Universal set

Universal set

In set theory, a universal set is a set which contains all objects, including itself. [Forster 1995 p. 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, [Church 1974 p. 308, but see also Forster 1995 p. 136 or 2001 p. 17.] but this is not possible for Oberschelp’s, since in it the singleton function is provably a set, [Oberschelp 1973 p. 40.] which leads immediately to paradox in New Foundations. [ Holmes 1998 p. 110.]

In probability theory and random variables, the Universal set, also called the sample space, is the one that contains all conceivable events that might possibly happen.

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.

See also

* Universe
* Set of all sets

References

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.

*

* [http://www.dpmms.cam.ac.uk/~tf/ T. E. Forster] (2001). [http://www.dpmms.cam.ac.uk/~tf/church2001.ps “Church’s Set Theory with a Universal Set.”]

* [http://math.boisestate.edu/~holmes/holmes/setbiblio.html Bibliography: Set Theory with a Universal Set,] originated by T. E. Forster and maintained by Randall Holmes at Boise State University.

* [http://math.boisestate.edu/~holmes Randall Holmes] (1998). " [http://math.boisestate.edu/~holmes/holmes/head.ps Elementary Set theory with a Universal Set,] " volume 10 of the Cahiers du Centre de Logique, Academia, Louvain-la-Neuve (Belgium).

* Arnold Oberschelp (1973). “Set Theory over Classes,” "Dissertationes Mathematicae" 106.

* Willard Van Orman Quine (1937) “New Foundations for Mathematical Logic,” "American Mathematical Monthly" 44, pp. 70-80.


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