Elementary epimorphisms between models of set theory

Abstract

We show that every $\varPi _1$-elementary epimorphism between models of $\mathrm{ZF}$ is an isomorphism and hence, trivial. On the other hand, nonisomorphic $\varSigma _1$-elementary epimorphisms between models of $\mathrm{ZF}$ can be constructed, as can fully elementary epimorphisms between models of $\mathrm{ZFC}^-$ (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 $\mathrm{ZFC}^-$ 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 $f{: } M \rightarrow N$ 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 $\varGamma$, a $\varGamma$-elementary epimorphism is defined by restricting the above definition to this class of formulas.