Monica M. VanDieren

GALOIS-STABILITY FOR TAME ABSTRACT ELEMENTARY CLASSES

We introduce tame abstract elementary classes as a generalization of all cases of abstract elementary classes that are known to permit development of stability-like theory. In this paper, we explore stability results in this new context. We assume that is a tame abstract elementary class satisfying the amalgamation property with no maximal model. The main results include:. Theorem 0.1. Suppose that is not only tame, but -tame. If and is Galois stable in μ, then , where is a relative of κ(T) from first order logic. is the Hanf number of the class . It is known that . The theorem generalizes a result from [17]. It is used to prove both the existence of Morley sequences for non-splitting (improving [22, Claim 4.15] and a result from [7]) and the following initial step towards a stability spectrum theorem for tame classes:. Theorem 0.2. If is Galois-stable in some , then is stable in every κ with κμ=κ. For example, under GCH we have that Galois-stable in μ implies that is Galois-stable in μ+n for all n < ω.

CATEGORICITY FROM ONE SUCCESSOR CARDINAL IN TAME ABSTRACT ELEMENTARY CLASSES

We prove that from categoricity in λ+ we can get categoricity in all cardinals ≥ λ+ in a χ-tame abstract elementary classe

Categoricity in abstract elementary classes with no maximal models

\mathcal{K}

Superstability and symmetry

which has arbitrarily large models and satisfies the amalgamation and joint embedding properties, provided

Symmetry in abstract elementary classes with amalgamation

\lambda > {\rm LS}(\mathcal{K})

Symmetry and the union of saturated models in superstable abstract elementary classes

and λ ≥ χ. For the missing case when

Superstability from categoricity in abstract elementary classes

\lambda ={\rm LS}(\mathcal{K})

Union of Saturated Models in Superstable Abstract Elementary Classes

, we prove that

Effectiveness of Online Advising on Honors Student Retention and Engagement

\mathcal{K}

Shelah's Categoricity Conjecture from a Successor for Tame Abstract Elementary Classes

is totally categorical provided that