$$\left| \prod_{i \in I} X_i \right| \lt \kappa$$

Properties

Every inaccessible cardinal is product-regular. Every uncountable product-regular cardinal is an inaccessible cardinal, but the finite cardinals $0$ representing the empty set and the countable cardinal $\aleph_0$ representing the natural numbers are product-regular cardinals which are not inaccessible cardinals.

See also

• regular cardinal

• inaccessible cardinal

References

• Tom Leinster, Large Sets 9, web