Sh:88r

• Shelah, S. (2009). Abstract elementary classes near \aleph_1. In Classification theory for abstract elementary classes, Vol. 18, College Publications, London, p. vi+813. arXiv: 0705.4137
  Ch. I of [Sh:h]
• Abstract:
  We prove, in ZFC, that no \psi \in \mathbb{L}_{\omega_1,\omega}[\mathbf{Q}] have unique models of uncountable cardinality; this confirms the Baldwin conjecture. But we analyze this in more general terms. We introduce and investigate AECs and also versions of limit models, and prove some basic properties like representation by a PC class, for any AEC.

  For PC_{\aleph_0}-representable AECs we investigate the conclusion of having not too many non-isomorphic models in \aleph_1 and \aleph_2, but we have to assume 2^{\aleph_0} < 2^{\aleph_1} and even 2^{\aleph_1} < 2^{\aleph_2}.

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@incollection{Sh:88r,
 author = {Shelah, Saharon},
 title = {{Abstract elementary classes near $\aleph_1$}},
 booktitle = {{Classification theory for abstract elementary classes}},
 series = {Studies in Logic (London)},
 volume = {18},
 year = {2009},
 pages = {vi+813},
 isbn = {978-1-904987-71-0},
 publisher = {College Publications, London},
 mrclass = {03-02 (03C45 03C48)},
 note = {\href{https://arxiv.org/abs/0705.4137}{arXiv: 0705.4137} Ch. I of [Sh:h]},
 arxiv_number = {0705.4137},
 refers_to_entry = {Ch. I of [Sh:h]}
}