### Abstract elementary classes near $\aleph_1$

by Shelah. [Sh:88r]

We prove in ZFC, no psi in L_{omega_1, omega}[mathbf Q]
have unique model of uncountable cardinality, this confirms the
Baldwin conjecture. But we analyze this in more general terms. We
introduce and investigate a.e.c. and also versions of limit models,
and prove some basic properties like representation by PC class, for
any a.e.c. For PC_{aleph_0}-representable a.e.c. we investigate
the conclusion of having not too many non-isomorphic models in
aleph_1 and aleph_2, but have to assume 2^{aleph_0} <
2^{aleph_1} and even 2^{aleph_1} < 2^{aleph_2} .

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