Sh:1261

• Shelah, S. Potentially Isomorphic; Only after forcing. 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. We can demand that the models are not just non-isomorphic, but far from each other (in a suitable sense). We give sufficient conditions: first, 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 forcings. Best would be to characterize the pairs (T,\mathbb{P}) for which we have such models.

  Then we prove this for unstable theories, but for a smaller class of cardinals.

  The case of (e.g.) Random Real forcing remains opaque, even for independent T-s, but presently we have solved it for the so-called \omega-independent case.

  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). This is part of the Main Gap program.

  While stable and unstable theories are discussed, no knowledge of stability theory is required to read this paper.

• No downloadable versions available.

@article{Sh:1261,
 author = {Shelah, Saharon},
 title = {{Potentially Isomorphic; Only after forcing}}
}