For some $L_\kappa$ in the constructible hierarchy, does there exist a $\kappa$ such that $\kappa$ is a worldly cardinal and that $L_\kappa$ contains all of the constructible reals?
The motivation for the question is as follows: in order to understand whether 'adding Cohen reals' is a "legitimate expansion of the mathematical ontology" (this a direct quote from Linnebo's "The Potential Hierarchy of Sets") according to the potentialist conception of the cumulative hierarchy, I email Prof. Linnebo with an example that wrongly assumes that from the potentialist point of view, L is completed. Prof. Linnebo kindly points this out to me and, in a later email, states that "We can complete something iff that something is all contained in some $V_\alpha$." If one wishes to identify a possible world as a model of ZFC (for the purposes of my example--one has to have a ground model in order to have a forcing extension, right?) then having a stage $L_\kappa$ where $\kappa$ is a worldly cardinal large enough to have $L_\kappa$ contain all of the constructible reals would seem to produce a model of ZFC+(V=L) to which one can add Cohen reals to form the forcing extension M[G] which violates CH however one wishes.
The answer is yes for trivial reasons: every real in $L$ appears in $L_{\omega_1}$, so if $\kappa$ is worldly then of course every real in $L$ lives in $L_\kappa$.
If I understand your motivating paragraph correctly, you also want to argue that if $\kappa$ is worldly then it is worldly in $L$ and that in this case $L_\kappa\models \mathrm{ZFC}$. Both of these things are true, but showing them might be a good exercise for you.