ZFC has no formal notion of "proper class," but informally, everyone uses the term anyway. $V$, $Ord$, etc are said to be proper classes. Similarly, although in ZFC, one can only take the "union" of some collection of sets if the entire collection is also a set, we have no problem saying $V$ is the "union" of the various $V_\alpha$. All of this would seem to be fairly self-evident and harmless.

But there is a peculiar subtlety about which "non-set-collections" of the universe get this label of "proper class," and I am curious to get some clarification. Here is what I mean, ordered from least to most controversial:

• We have the various ordinals. Each one is a set, but their union, $Ord$, isn't a set. Is this union a proper class? (Yes.)
• We have the various sets $V_a$ from the Von Neumann heirarchy. Each of these is a set, but their union, $V$, is not a set. Is this union a proper class? (Yes.)
• Suppose we have ZF with the negation of the axiom of infinity. We have some infinite collection $S$ of natural numbers. Each finite subset of $S$ is a set, but their union, $S$, is not a set. Is this union a proper class? (I think so, relative to this theory.)
• Suppose we have some model of ZFC which is missing some subset of the naturals called $S$. Each finite subset of $S$ is a set, but their union, $S$, is not a set. Is this union a proper class, even though it is now a subclass of a set? (Uh, yes? No?)
• Suppose we have some countable model of ZFC. The model thinks that "$\omega_1$" is uncountable and regular, but there exists some countable set $S$ of ordinals that is cofinal in "$\omega_1$" which the model doesn't know about. Each finite subset of $S$ is a set, but their union, $S$, is not a set. Is this union a proper class, even though it is now a subclass of a set? (Maybe?)
• We are in positive set theory, which has a set of all sets. There are many sets in here which are well-founded, but the union on all of these is not a set. Is this union a proper class, even though it is now a subclass of a set? (I'm sure I've heard the term used this way.)

We can go a lot further, but hopefully the point is clear. There can be collections of elements which fail to be a set for reasons other than being "too large." This happens all the time, particularly in ZF without choice, where there seems to be no limit to how bizarre some of these models can get. Do we think of these things as "proper classes?" Maybe "semisets?"