Abstract classes with few models have ‘homogeneous-universal’ models

Abstract

This paper is concerned with a class K of models and an abstract notion of submodel ≤. Experience in first order model theory has shown the desirability of finding a ‘monster model’ to serve as a universal domain for K. In the original constructions of Jónsson and Fraïssé, K was a universal class and ordinary substructure played the role of ≤. Working with a cardinal λ satisfying λ<λ = λ guarantees appropriate downward Löwenheim-Skolem theorems; the existence and uniqueness of a homogeneous-universal model appears to depend centrally on the amalgamation property. We make this apparent dependence more precise in this paper.The major innovation of this paper is the introduction of a weaker notion (chain homogeneous-universal) to replace the natural notion of (K, <)-homogeneous-universal model. Modulo a weak extension of ZFC (provable if V = L), we show (Corollary 5.24) that a class K obeying certain minimal restrictions satisfies a fundamental dichotomy. For arbitrarily large λ, either K has the maximal number of models in power λ or K has a unique chain homogeneous-universal model of power λ. We show (5.25) in a class with amalgamation this dichotomy holds for the notion of K-homogeneous-universal model in the more normal sense.The methods here allow us to improve our earlier results [5] in two other ways: certain requirements on all chains of a given length are replaced by requiring winning strategies in certain games; the notion of a canonically prime model is avoided. A full understanding of these extensions requires consideration of the earlier papers but we summarize them quickly here.