Adding highly undefinable sets over $L$

Abstract

We sow that there exists a generic extension of the G\"{o}del'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''.