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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