Sh:600

• Shelah, S. (2009). Categoricity in abstract elementary classes: going up inductively. In Classification Theory for Abstract Elementary Classes. arXiv: math/0011215
  Ch. II of [Sh:h]
• Abstract:
  We deal with beginning stability theory for “reasonable" non-elementary classes without any remnants of compactness like dealing with models above Hanf number or by the class being definable by \mathbb L_{\omega_1,\omega}. We introduce and investigate good \lambda-frame, show that they can be found under reasonable assumptions and prove we can advance from \lambda to \lambda^+ when non-structure fail. That is, assume 2^{\lambda^{+n}} < 2^{\lambda^{+n+1}} for n < \omega. So if an a.e.c. is cateogorical in \lambda,\lambda^+ and has intermediate number of models in \lambda^{++} and 2^\lambda < 2^{\lambda^+} < 2^{\lambda^{++}}, LS(\mathfrak{K}) \le \lambda). Then there is a good \lambda-frame \mathfrak{s} and if \mathfrak{s} fails non-structure in \lambda^{++} then \mathfrak{s} has a successor \mathfrak{s}^+, a good \lambda^+-frame hence K^\mathfrak{s}_{\lambda^{+3}} \ne \emptyset, and we can continue.
• Version 2023-06-18 (101p)

@inbook{Sh:600,
 author = {Shelah, Saharon},
 title = {{Categoricity in abstract elementary classes: going up inductively}},
 booktitle = {{Classification Theory for Abstract Elementary Classes}},
 year = {2009},
 note = {\href{https://arxiv.org/abs/math/0011215}{arXiv: math/0011215} Ch. II of [Sh:h]},
 arxiv_number = {math/0011215},
 refers_to_entry = {Ch. II of [Sh:h]}
}