# Universal set

In set theory, a **universal set** is a set which contains all objects, including itself.^{[1]} 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.

## Reasons for nonexistence

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.

### Russell's paradox

#### Regularity and pairing

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.^{[3]}

#### Comprehension

Russell's paradox prevents the existence of a universal set in set theories that include

^{[2]}

If this axiom could be applied to a universal set , with defined as the predicate ,
it would state the existence of Russell's paradoxical set, giving a contradiction.
It was this contradiction that led the axiom of comprehension to be stated in its restricted form, where it asserts the existence of a subset of a given set rather than the existence of a set of all sets that satisfy a given formula.^{[2]}

When the axiom of restricted comprehension is applied to an arbitrary set , with the predicate , it produces the subset of elements of 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. In this way, it is possible to construct a witness to the non-universality of , even in versions of set theory that allow sets to contain themselves. 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.

## Theories of universality

The difficulties associated with a universal set can be avoided either by using a variant of set theory in which the axiom of comprehension is restricted in some way, or by using a universal object that is not considered to be a set.

### Restricted comprehension

There are set theories known to be

^{[4]}but this is not possible for Oberschelp's, since in it the singleton function is provably a set,

^{[5]}which leads immediately to paradox in New Foundations.

^{[6]}

Another example is

### Universal objects that are not sets

The idea of a universal set seems intuitively desirable in the

The category of sets can also be considered to be a universal object that is, again, not itself a set. It has all sets as elements, and also includes arrows for all functions from one set to another. Again, it does not contain itself, because it is not itself a set.

## See also

- Universe (mathematics)
- Grothendieck universe
- Domain of discourse
- Von Neumann–Bernays–Gödel set theory — an extension of ZFC that admits the class of all sets

## Notes

**^**Forster (1995), p. 1.- ^
^{a}^{b}^{c}Irvine & Deutsch (2021). **^**Cenzer et al. (2020).**^**Church (1974, p. 308). See also Forster (1995, p. 136), Forster (2001, p. 17), and Sheridan (2016).**^**Oberschelp (1973), p. 40.**^**Holmes (1998), p. 110.

## References

- Cenzer, Douglas; Larson, Jean; Porter, Christopher; Zapletal, Jindrich (2020).
*Set Theory and Foundations of Mathematics: An Introduction to Mathematical Logic*. World Scientific. p. 2.S2CID 208131473. - Church, Alonzo (1974). "Set theory with a universal set".
*Proceedings of the Tarski Symposium: An international symposium held at the University of California, Berkeley, June 23–30, 1971, to honor Alfred Tarski on the occasion of his seventieth birthday*. Proceedings of Symposia in Pure Mathematics. Vol. 25. Providence, Rhode Island: American Mathematical Society. pp. 297–308. . - Forster, T. E. (1995).
*Set Theory with a Universal Set: Exploring an Untyped Universe*. Oxford Logic Guides. Vol. 31. Oxford University Press. . - Forster, Thomas (2001). "Church's set theory with a universal set". In Anderson, C. Anthony; Zelëny, Michael (eds.).
*Logic, Meaning and Computation: Essays in Memory of Alonzo Church*. Synthese Library. Vol. 305. Dordrecht: Kluwer Academic Publishers. pp. 109–138.MR 2067968. - Holmes, M. Randall (1998).
*Elementary set theory with a universal set*. Cahiers du Centre de Logique [Reports of the Center of Logic]. Vol. 10. Université Catholique de Louvain, Département de Philosophie, Louvain-la-Neuve. . - Irvine, Andrew David; Deutsch, Harry (Spring 2021). "Russell's Paradox". In Zalta, Edward N. (ed.).
*The Stanford Encyclopedia of Philosophy*. - MR 0319758.
- Willard Van Orman Quine (1937) "New Foundations for Mathematical Logic,"
*American Mathematical Monthly*44, pp. 70–80. - Sheridan, Flash (2016). "A variant of Church's set theory with a universal set in which the singleton function is a set" (PDF).
*Logique et Analyse*.**59**(233): 81–131.MR 3524800.

## External links

- Weisstein, Eric W. "Universal Set".
*MathWorld*. - Bibliography: Set Theory with a Universal Set, originated by T. E. Forster and maintained by Randall Holmes.