Suppose $\kappa$ is an uncountable cardinal, with $L_\kappa$ an admissible set (i.e. a model of Kripke–Platek set theory). Let $<_\gamma \subseteq \kappa \times \kappa$ denote a wellordering of $\kappa$ (equivalently, of $L_\kappa$) such that $\mathrm{ot}(<_\gamma) = \gamma$.
What is the least ordinal $\delta$ such that $<_\delta$ is not first order definable (i.e. in $\mathcal{L}_\in$) over $L_\kappa$?
Variations on this question include:
- What is least such ordinal when we consider $L_\alpha$ for any admissible $\alpha$?
- How is the answer affected by restricting definability, e.g. to $\Delta^0_1$, or strengthening it to $\mathcal{L}^2_\in$?
- This is all parameter-free, does allowing parameters substantially change the answer?
I'm afraid I don't know much about $L$ or admissibility so these may be naïve questions, sorry!