Isomorphism property in nonstandard extensions of ZFC universe

Abstract

We study models of HST, a nonstandard set theory which includes, in particular, the ZFC Replacement and Separation schemata in the language containing the membership and standardness predicates, and Saturation for well-orderable families of internal sets. This theory admits an adequate formulation of the isomorphism property IP: "any two elementarily equivalent internally presented structures of a wellorderable language are isomorphic." IP implies, for instance, that all infinite internal sets are equinumerous, and there exists a unique (modulo isomorphisms) internal elementary extension of the standard reals. We prove that IP is independent of HST (using the class of all sets constructible from internal sets) and consistent with HST (using generic extensions of $\HST$ models by a sufficient number of generic isomorphisms).