Categoricity and solvability of A.E.C., quite highly

by Shelah. [Sh:734]

We investigate in ZFC what can be the family of large enough cardinals mu in which an a.e.c. K is categorical or even just solvable. We show that for not few cardinals lambda < mu there is a superlimit model in K_lambda . Moreover, our main result is that we can find a good lambda-frame s categorical in lambda such that K_s subseteq K_lambda . We then show how to use {705} to get categoricity in every large enough cardinality if K has cases of mu-amalgamation for enough mu and 2^mu < 2^{mu^{+1}} < ... < 2^{mu^{+n}} ... for enough mu .

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