Abstract elementary classes near aleph_1

TL;DR

This paper proves the Baldwin conjecture in ZFC, showing no models of uncountable size are unique in certain logics, and explores properties of abstract elementary classes and limit models.

Contribution

It introduces and studies a.e.c. and limit models, establishing their basic properties and representation, and investigates model diversity in aleph_1 and aleph_2 under set-theoretic assumptions.

Findings

No unique models of uncountable size in L_{omega_1,omega}[Q] in ZFC.

Basic properties of a.e.c. and limit models are established.

Analysis of non-isomorphic models in aleph_1 and aleph_2 under cardinal assumptions.

Abstract

We prove in ZFC, no psi in L_{omega_1,omega}[Q] have unique model of uncountable cardinality, this confirms theBaldwin conjecture. But we analyze this in more general terms. We introduce and investigate a.e.c. and also versions of limit models, and prove some basic properties like representation by PC class, for any a.e.c. For PC_{aleph_0}-representable a.e.c. we investigate the conclusion of having not too many non-isomorphic models in aleph_1 and aleph_2, but have to assume 2^{aleph_0}<2^{aleph_1} and even 2^{aleph_1}<2^{aleph_2}.