Definition 1

A cardinal $\alpha$ is said to be finite if $\alpha \ne \alpha + 1$. A finite cardinal is also called a natural integer (or simply an integer if there is no risk of confusion [^1]). A set $E$ is said to be finite if $\operatorname{Card}(E)$ is a finite cardinal; and $\operatorname{Card}(E)$ is then called the number of elements of $E$.

A family (Chapter II, § 3, no. 4) is said to be finite if its index set is finite.

When we say that the number of objects of a certain type is an integer $m$, we mean that these objects are elements of a finite set whose number of elements is $m$. A set whose number of elements is $m$ is also called a set of $m$ elements.