### 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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