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Working in ZFC, are there collections of sets that are neither a set nor a definable proper class?

I mean if some collection of sets is not a set can we necessarily conclude that it is a definable proper class?

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    $\begingroup$ If it is not a set, then you are no longer quite in ZFC anyway. $\endgroup$ – Tobias Kildetoft Jun 16 '15 at 11:44
  • $\begingroup$ @TobiasKildetoft Formally yes but we manage to deal with proper classes in ZFC. I think one can easily formalize my question in ZFC. $\endgroup$ – user248567 Jun 16 '15 at 11:47
  • $\begingroup$ You can only really formalize questions about proper classes in ZFC when they are definable proper classes. (The way that you formalize these questions is by using the definition of the definable proper class!) So in a sense, the answer to your question is "no". $\endgroup$ – Paul McKenney Jun 16 '15 at 13:57
  • $\begingroup$ @PaulMcKenney But it seems quite meaningful question to me! In several cases I came across collections of sets that are too large to form a set but it is not immediately clear that they are definable with a formula with parameters. So what are they?! $\endgroup$ – user248567 Jun 16 '15 at 14:24
  • $\begingroup$ I'd be interested to see some examples. But for now I think Asaf's answer is probably what you want. $\endgroup$ – Paul McKenney Jun 16 '15 at 14:37
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This question makes sense when working "outside" a model of $\sf ZFC$. In which case the answer is almost trivially "yes".

Before getting into that, just note that working inside the model there are only sets, and formulas from the meta-theory which you can use to "approximate" collections which are not sets. So if you want to get outside this context you need to either work in "a larger universe" or in something which substantially augments $\sf ZFC$.

If you have a model of $\sf ZFC$, it is a set, and it has only "so many" [read: as the cardinality of the model] classes, which are definable (with parameters), but Cantor's theorem says that there are more subsets than that. So these give you an example.

Specific, and perhaps interesting, examples, are if you have a model $M$ and $N$ is a generic extension of $M$, then there is some $P\in M$ and $G\in N\setminus M$ such that $G\subseteq P$. This gives a "subcollection" of $M$ which is neither a class nor an element of $M$.

Another such example could be non-well founded models, where you cannot "locate" the well-founded part. It is a subset of some ordinal of the model, so it is a collection of ordinals of our model, but it is neither a set there nor a definable class (since it is an inductive class of ordinals which is not all the ordinals of the model, it cannot be a definable class there).

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