Will Boney

TAMENESS FROM LARGE CARDINAL AXIOMS

We show that Shelahs Eventual Categoricity Conjecture follows from the existence of class many strongly compact cardinals. This is the first time the consistency of this conjecture has been proven. We do so by showing that every AEC with

Canonical forking in AECs

LS(K)

Tameness and extending frames

below a strongly compact cardinal

Chains of saturated models in AECs

\kappa

TAMENESS AND FRAMES REVISITED

is

Forking in Short and Tame Abstract Elementary Classes

< \kappa

Superstability from categoricity in abstract elementary classes

tame and applying the categoricity transfer of Grossberg and VanDieren. These techniques also apply to measurable and weakly compact cardinals and we prove similar tameness results under those hypotheses. We isolate a dual property to tameness, called \emph{type shortness}, and show that it follows similarly from large cardinals.

A presentation theorem for continuous logic and Metric Abstract Elementary Classes

Boney and Grossberg (BG) proved that every nice AEC has an in- dependence relation. We prove that this relation is unique: In any given AEC, there can exist at most one independence relation that satisfies existence, exten- sion, uniqueness and local character. While doing this, we study more generally properties of independence relations for AECs and also prove a canonicity re- sult for Shelahs good frames. The usual tools of first-order logic (like the finite equivalence relation theorem or the type amalgamation theorem in simple theo- ries) are not available in this context. In addition to the loss of the compactness theorem, we have the added difficulty of not being able to assume that types are sets of formulas. We work axiomatically and develop new tools to understand this general framework.

Computing the Number of Types of Infinite Length

We combine two notions in AECs, tameness and good

μ-Abstract elementary classes and other generalizations ☆

\lambda