Proposition 2

For every family $(\mathfrak{a}_\iota)_{\iota \in I}$ of cardinals, there exists a unique cardinal $\mathfrak{b}$ such that $\mathfrak{a}_\iota \leqslant \mathfrak{b}$ for all $\iota \in I$ and such that every cardinal $\mathfrak{c}$ for which $\mathfrak{a}_\iota \leqslant \mathfrak{c}$ for all $\iota \in I$ is $\geqslant \mathfrak{b}$.

There exists a set E containing all the sets $\mathfrak{a}_\iota$ (e.g., the sum of these sets (Chapter II, § 4, no. 8)), whence $\mathfrak{a}_\iota \leqslant \mathfrak{a} =$ Card (E) for all $\iota \in I$. The set F of cardinals $\leqslant \mathfrak{a}$ is well-ordered and contains all the $\mathfrak{a}_\iota$, and therefore the family $(\mathfrak{a}_\iota)_{\iota \in I}$ has a least upper bound $\mathfrak{b}$ in F. Let $\mathfrak{c}$ be a cardinal $\geqslant \mathfrak{a}_\iota$ for all $\iota \in I$; if $\mathfrak{c} < \mathfrak{b} \leqslant \mathfrak{a}$, then $\mathfrak{c} \in F$, and the inequality $\mathfrak{a}_\iota \leqslant \mathfrak{c}$ contradicts the definition of the least upper bound of the family $(\mathfrak{a}_\iota)$ in the ordered set F; hence the result.

¶ By abuse of language, the cardinal $\mathfrak{b}$ is called the least upper bound of the family $(\mathfrak{a}_\iota)_{\iota \in I}$ and is denoted by $\sup_{\iota \in I} \mathfrak{a}_\iota$.