ABSTRACT
Thus, we must abandon the ultimate axiom and settle for weaker forms. Namely, we may require that some definable subclass W of V, which is seen as an approximation to V, is sufficiently elementarily equivalent to W as computed in some generic extensions of V. Whenever W is a substructure of W as computed in the generic extensions, then one can even require that W be an elementary substructure of the generic W. That is, we may have elements of the ground-model W as parameters. Many axioms of this sort have been studied in the literature and are currently one of the main topics in the foundations of Set Theory. These are the axioms of generic absoluteness, which may also be called axioms of generic invariance, that is, they assert that whatever can be forced is true, subject only to the restrictions that are strictly necessary for them to be consistent with the axioms of ZFC.
