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Concept
Universal set
Summary
In set theory, a universal set is a set which contains all objects, including itself. In set theory as usually formulated, it can be proven in multiple ways that a universal set does not exist. However, some non-standard variants of set theory include a universal set.
Many set theories do not allow for the existence of a universal set. There are several different arguments for its non-existence, based on different choices of axioms for set theory.
In Zermelo–Fraenkel set theory, the axiom of regularity and axiom of pairing prevent any set from containing itself. For any set , the set (constructed using pairing) necessarily contains an element disjoint from , by regularity. Because its only element is , it must be the case that is disjoint from , and therefore that does not contain itself. Because a universal set would necessarily contain itself, it cannot exist under these axioms.
Russell's paradox
Russell's paradox prevents the existence of a universal set in set theories that include Zermelo's axiom of comprehension.
This axiom states that, for any formula and any set , there exists a set
that contains exactly those elements of that satisfy .
As a consequence of this axiom, to every set there corresponds another set consisting of the elements of that do not contain themselves. cannot contain itself, because it consists only of sets that do not contain themselves. It cannot be a member of , because if it were it would be included as a member of itself, by its definition, contradicting the fact that it cannot contain itself. Therefore, every set is non-universal: there exists a set that it does not contain. This indeed holds even with predicative comprehension and over intuitionistic logic.
Cantor's theorem
Another 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.
Official source
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