Twins: non-isomorphic models forced to be isomorphic Part I

For which (first-order complete, usually countable) T do there exist non-isomorphic models of T which become isomorphic after forcing with a forcing notion P? Necessarily, P is non-trivial; i.e it adds some new set of ordinals. It is best if we also demand that it collapses no cardinal. It is better if we demand on the one hand that the models are non-isomorphic, and even far from each other (in a suitable sense), but on the other hand, L-equivalent in some suitable logic L. In this part we give sufficient conditions: for theories with the independence property, we prove this when P adds no new ω-sequence. We may prove it “for some P," but better would be for some specific forcing notions, or a natural family. Best would be to characterize the pairs (T,P) for which we have such models. The results say (e.g.) that there are models M1,M2 which are not isomorphic (and even far from being isomorphic, in a rigorous sense) which become isomorphic when we extend the universe by adding a new branch to the tree (θ>2,⊲). We shall mention some specific choices of P: mainly (θ>2,⊲) with θ=θ<θ. This work does not require any serious knowledge of forcings, nor of stability theory, though they form the motivation. Concerning forcing, the reader just has to agree that starting with a universe V of set theory (i.e. a model of ZFC) and a quasiorder P, there are a new directed G⊆P meeting every dense subset D of P and a universe V[G] (so it satisfies ZFC) of which the original V is a transitive subclass. We may say that V[G] (also denoted VP) is the universe obtained by forcing with P. This is part of the classification and so-called Main Gap programs.