Introduction to: classification theory for abstract elementary class

TL;DR

This paper introduces the classification theory for abstract elementary classes, extending model theory concepts beyond first-order logic to more general classes of structures, aiming to develop a non-elementary classification framework.

Contribution

It provides an introductory overview of classification theory for abstract elementary classes, aiming to establish foundational results without relying on first-order logic assumptions.

Findings

Framework for non-elementary classification theory

Extension of stability and categoricity concepts

Foundational overview for future detailed development

Abstract

Classification theory of elementary classes deals with first order (elementary) classes of structures (i.e. fixing a set T of first order sentences, we investigate the class of models of T with the elementary submodel notion). It tries to find dividing lines, prove their consequences, prove "structure theorems, positive theorems" on those in the "low side" (in particular stable and superstable theories), and prove "non-structure, complexity theorems" on the "high side". It has started with categoricity and number of non-isomorphic models. It is probably recognized as the central part of model theory, however it will be even better to have such (non-trivial) theory for non-elementary classes. Note also that many classes of structures considered in algebra are not first order; some families of such classes are close to first order (say have kind of compactness). But here we shall deal with a classification theory for the more general case without assuming knowledge of the first order case (and in most parts not assuming knowledge of model theory at all). The present paper includes an introduction to the forthcoming book on Classification Theory for Abstract Elementary Classes