A Model of Set Theory with a Universal Set.

Let T be the theory of the language of set theory saying that: (1) Sets are extensional; (2) Every set has a universal complement, i.e. given a set x there is a set y such that every set z is a member of y if and only if it is not a member of x; (3) Every set x has a power set containing exactly the subsets of x; (4) The result of replacing every member of a well-founded set by some set is a set; (5) The well-founded sets form a model of Zermelo-Fraenkel set theory. Then within the universe V of Zermelo-Fraenkel set theory there is a definable internal model of T. The members of the internal model are chosen by an inductive definition within V, and then a new membership relation is inductively defined such that the members of the internal model with the defined membership relation satisfy T. It happens that the members of the internal model which are well-founded on the defined membership relation form an isomorphic copy of V. Thus one can regard the construction as an extension of V to a model of T. Corollary: T is consistent if and only if Zermelo-Fraenkel set theory is consistent.