The Cardinality of Henkin-Keisler Models

Abstract

Let α be an infinite cardinal and T={λλ=α} be a family of interpretations of type K where for each λ=α card Aλ =βλ. Considers the problem of constructing a maximal entension of Th(T ^*) whose Henkin-Keisler model is of cardinality Пβλ. By results of the last chapter, this chapter problem is equivalent to the analogous problem for ultraproducts. while the techniques developed here for solving the first problem are essentially those for solving the second, the general approach is analogous to that used to construct a Henkin model of a prescribed cardinality. To get something of the flavor of this approach, we consider a problem which can be solved by using both Henkin models and Henkin- Keisler models.