Remark

Given any set $A$, there exists a set whose elements are the cardinals $\operatorname{Card}(X)$ for all the subsets $X$ of $A$, namely, the set of objects of the form $\operatorname{Card}(X)$ for $X\in\mathfrak{P}(A)$ (Chapter II, § 1, no. 6). For every cardinal $\alpha$ the relation “$\xi$ is a cardinal and $\xi\leq\alpha$” is therefore collectivizing in $\alpha$ (Chapter II, § 1, no. 4), because it is equivalent to the relation "$\mathfrak{x}$ is of the form Card (X) for $X \subset \mathfrak{a}$"; the set of all $\mathfrak{x}$ satisfying this relation is called the set of cardinals $\leqslant \mathfrak{a}$.