Small models

Abstract

This paper surveys some problems in the model theory of uncountable first order languages. These problems were first raised in Mal'cev. Their solutions involve the construction of infinite models which are "small" relative to the cardinality of their language. The most important of these problems concern extending the Upward Löwenheim- Skolem Theorem for uncountable languages. The earliest results relevant to this problem used the ultraproduct construction to obtain "small" elementary extensions. Stronger results have been obtained more recently by other methods. These are used to construct "small" elementary subsystems.