Contemporary Mathematics
Volume 380, 2005
Ehrenfeucht-Mostowski models in Abstract Elementary
Classes
John T. Baldwin
ABSTRACT.
Let K be an abstract elementary class. We give a unified and
often simplified exposition of a number of results of Shelah. In particular,
we prove the presentation theorem.
If
K has the amalgamation property,
the joint embedding property and arbitrarily large models, then an analog of
Morley's omitting types theorem for holds for Galois types. Further if K is
.\-categorical for a regular cardinal
.X, K
is stable in all cardinalities less than
.X.
We conclude by explaining the relevance of the notion of tameness.
We work in the context of an abstract elementary class (AEC) with the amalga-
mation and joint embedding properties and arbitrarily large models. We prove
two results using Ehrenfeucht-Mostowski models: 1) Morley's omitting types the-
orem- for Galois types. 2) If an AEC (with amalgamation) is categorical in some
uncountable power
J1,
it is stable in (every) ,\
Jl,.
These results are lemmas towards Shelah's consideration
[14]
of the downward
transfer of categoricity, which we discuss in Section 6. This paper expounds some
of the main ideas of
[14],
filling in vague allusions to earlier work and trying to
separate those results which depend only on the Ehrenfeucht-Mostowski method
from those which require more sophisticated stability theoretic tools.
In
[17],
Shelah proclaims the aim of reconstructing model theory, 'with no use
of even traces compactness'. We analyze here one aspect of this program. Keisler
organizes
[10]
around four kinds of constructions: the Henkin method, Ehrenfeucht-
Mostowski models, unions of chains, and ultraproducts. The later history of model
theory reveals a plethora of tools arising in stability theory. Fundamental is a notion
of dependence which arises from Morley's study of rank, and passes through various
avatars of splitting, strong splitting, and dividing before being fully actualized in
the first order setting as forking. We eschew this technique altogether in this paper-
to isolate its role. For more general accounts of Abstract Elementary Classes, see
[1, 8].
Partially supported by NSF grant DMS-0100594.
©
2005 American Mathematical Society
http://dx.doi.org/10.1090/conm/380/07103
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