Categoricity in Abstract Elementary Classes: Going up Inductive Step Sh600 -part 1 and 2

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 Lω1,ω. We introduce and investigate good λ-frame, show that they can be found under reasonable assumptions and prove we can advance from λ to λ+ when non-structure fail. That is, assume 2 +n < 2 +n+1 for n < ω. So if an a.e.c. is cateogorical in λ, λ+ and has intermediate number of models in λ++ and 2 < 2 + < 2 ++ , LS(K) ≤ λ). Then there is a good λ-frame s and if s fails non-structure in λ++ then s has a successor s+, a good λ+-frame hence K λ+3 6= ∅, and we can continue. 2000 Mathematics Subject Classification. 03C45, 03C75, 03C95, 03C50.