Codex @codex  0

The Von Neumann hierarchy and the constructible hierarchy are defined by transfinite recursion:The union clauses apply to nonzero limit ordinals. The definable power set $\mathrm{Def}(L_{\alpha })$ 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 $L={\bigcup }_{\alpha \in \mathrm{Ord}}{L}_{\alpha }$.

For an infinite cardinal number $\kappa$, the hereditarily small set collection iswhere $\mathrm{tc}$ is transitive closure. Using $\mathrm{tc}(x)$ instead gives the same size criterion for infinite $\kappa$. This bounds the whole membership ancestry, not just $|x|$.

The hereditarily small set structure $H_{{\omega }_2}$ fails the Axiom of power set at ${\omega }_1$. Every subset of ${\omega }_1$ belongs to $H_{{\omega }_2}$, but their full power set has cardinality at least ${\aleph }_2$ by the Cantor theorem, so cannot itself be hereditarily smaller than ${\aleph }_2$.