A universal set, or a set that contains all sets, exists.

False. In standard set theory (ZF/ZFC), a universal set leads to contradictions. Russell’s paradox considers R = {x | x ∉ x}; asking whether R ∈ R yields an impossibility. Cantor’s theorem also shows the power set P(U) is strictly larger than U, but if U contained all sets it would have to contain P(U), contradicting “larger than.” To avoid these paradoxes, mathematicians distinguish sets