Elementary epimorphisms between models of set theory

We show that every _1 -elementary epimorphism between models of ZF is an isomorphism and hence, trivial. On the other hand, nonisomorphic _1 -elementary epimorphisms between models of ZF can be constructed, as can fully elementary epimorphisms between models of ZFC^- (a formulation of ZFC without powerset). We construct examples of such elementary epimorphisms. We also construct an inverse-directed system of elementary epimorphisms between models of ZFC^- and determine the inverse limit of this system. Elementary epimorphisms were introduced by Rothmaler (J Symb Log 70(2):473–488, 2005 ). An elementary epimorphism is somewhat like an elementary embedding taken in reverse. To be precise, a surjective homomorphism f: M → N between two model-theoretic structures is an elementary epimorphism if and only if every formula with parameters satisfied by N is satisfied in M using a preimage of those parameters. Given a class of formulas , a -elementary epimorphism is defined by restricting the above definition to this class of formulas.