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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 cannot 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 façon 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) "