= Solution
A <cardinal number> is an <initial ordinal>: an <ordinal> not in <bijection> with any smaller <ordinal>. In $ZFC$ every set has the <cardinality> of a unique such <ordinal>.
For an infinite <cardinal> $\kappa$, its <cofinality> is the least <cardinality> of an unbounded subset of the <ordinal> $\kappa$. Equivalently,
$$
\boxed{\operatorname{cf}(\kappa)=\min\{\theta:\exists\text{ a cofinal map }f:\theta\to\kappa\}.}
$$
A subset is unbounded when no <ordinal> below $\kappa$ bounds all its elements. A <cofinal> sequence can be taken increasing, and $\operatorname{cf}(\kappa)$ is an infinite <regular cardinal>.
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