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There's no largest integer, but the integers still form a set with an (infinite) cardinality, namely aleph-zero.

Likewise, there's no largest cardinal, but you'd think that the cardinals also formed a set with an (infinite) cardinality of its own.

Fun fact: they don't. There are too many of them.

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Like I said in my response to pndmnm, in my view if someone "[would] think that the cardinals also formed a set with an (infinite) cardinality of its own" then they haven't really grasped the theorem yet. It's built into it that if we're always allowed to form the powerset of a set then regardless of how large a cardinal you can find, you can always take its powerset and obtain a larger cardinal.

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