Definition 4.7.1.28 (Cofinality). Let $(T, \leq )$ be a linearly ordered set. We let $\mathrm{cf}(T)$ denote the smallest ordinal $\alpha $ for which there exists a well-ordered set $(S, \leq )$ of order type $\alpha $ and a cofinal function $f: S \rightarrow T$. We refer to $\mathrm{cf}(T)$ as the cofinality of the linearly ordered set $T$.
If $\beta $ is an ordinal, let $\mathrm{cf}(\beta )$ denote the cofinality $\mathrm{cf}(T)$, where $(T, \leq )$ is any well-ordered set of order type $\beta $. We refer to $\mathrm{cf}(\beta )$ as the cofinality of $\beta $.