Isomorphism property in nonstandard extensions of the ZFC universe Original Research Article

We study models of HST (a nonstandard set theory which includes, in particular, the Replacement and Separation schemata of ZFC 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 propertyIP, which postulates that any two elementarily equivalent internally presented structures of a well-orderable language are isomorphic. 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 a constructible model of HST by a sufficient number of generic isomorphisms).