Here, I want to delve into what do we exactly feel about what constitutes a platonic existence of a set? Or what makes us think or actually a kind of feel or sense the existence of a set in the platonic realm of sets. The platonic realm of sets is supposed to be the world of truly existing sets, it is satisfaction in that world that we'd label as set theoretic truth. Now, the question is about existence of big sets, like the universe, Frege cardinals, Frege ordinals, the set of all of those, etc.. From the viewpoint of $\sf ZF$ existence of such entities is refuted altogether. However, what make us feel that $\sf ZFC$ is saying the truth of the whole set world in the platonic sense? I see the rules of $\sf ZF$ as being true of the Platonic realm of Well-founded sets, their motivation is the Cumulative Hierarchy, and the underpinning of its rules lies in the iterative conception of sets. The latter is most elegantly captured by the Cumulative Hierarchy. I don't see a clear motivation of it beyond this iterative picture. But, why should we think that the Platonic realm obeys such a trend. Why not restrict the rules of $\sf ZFC$ to the realm of well-founded sets, as it appears to be heavily ingrained in, and so permit existence of other kinds of sets with different rules.

For example if we have all rules of $\sf NFI$ to govern all sets, then simply add the schema of replacement restricted to the well-founded realm of sets, also add an axiom of existence of a set of all von Neumann naturals. Add an axiom of Choice over all sets. Also, to preserve some intuitive glimpse we add the axiom that all sets are strongly Cantorian.

$\forall X \exists f: f=\{ \langle x, \{x\} \rangle \mid x \in X \} $

And so we'd have the Category of sets being Cartesian closed!

All of those seems to be a way of welcoming the big sets to the platonic realm of sets with them having nice properties that we like. But, if there is a real world of sets, it need not obey these nice features. Actually they might defy Choice and Cartesian closedness. Now that $\sf Con(NF)$ proof of Randall Holmes has been Lean-verified. Then we might as well contemplate these big sets having awkward characteristics that defy basic intuitions about sets. And so instead of $\sf NFI$ we may take the whole of $\sf NF$ instead. And of course restrict Choice to the well-founded realm, and shun strong Cantorian axiom.

I chose $\sf NFI$ above because it is the strongest in comprehending over such sets in keeping with nice properties and intuitive expectations. $\sf NF$ may be an alternative because it is in some sense the maximal we have about comprehending over such sets.

So my question is about what really underpins our conception about existence of sets in the Platonic world should such a realm be there? Is it their formal mileage? Or is it some vague intuitive sense of them being real? To what extent we may accept a whole exotic world of sets lying there beyond the ardent cumulative hierarchy world of sets. What is the big picture here?

philosophicalconsequences beyond what we can already get from the consistency of NFU, which was established decades ago. In this article, Holmes addresses some of the objections articulated by Joel David Hamkins, and tries to give an intuitive picture of what NFU is all about. $\endgroup$1more comment