Adding highly undefinable sets over $L$

We sow that there exists a generic extension of the Gödel’s constructible universe in which diamond holds and there exists a subset $Y\subseteq {\omega }_1$ such that for stationary many $\delta <{\omega }_1,$ the set $Y\cap \delta$ is not definable in the structure $(L_{F(\delta )},\in )$, where $F(\delta )>\delta$ is the least ordinal such that $L_{F(\delta )}\models$``$\delta$ is countable’'.