Good Frames With A Weak Stability

TL;DR

This paper establishes conditions under which models in an abstract elementary class can be extended across increasing cardinalities, assuming limited models at higher levels, thus advancing understanding of model existence in such classes.

Contribution

It introduces new conditions on the base class models that ensure the existence of models at all higher cardinalities, extending previous stability results.

Findings

Models exist at all higher cardinalities under specified conditions.

Conditions on K_λ are inherited by subclasses, ensuring model existence.

Provides a framework for understanding model extension in abstract elementary classes.

Abstract

Let K be an abstract elementary class of models. Assume that there are less than the maximal number of models in K_{\lambda^{+n}} (namely models in K of power \lambda^{+n}) for all n. We provide conditions on K_\lambda, that imply the existence of a model in K_{\lambda^{+n}} for all n. We do this by providing sufficiently strong conditions on K_\lambda, that they are inherited by a properly chosen subclass of K_{\lambda^+}.