Sh:300c

• Shelah, S. (2009). Universal Classes: A frame is not smooth or not \chi-based. In Classification Theory for Abstract Elementary Classes II.
  Ch. V (C) of [Sh:i]
• Abstract:
  Our main aim is to prove that if a framework {\mathfrak s} (satisfying AxFr_1, of course) fail smoothness or being \chi_{\mathfrak s}-based then we have strong non-structure results. Smoothness say that the union of an increasing chain of \le_{\mathfrak s}-submodel of N is a \le_{\mathfrak s}-submodel of N. Now smoothness is obvious in elementary classes and even in abstract elementary classes. The second, \chi_{\mathfrak s}-based, saying there are many non-forking quadruples, for elementary classes it follows from the non-order property, but here also it becomes a dividing line. To enable us to prove the non-structure results, we have to develop some positive theory assuming only AxFr_1, using the closure instead of smoothness. Some more positive results are proved for {\mathfrak s} which satisfies smoothness (and AxFr_1).
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@inbook{Sh:300c,
 author = {Shelah, Saharon},
 title = {{Universal Classes: A frame is not smooth or not $\chi$-based}},
 booktitle = {{Classification Theory for Abstract Elementary Classes II}},
 year = {2009},
 note = {Ch. V (C) of [Sh:i]},
 refers_to_entry = {Ch. V (C) of [Sh:i]}
}