An abstract elementary class (AEC) is a class of structures of a fixed
vocabulary satisfying some natural closure properties. These classes encompass
the normal classes defined in model theory and natural
examples arise from mathematical practice, e.g. in algebra
not to mention first order and infinitary logics.
An AEC is always
endowed with a special substructure relation which is not always the obvious
one. Abstract
elementary classes provide one way out of the cul de sac of the model theory of
infinitary
languages which arose from over-concentration on syntactic criteria.
This is the second volume of a two-volume monograph on abstract elementary
classes. It is quite
self-contained and deals with three separate issues. The first is the topic of
universal classes,
i.e. classes of structures of a fixed vocabulary such that a structure belongs
to the class if and
only if every finitely generated substructure belongs. Then we derive
from an assumption on the number of models, the existence of an (almost) good
frame. The notion of frame is a natural generalization of the first order
concept of superstability to this context. The assumption
says that the weak GCH holds for
a cardinal $\lambda$, its successor and double successor, and the class is
categorical in the
first two, and has an intermediate value for the number of models in the third.
In particular, we can conclude from this argument
the existence of a model in the next cardinal. Lastly we deal with the
non-structure part of the
topic, that is, getting many non-isomorphic models in the double successor of $
\lambda$ under
relevant assumptions, we also deal with almost good frames themselves and some
relevant set theory.