Set Theories With A Universal Set
There are set theories known to be consistent (if the usual set theory is consistent) in which the universal set V does exist (and is true). In these theories, Zermelo's axiom of separation does not hold in general, and the axiom of comprehension of naive set theory is restricted in a different way.
The most widely studied set theory with a universal set is Willard Van Orman Quine’s New Foundations. 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, but this is not possible for Oberschelp’s, since in it the singleton function is provably a set, which leads immediately to paradox in New Foundations.
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.
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