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How is the Cantor's paradox resolved in the ZFC system?

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Assuming that ZF is consistent, the axioms of ZF, with or without the axiom of choice, simply do not permit a ‘set of all cardinalities’ to be formed; that collection is a proper class, and in ZF there are no proper classes as formal objects. They are rather to be identified with predicates. For example, what is (informally in ZF, formally in, e.g., NBG) the proper class of all sets can be identified with the predicate $x=x$. The discussion of Cantor’s paradox in Wikipedia is a reasonable starting point.

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