### Categoricity of an abstract elementary class in two successive cardinals

by Shelah. [Sh:576]

Israel J Math, 2001

We investigate categoricity of abstract elementary classes
without any remnants of compactness (like non-definability of well
ordering, existence of E.M. models or existence of large cardinals).
We prove (assuming a weak version of GCH around lambda) that if
K is categorical in lambda, lambda^+, LS(K)
<= lambda and 1 <= I(lambda^{++}, K)< 2^{lambda^{++}}
then K has a model in lambda^{+++} .

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