In this way Solovay showed that in the proof of the existence of a non-measurable set from ZFC (Zermelo–Fraenkel set theory plus the axiom of choice), the axiom of choice is essential, at least granted that the existence of an inaccessible cardinal is consistent with ZFC.
The second step is to construct Solovay's model N as the class of all sets in M[G] that are hereditarily definable over a countable sequence of ordinals.
Several authors proved weaker versions of Solovay's result without assuming the existence of an inaccessible cardinal.
371 Finally, Shelah (1984) showed that consistency of an inaccessible cardinal is also necessary for constructing a model in which all sets of reals are Lebesgue measurable.
Shelah also showed that the Σ13 condition is close to the best possible by constructing a model (without using an inaccessible cardinal) in which all Δ13 sets of reals are measurable.