A Model of Set Theory with a Universal SetLet 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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Alonzo Church Aooo(x argument Arts UW Assume axiom scheme Azja binary relation Church's construction claim class model defined class of E-members clause g clauses viii complement defined membership relation DF VW E-subsets EMERSON CLARK MITCHELL empty set extensionality g by clause imply induction hypothesis induction step inductive definition internal model j-cardinal numbers MODEL OF SET model of Z-F notation pair is decided paper power object power set predicate Proof proper class Reed College relativize all quantifiers replacement object set of A-objects Sublemma subset comprehension schema theorem thesis trivial unions and intersections UNIVERSAL SET universe of Z-F UNIVERSITY OF WISCONSIN-MADISON Vw wEx Vx Vy(A(x Vz zEx Vzem well-founded set WF(u WF(x x'Sx x'Sy Xi+m xSy or xEy y,EPx Y₁ y₁EPx y₁Ex y₁Sx Z-F thinks Zermelo-Fraenkel set theory zEx by clause zEy by contraposition zły ʼn R(a)²