Various set\class theories present different kinds of ontology, broadly speaking there is the dichotomy of classes versus Ur-elements, and the former can be further subdivided into sets and proper classes, and the former can even be further subdivided into big and small sets. Now a class can be characterized in a loose sense as an object that possesses a certain nature that makes it relevant to speak of it having or not having members, an Ur-element is an object that is not a class, for although an Ur-element can be spoken of as being empty of members, but it appears that it can more appropriately be spoken about as being an entity that speaking of it having or not having members seem to be irrelevant to its nature, like most ordinary objects we have around us, for although one can say a "mountain doesn't have members" however that phrase from the outset seems to be vacant, unlike having an empty class, which is not vacant. Now classes are often divided into "sets" which can themselves be members of other classes, and into proper classes which are non-set classes. Those are spoken about in theories like NBG and MK which seems to extend the standard line of thought about sets. However, my question here is more related to the division of big and small sets, where the later ones are those sets of ZF and related theories, while the formers ones are sets like the set of all sets, Frege's cardinals and ordinals and the set of all of those, etc.. those are spoken about in theories like NF(U), or another line of theories of Church and Oswald and related theories.
Now clearly the most precious of those entities is the sets and more specifically the well-founded sets. Those seem to give rise to much mathematics and formal work, while the other sorts seem to be much less fertile.
In this article the author speaks about a principle due to Quine that of "No entity without identity!", i.e. we cannot speak of an entity if we cannot have a procedure that verifies the identity of objects possessing that entity. He speaks of the well-founded sets being entities that serve to meets this principle, having only finite membership chains, so their identity can be verified after the identity of their members since there is an end to that recursive verification of identity, and he outlines that this is one of the most attractive aspects that justifies the belief in the existence of those sets! however, the author strives to define ambiance that can subsume big sets to meeting this principle through the iterative conception of sets, but it doesn't seem to be an easy task, even though shown to be not impossible or at least not far from that principle.
The point is although the well-founded sets of Z and ZF do enjoy nice properties, yet why should that entitle them to be the "ONLY" sets being there? I mean truly the big sets are hard to comprehend, but still, this doesn't seem to be a sufficient cause to reject their existence.
My question is: IF one holds a Platonistic view of all sets preexisting in some platonic realm, then is it NATURAL to think of the existence of the big sets, or is it more appropriate to consider them as fake objects even though one can provide a consistent discourse about them (relative to the known theories of ZF and its extensions)?
To emphasize my question, we all know that Ur-elements exist, but ZFC [the main theory of sets and their elements] doesn't mention them, however, ZFC not mentioning them, doesn't mean that they don't exist. In a similar manner I'm speaking about the "EXISTENCE" of big sets and also of proper classes, is it natural to hold that these entities "exist" (especially from a single platonic universe perspective) or the natural tendency is to reject the claim of their existence?
Though what I mean by "natural" is difficult to define here, I don't know really how to spell it out, but still, I think there is some intuitive aspect to it that can be commonly touched and shared and I'm relying on this glimpse.
I personally feel it natural to have big sets, despite their low formal mileage. I think a theory that displays full ontology of classes and their elements would be a theory that has all four ontologies that I've spoken about here, like ML of Quine. In that sense, ZF would be true of the well-founded set realm, and not of all sets.
I mean is that "natural" sense of mine rejected?
If so then in what sense it is so rejected?
Note: This question first appeared at MathOverflow, however it was considered a question belonging to philosophy rather than mathematics, thus was shifted here. However there was some answers to this question at that site, and I made a brief summary of this question there, see it here: