Categoricity in abstract elementary classes: going up inductively

by Shelah. [Sh:600]

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 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 (K) <= lambda) . Then there is a good lambda-frame s and if s fails non-structure in lambda^{++} then s has a successor s^+, a good lambda^+-frame hence K^s_{lambda^{+3}} ne emptyset, and we can continue.


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