Sh:363

• Shelah, S. On spectrum of \kappa-resplendent models. Preprint. arXiv: 1105.3774
  Ch. V of [Sh:e]
• Abstract:
  We prove that some natural “outside" property of counting models up to isomorphism is equivalent (for a first order class) to being stable.

  For a model, being resplendent is a strengthening of being \kappa-saturated. Restricting ourselves to the case \kappa > |T| for transparency, to say a model M is \kappa-resplendent means:

  when we expand M by < \kappa individual constants \langle c_i : i < \alpha \rangle, if (M, c_i)_{ < \alpha } has an elementary extension expandable to be a model of T' where {\rm Th}((M, c_i)_{i < \alpha} ) \subseteq T', |T'| < \kappa then already (M, c_i)_{i < \alpha} can be expanded to a model of T' .

  Trivially, any saturated model of cardinality \lambda is \lambda-resplendent. We ask: how many \kappa-resplendent models of a (first order complete) theory T of cardinality \lambda are there? We restrict ourselves to cardinals \lambda = \lambda^\kappa + 2^{|T|} and ignore the case \lambda = \lambda^{<\kappa} + |T| < \lambda^\kappa. Then we get a complete and satisfying answer: this depends only on T being stable or unstable. In this case proving that for stable T we get few, is not hard; in fact, every resplendent model of T is saturated hence it is determined by its cardinality up to isomorphism. The inverse is more problematic because naturally we have to use Skolem functions with any \alpha < \kappa places. Normally we use relevant partition theorems (Ramsey theorem or Erdős-Rado theorem), but in our case the relevant partitions theorems fail so we have to be careful.

• Version 2023-11-28 (29p)

@article{Sh:363,
 author = {Shelah, Saharon},
 title = {{On spectrum of $\kappa $-resplendent models}},
 note = {\href{https://arxiv.org/abs/1105.3774}{arXiv: 1105.3774} Ch. V of [Sh:e]},
 arxiv_number = {1105.3774},
 refers_to_entry = {Ch. V of [Sh:e]}
}