Re: Wait, more than one collective noun?
"Yes, as far as I know (which admittedly isn't very far), DavCrav's claim is incorrect under any version of ZF. ZF doesn't formalize the notion of classes. Wikipedia says "In some extensions of ZFC, objects like R [Russel's set-of-all-sets-that-contain-themselves] are called proper classes". But that usage isn't universal.
I don't think the AoC is required under ZF for sets to contain themselves in general (that is, sets can contain themselves under ZF and ZFC), but I may well be wrong about that."
Yes, you are wrong. There is no model of ZF in which there is a set consisting of all sets, because it wouldn't satisfy the axiom of separation. Therefore it cannot exist in any (consistent) extension of ZF, such as ZFC, ZF+CH, and anything else. Under intensional dependent type theory one can do better, but that's still under development.
So the object containing all sets, or the object containing all ordinal numbers, etc. can never be a set. It is called a proper class in general mathematical parlance, but it is not constructible in first-order logic. But then, lots of things aren't constructible there, that's why we don't always use it. It's nice to have Cat, for example.
(Source: me, a professional pure mathematician.)