Sh:300a

• Shelah, S. (2009). Universal Classes: Stability theory for a model. In Classification Theory for Abstract Elementary Classes II.
  Ch. V of [Sh:i]
• Abstract:
  We deal with a universal class of models K, (i.e. a structure \in K iff any finitely generated substructure \in K). We prove that either in K there are long orders (hence many complicated models) or K, under suitable order \le_{\mathfrak s} is an a.e.c. with some stability theory built in. For this we deal with the existence of indiscernible sets and (introduce and prove existence) of convergence sets. Moreover, improve the results on the existence of indiscernible sets such that for some first order theories, we get strong existence results for set of elements, whereas possibly for some sets of n-tuples this fails. In later sub-chapters we continue going up in a spiral - getting either non-structure or showing closed affinity to stable, but the dividing lines are in general missing for first order classes.
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@inbook{Sh:300a,
 author = {Shelah, Saharon},
 title = {{Universal Classes: Stability theory for a model}},
 booktitle = {{Classification Theory for Abstract Elementary Classes II}},
 year = {2009},
 note = {Ch. V of [Sh:i]},
 refers_to_entry = {Ch. V of [Sh:i]}
}