Tame abstract elementary class
In model theory, a discipline within the field of mathematical logic, a tame abstract elementary class is an abstract elementary class which satisfies a locality property for types called tameness. Even though it appears implicitly in earlier work of Shelah, tameness as a property of AEC was first isolated by Grossberg and VanDieren, who observed that tame AECs were much easier to handle than general AECs.
Definition
Let K be an AEC with joint embedding, amalgamation, and no maximal models. Just like in first-order model theory, this implies K has a universal model-homogeneous monster model. Working inside, we can define a semantic notion of types by specifying that two elements a and b have the same type over some base model if there is an automorphism of the monster model sending a to b fixing pointwise. Such types are called Galois types.One can ask for such types to be determined by their restriction on a small domain. This gives rise to the notion of tameness:
- An AEC is tame if there exists a cardinal such that any two distinct Galois types are already distinct on a submodel of their domain of size. When we want to emphasize, we say is -tame.
Discussion and motivation
While there are examples of non-tame AECs, most of the known natural examples are tame. In addition, the following sufficient conditions for a class to be tame are known:Tameness is a large cardinal axiom: There are class-many almost strongly compact cardinals if and only if any abstract elementary class is tame.Some tameness follows from categoricity: If an AEC with amalgamation is categorical in a cardinal of high-enough cofinality, then tameness holds for types over saturated models of size less than.Conjecture 1.5 in : If K is categorical in some λ ≥ Hanf then there exists χ < Hanf such that K is χ-tame.Many results in the model theory of AECs assume weak forms of the Generalized continuum hypothesis and rely on sophisticated combinatorial set-theoretic arguments. On the other hand, the model theory of tame AECs is much easier to develop, as evidenced by the results presented below.