Almost Galois $\omega$-Stable classes

by Baldwin and Larson and Shelah. [BlLrSh:1003]

Theorem. Suppose that an aleph_0-presentable Abstract Elementary Class (AEC), boldmath {K}, has the joint embedding and amalgamation properties in aleph_0 and <2^{aleph_1} models in aleph_1 . If boldmath {K} has only countably many models in aleph_1, then all are small. If, in addition, boldmath {K} is almost Galois omega-stable then boldmath {K} is Galois omega-stable.

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