### Toward classification theory of good $\lambda$ frames and abstract elementary classes

by Shelah. [Sh:705]

Our main aim is to investigate a good lambda-frame s which is as in the end of {600}, i.e. s is n-successful for every n (i.e. we can define a good lambda^{+n}-frame s^{+n} such that s^{+0} = s, s^{+(n+1)} = (s^{+n})^+). We would like to prove then K^s has model in every cardinal > lambda, and it is categorical in one of them iff it is categorical in every one of them. For this we shall show that K_{s^{+n}} 's are similar to superstable elementary classse with prime existence. (Actually also K^s_{>= lambda^{+ omega}}, but the full proof are delayed).

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