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I have read in my university's lecture notes, that a class as usually defined, can only contain sets as elements and not proper classes. Is there a reason for this that I am missing. Is it to do with a paradox/contradiction that can be derived using the Replacement Axiom.

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ZFC is a first-order theory whose domain of discourse is the universe of sets, so it doesn't formally talk about classes at all. We use the term class to refer informally to a collection of sets, such as the collection of all sets satisfying a given first-order formula. You can also consider collections of classes, but then you're doing a sort of naive set theory and are susceptible to things like Russel's paradox, which axiomatic set theory was invented to avoid.

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    $\begingroup$ There are also a few approaches to considering collections of classes responsibly (though I can’t personally vouch for their usefulness beyond whatever intrinsic interest one might have in the problem). $\endgroup$ Commented May 1 at 23:28

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