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 Russell's paradox and is consequently not allowed. However, some non-standard variants of set theory include a 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 Russell's paradox and is consequently not allowed. However, some non-standard variants of set theory include a universal set. There is no standard notation for the universal set of a given set theory. Common symbols include V, U and ξ. 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. It is directly contradicted by the axiom of regularity, and 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 φ ( x ) {displaystyle varphi (x)} and any set A, there exists a set that contains exactly those elements x of A that satisfy φ {displaystyle varphi } . If a universal set V existed and the axiom of comprehension could be applied to it, thenthere would also exist a set { x ∈ V ∣ x ∉ x } {displaystyle {xin Vmid x ot 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. A second difficulty with the idea of a universal set concerns the power set of the set of all sets. Because this power set is a set of sets, it would necessarily be a subset of the set of all sets, provided that both exist. However, this conflicts with Cantor's theorem that the power set of any set (whether infinite or not) always has strictly higher cardinality than the set itself.