set theory – Upper bound for constructibility orders of elements of the continuum

In a constructible universe (ZFC + V=L), is there any known upper bound for the constructibility orders of all elements of the continuum, i.e. some separately described ordinal $alpha$ such that we can prove $mathcal{P}(omega)subset L_alpha$ ? For example (under some large cardinal axiom), can it be proven that the first inaccessible cardinal is such an upper bound, or can this cardinal still fail at this ? I intuitively suspect undecidabilities in this matter but am no expert in the field. Thanks.