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- The concept of a "universal set" is problematical in rigorous versions of set theory. Do such versions leave ##\emptyset ^ C## undefined?

In an elementary school version of set theory, we can take the complement of the empty set to obtain ##\emptyset ^ C = \mathbb{U}## However, in a sophisticated version of set theory, the concept of a "universal set" ##\mathbb{U}## is problematical. ( So says the current Wikipedia article on "unversal set" https://en.wikipedia.org/wiki/Universal_set ).

How do sophisticated versions of set theory treat the concept of a complement to the empty set. Do they leave it undefined?

How do sophisticated versions of set theory treat the concept of a complement to the empty set. Do they leave it undefined?