### A.E.C. with not too many models

by Shelah. [Sh:893]

Consider an a.e.c. K and the class of
{bold C}_{aleph_0} cardinals of cofinality aleph_0 .
A nicely stated consequence of this work is for some closed
unbounded class C of cardinals we have
(a) quad dot I(lambda, K) >= lambda for
lambda in C cap {bold C_{aleph_0}}
or
(b) quad if M in K_lambda and lambda in {bold
C}_{aleph_0}, then M has le_K-extension (so
in K) of arbitrarily large cardinals.

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