= Solution
The <Von Neumann hierarchy> and the <constructible hierarchy> are defined by <transfinite recursion>:
$$
\boxed{V_0=\varnothing,\qquad V_{\alpha+1}=\mathcal P(V_\alpha),\qquad
V_\lambda=\bigcup_{\alpha<\lambda}V_\alpha},
$$
$$
\boxed{L_0=\varnothing,\qquad L_{\alpha+1}=\operatorname{Def}(L_\alpha),\qquad
L_\lambda=\bigcup_{\alpha<\lambda}L_\alpha}.
$$
The union clauses apply to nonzero <limit ordinals>. The <definable power set> $\operatorname{Def}(L_\alpha)$ consists of subsets definable over $(L_\alpha,\in)$ by a <first-order formula> with finitely many parameters from $L_\alpha$. Definability over that structure is essential; it is not unrestricted definability in the ambient universe. The full <constructible universe> is $L=\bigcup_{\alpha\in\operatorname{Ord}}L_\alpha$.
For an infinite <cardinal number> $\kappa$, the <hereditarily small set> collection is
$$
\boxed{H_\kappa=\{x:|\operatorname{tc}(\{x\})|<\kappa\}},
$$
where $\operatorname{tc}$ is <transitive closure>. Using $\operatorname{tc}(x)$ instead gives the same size criterion for infinite $\kappa$. This bounds the whole membership ancestry, not just $|x|$.
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