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Theorem. Suppose that k = (K, $$\prec_k$$ ) is an 0-presentable abstract elementary class with Löwenheim–Skolem number 0, satisfying the joint embedding and amalgamation properties in 0. If K has only countably many models in 1, then all are small. If, in addition, k is almost Galois ω-stable then k is Galois ω-stable. Suppose that k = (K, $$\prec_k$$ ) is an 0-presented almost Galois ω-stable AEC satisfying amalgamation for countable models, and having a model of cardinality 1. The assertion that K is 1-categorical is then absolute.



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