Universal set

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For "universal set" in the sense of the universe with respect to which an absolute set complement is taken, see Universe (mathematics).

In set theory, a universal set is a set which contains all objects, including itself.^[1] The most widely-studied set theory with a universal set is Willard Van Orman Quine’s New Foundations, but Alonzo Church and Arnold Oberschelp also published work on such set theories. Church speculated that his theory might be extended in a manner consistent with Quine’s,^[2] but this is not possible for Oberschelp’s, since in it the singleton function is provably a set,^[3] which leads immediately to paradox in New Foundations.^[4]

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.

[edit] See also

[edit] References

^ Forster 1995 p. 1.
^ Church 1974 p. 308, but see also Forster 1995 p. 136 or 2001 p. 17.
^ Oberschelp 1973 p. 40.
^ Holmes 1998 p. 110.

[edit] Bibliography

Alonzo Church (1974). “Set Theory with a Universal Set,” Proceedings of the Tarski Symposium. Proceedings of Symposia in Pure Mathematics XXV, ed. L. Henkin, American Mathematical Society, pp. 297-308.

T. E. Forster (1995). Set Theory with a Universal Set: Exploring an Untyped Universe (Oxford Logic Guides 31). Oxford University Press. ISBN 0-19-851477-8.

T. E. Forster (2001). “Church’s Set Theory with a Universal Set.”

Bibliography: Set Theory with a Universal Set, originated by T. E. Forster and maintained by Randall Holmes at Boise State University.

Randall Holmes (1998). Elementary Set theory with a Universal Set, volume 10 of the Cahiers du Centre de Logique, Academia, Louvain-la-Neuve (Belgium).