= Solution
An <initial ordinal> is an ordinal $\kappa$ that is not equipotent to any smaller ordinal. Equivalently, it is the least ordinal having its cardinality. The <aleph number>s enumerate the infinite initial ordinals:
$$
\aleph_0=\omega,
$$
$\aleph_{\alpha+1}$ is the least initial ordinal greater than $\aleph_\alpha$, and at a limit $\lambda$, $\aleph_\lambda$ is the least initial ordinal greater than every $\aleph_\beta$ for $\beta<\lambda$.
Each $\aleph_\alpha$ is infinite and initial by construction. Conversely, let $\kappa$ be an infinite initial ordinal. The infinite initial ordinals below $\kappa$ form a set and therefore have an ordinal order type $\alpha$. The recursive enumeration above lists exactly those predecessors before stage $\alpha$, so its next value is $\aleph_\alpha=\kappa$.
Read literally, “has cardinality $\aleph_\alpha$” is too weak: $\omega+1$ has cardinality $\aleph_0$ but is not initial. The valid characterization is
$$
\kappa\text{ is an infinite initial ordinal}
\quad\Longleftrightarrow\quad
\kappa=\aleph_\alpha\text{ for some }\alpha.
$$
If “has cardinality” in the question means “is the cardinal represented by,” this is precisely the requested statement.
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