Universal set

In set theory, a universal set is a set which contains all objects, including itself. In set theory as usually formulated, the conception of a universal set leads to a paradox (Russell's paradox) and is consequently not allowed. However, some non-standard variants of set theory include a universal set. It is often symbolized by the Greek letter xi.

Zermelo–Fraenkel set theory and related set theories, which are based on the idea of the cumulative hierarchy, do not allow for the existence of a universal set. Its existence would cause paradoxes which would make the theory inconsistent.

Russell's paradox prevents the existence of a universal set in Zermelo–Fraenkel set theory and other set theories that include Zermelo's axiom of comprehension. This axiom states that, for any formula ${\displaystyle \varphi (x)}$ and any set A, there exists another set

that contains exactly those elements x of A that satisfy ${\displaystyle \varphi }$. If a universal set V existed and the axiom of comprehension could be applied to it, then there would also exist another set ${\displaystyle \{x\in V\mid x\not \in x\}}$, the set of all sets that do not contain themselves. However, as Bertrand Russell observed, this set is paradoxical. If it contains itself, then it should not contain itself, and vice versa. For this reason, it cannot exist.

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