Sh:1261

• Shelah, S. Twins: non-isomorphic models forced to be isomorphic, Part I. Preprint.
• Abstract:
  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 \mathbb{P}? Necessarily, \mathbb{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, \mathcal{L}-equivalent in some suitable logic \mathcal{L}. We give sufficient conditions: for theories with the independence property, we proved this when \mathbb{P} adds no new \omega-sequence. We may prove it “for some \mathbb{P}," but better would be for some specific forcing notions (or for a natural family). Best would be to characterize the pairs (T,\mathbb{P}) for which we have such models.

  This work does not require any knowledge of forcings. The results say (e.g.) that there are models M_1,M_2 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 ({}^{\theta>}2,\lhd). No knowledge of stability theory is required to read this paper.

  This is part of the classification and Main Gap program.

• Version 2025-04-17_2 (34p)

@article{Sh:1261,
 author = {Shelah, Saharon},
 title = {{Twins: non-isomorphic models forced to be isomorphic, Part I}}
}