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.
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3w w e z 3z t m Alonzo Church Assume axiom scheme binary relation Church's construction claim class model 21 class model defined class of E-members clause g yl clauses viii completes the proof DOCTOR OF PHILOSOPHY E-subsets EMERSON CLARK MITCHELL empty set extensionality f and h Hence Vz imply inconsistent comprehension induction hypothesis induction step inductive definition internal model MODEL OF SET model of Z-F non-replacement A-object notation ordered pair paper power object power set predicate proper class Reed College replacement object set of objects Sublemma subset comprehension schema t m 3 zEx theorem trivial unions and intersections UNIVERSAL SET universe of Z-F Vw wEx Vz e m zSx Vz x e z Vz z e m Vz zEx well-founded set WF(u WF(x x y 3z zEx x y x y x'Sy y^Ex by clause Zermelo-Fraenkel set theory zEx1 zEy by contraposition