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In the definitions I have seen, a class is a collection of sets. By that, I cannot have a class A that contains a class B, because B is a class not a set.

But I know there are counterexamples, for example the collection of {1, 2} and {3, 4} can be described as the set {{1, 2}, {3, 4}}. Both {1, 2} and {3, 4} are classes... just not proper classes. So I can at least say there are some collections of classes which are, themselves, classes.

I'm curious if there's any examples like this involving proper classes. While I know that, in general, a collection of classes in category theory is called a conglomerate, but intuitively feels like a collection containing a finite number of classes (say, the class of groups and class of sets) doesn't seem like it should be obliged to be any larger than a class. Especially since those higher order concepts are sufficiently rare that there isn't a good name for a collection of conglomerates.

My intuition has failed me before, so I ask. Can a class ever contain a proper class as an element?

Cort Ammon
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  • Can you narrow this question down to a specific set theory? Because in NBG or MK or ZFC the answer is provably (and easily so) "no". – Malice Vidrine Oct 23 '18 at 06:05
  • @MaliceVidrine I tagged it with category-theory, hoping that that might specify it enough. If the answer is "it depends on which set-theory you use," that might be the answer I get. As I continued my search for an answer, I came across this question whose answer points out the sort of tricks that are pulled for the special case of categories being composed of sets, but that seemed to be a special case. – Cort Ammon Oct 23 '18 at 06:12
  • I know of only one category theory text that explicitly uses the set-classes-conglomerates approach instead of either being agnostic about the issue or using Grothendieck universes. And I've seen proposals for things as varied as a strengthened version of Ackermann, or NFU with a universe of wellfounded sets. "Category theory" does not determine the set theoretical framework by a long shot. – Malice Vidrine Oct 23 '18 at 06:24
  • This is a question about predicate logic and set theory. Classes aren't a concept of category theory. They come up when trying to provide set theoretic foundations for category theory, but they come up in many other contexts unrelated to category theory and there are approaches to formalizing category theory that don't rely on classes. – Derek Elkins left SE Oct 24 '18 at 22:32

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