A universal set is a set containing every object under consideration in a specified universe of discourse. The complement of every set is then taken relative to the universal set, and the complement of the universal set is the empty set.

In Zermelo-Fraenkel set theory, there is no set of all sets; the collection of all sets is a proper class. Indeed, if a set contained every set, then the axiom of separation would give the set

Since is universal, , and the definition would imply iff , which is Russell's antinomy. Some alternative axiomatic set theories instead admit a universal set (Holmes 1998).

See also

Axiomatic Set Theory, Empty Set, Proper Class, Russell's Antinomy, Set, Set Theory

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References

Fraenkel, A. A. and Bar-Hillel, Y. Foundations of Set Theory. Amsterdam, Netherlands, 1958.Holmes, M. R. Elementary Set Theory with a Universal Set. Cahiers du Centre de Logique, Vol. 10. Louvain-la-Neuve, Belgium: Academia, 1998.

Referenced on Wolfram|Alpha

Universal Set

Cite this as:

Weisstein, Eric W. "Universal Set." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/UniversalSet.html

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