Categoricity of an abstract elementary class in two successive cardinals, revisited

by Shelah. [Sh:E46]

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 has intermediate number of models in lambda^{++}, then K has a model in lambda^{+++} .

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