### 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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