Hereditarily countable constructible sets appear below omega-one ($H_{{\omega }_1}^L=L_{{\omega }_1}^L$)

Inside the constructible universe, every hereditarily countable set appears in a countable level of the constructible hierarchy. Take a countable elementary substructure of a sufficiently large $L_{\eta }$ containing its transitive closure pointwise. The condensation lemma for the constructible universe identifies the collapse with $L_{\beta }$ for countable $\beta$, and the collapse fixes the original set. Conversely, a countable-indexed level is countable inside the constructible universe.