- 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 isWillard Van Orman Quine ’sNew Foundations , butAlonzo 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 andrandom variables , the Universal set, also called thesample 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 thecumulative 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.

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

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