I would like to mention here that the notion of and *interpretation* of set theory as this is normally understood can usefully be liberalised.

First note that one of the relevant "interpretations", Gödel's constructible universe comes really as a family of interpretations
of various heights, one for each ordinal, of which infinitely many are models of ZFC.

Next note that the other interpretation mentioned by Gödel, sets as "arbitrary multitudes" regardless of definability, also
comes in many versions of differing height.
Its not clear that this is the view of Gödel who may well have thought it coherent to talk of a class of all such sets, but
since any such class would itself be an arbitrary multitude, incoherence is too close for comfort.