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Monica VanDieren - Rutgers Abstract
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E-mail:
monica@math.stanford.edu
Homepage:
http://math.stanford.edu/~monica/home.html.
Title:
Towards a Categoricity Theorem for Abstract Elementary Classes.
Abstract:
We continue work of Shelah
and Villaveces under GCH in abstract elementary classes with no
maximal models. We are particularly interested in the Los-Shelah
(categoricity transfer) Conjecture. Two main steps in partial results
have included
- proving the categoricity model is saturated (in the sense of
Galois-types) and
- proving the
uniqueness of saturated models.
While these statements are relatively easy to derive under the full
amalgamation property, we do not have the amalgamation property at our
disposal. Thus, an intermediary step of Shelah and Villaveces was
to prove the uniqueness of limit models (as a substitute for saturation).
While they were able to make progress towards the categoricity conjecture,
Shelah and Villaveces left open the problem of proving the uniqueness of
limit models.
We are interested in proving the uniqueness of limit models. While this
result has played a 'behind-the-scenes' role in Shelah's proof of the
categoricity conjecture for classes with the amalgamation property, the
uniqueness of limit models was critical in Kolman and Shelah's proof of
the amalgamation property in categorical $L_{\kappa,\omega}$ theories, when
$\kappa$ is a measurable cardinal.
Our attempt at proving the uniqueness of
limit models can be broken down into two parts, prove:
- If $\K$ is categorical, then the categoricity model
is weakly model homogeneous;
- The uniqueness of limit models under the assumption that
the categoricity model is weakly model homogeneous.
We introduce the notion of weak model homogeneity as a substitute for model
homogeneity and describe its connection with other conventional properties
such as saturation (with respect to Galois-types) and amalgamation.
We will provide a proof to (2) using a weak diamond. A summary of partial
results towards (1) will conclude the talk.
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Slides from the talk. Note: DVI file requires xy-pic.
Copyright Monica VanDieren.
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