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Classes can be considerd as "larger" than sets in the sense that any set is a class.

Is there mathematical object which is "larger" than classes ?

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    What types of things do you have in mind? The question has no context to tell what type of answer you are looking for.2010-08-06
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    There are things that are too big to be sets. Are there things that are too big to be too big to be sets ?2010-08-06
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    in ZFC there are only sets and no classes, but yes you get set theories which have classes and there are also type theories which have types, types of types, types of types of types of types... in an infinite hierarchy. The actual structure of the hierarchy doesn't seem to be very interesting, it's just necessary to avoid paradoxes.2010-08-06
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    Is there a Cantor-diagonal-style argument or something to show there are different sizes of classes, as there is for sets ?2010-08-06
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    No, if you have Cantor-diagonal-style machinery available you are dealing with sets. Classes are really there as a catching net for the paradoxes of set theory (so Russell's Paradox escapes being a paradox because the class of all sets where $x \notin x$ is not itself a set).2010-08-06

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