Without a metatheory one does not have a truth predicate which complicates things for realism because the whole point of that philosophy is that sets really do exist and actually do have the properties ascribed to them One could argue in favor of a disquotational definition of truth but that merely pushes the problem around leading to the same problems as full second order semantics for arithmetic Also not having a metatheory means that one must treat set theory differently than other branches of mathematics such as group theory or linear ordering theory This seems like a difficult distinction to justify I do think that the no metatheory interpretation of ZFC or Zermelian set theory if you prefer that is somewhat common because it is one way of avoiding the issue that what is called V in the object theory can not be distinguished from a set in the metatheory One consequence of my original point is that it would also be coherent to interpret this no metatheory language merely as a way of making statements from the viewpoint of a person inside a particular model of set theory Moreover this interpretation is entirely compatible with the viewpoint that there is an objectively defined collection consisting of all sets if no additional meta level principles such as the ability to force over V are added However if one already viewed all sets as merely a fa on de parler with no objective referent as I think you are proposing then it s not clear why one sort of disquotation all sets all sets is better than another all sets all sets in the model at hand Carl talk