Sh:331

• Shelah, S. A complicated family of members of trees with \omega +1 levels. Preprint. arXiv: 1404.2414
  Ch. VI of The Non-Structure Theory" book [Sh:e]
• Abstract:
  Our main aim is to prove that if T is a complete first-order theory, which is not superstable (no knowledge on this notion is required), included in a first-order theory T_1 then for any \lambda > |T_1| there are 2^\lambda models of T_1 such that for any two of them, the \tau(T)-reducts of one is not elementarily embeddable into the \tau(T)-reduct of the other, thus completing the investigation of the 1978 author’s book “Classification Theory and the Number of Non-Isomorphic Models”. Note the difference with the case of unstable T: there \lambda \ge |T_1| + \aleph_{1} suffices for the existence of 2^{\lambda} pairwise non-isomorphic such models.

  As earlier, it suffices for every such \lambda to find a complicated enough family of trees with \omega + 1 levels of cardinality \lambda. If \lambda is regular this is done already in Chapter VIII of the author’s book. The proof here (in sections 1 and 2) goes by dividing into cases, each with its own combinatorics. In particular, we have to use guessing clubs which was discovered for this aim.

  In §3 we improve the combinatorics, an aim is to consider strongly \aleph_\epsilon-saturated models of stable T (so if you do not know stability better just ignore this). We also deal with separable reduced Abelian p-groups. We then deal with various improvements of the earlier combinatorial results.

• Version 2026-04-21_2 (57p)

@article{Sh:331,
 author = {Shelah, Saharon},
 title = {{A complicated family of members of trees with $ \omega +1 $ levels}},
 note = {\href{https://arxiv.org/abs/1404.2414}{arXiv: 1404.2414} Ch. VI of The Non-Structure Theory" book [Sh:e]},
 arxiv_number = {1404.2414},
 refers_to_entry = {Ch. VI of The Non-Structure Theory" book [Sh:e]}
}