 Countable-ordinal correctness under constructibility

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.