Set of all sets

Referring to the set of all sets typically leads to a paradox. Is the set an element of itself, or not? Usually, formal logic disallows the construction of such a set. You can refer to "all sets" in the metalanguage.

The statement "no set is an element of itself" has a quantifier that ranges over all sets.

Although in naive set theory, we can talk of the set of all sets that are not elements of themselves, there appear to be set theories with no contradiction found so far where such a thing as the set of all set exists but the set of all sets that are not elements of themselves doesn't.

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