Assuming $V=L$, every uncountable <transitive model> of <ZFC> contains all ambient <countable ordinals> and witnesses their countability internally. Its ordinal height is at least $\omega_1$, and <absoluteness of constructible levels> puts $L_{\omega_1}$ inside it. Every countability witness for a countable ordinal can be chosen in $L_{\omega_1}$ by <hereditarily countable constructible sets appear below omega-one>.


Countable-ordinal correctness under constructibility

{title2=$H_{\omega_1}^L=L_{\omega_1}^L$}

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>.


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 Incoming links: Hereditarily countable constructible sets appear below omega-one