Godel's constructable universe ($L$) is defined using definable power set operator in first order logic ($\mathcal{L}_{\omega ,\omega}$). One can produce such a universe in infinitary logics in the same way using corresponding notions of formulas and definability. Obviously $L$ becomes larger when the logic has more expression power.

For each cardinal $\kappa$ define $L_{\kappa}$ to be Godel's constructable universe in the infinitary logic $\mathcal{L}_{\kappa,\kappa}$ and $L_{\infty}$ is Godel's constructable universe in $\mathcal{L}_{\infty,\infty}$.

1. Is $L_{\kappa}$ a model of $ZFC$ for each cardinal $\kappa$? What about $ZFC+GCH$?

2. What is $L_{\infty}$?

3. Is there any (possibly large) cardinal $\kappa$ such that $L_{\kappa}$ is Dodd-Jensen core model or $L[U]$?