Elementary Subsystems of Henkin-Keisler Models I

Abstract

In this chapter we investigate elementary subsystems of Henkin-Kesler models. Since finite interpretations have no elementary subsystems different from themselves, attention is focused on infinite Henkin-Keisler models. Given any infinite interpretation of type K, the Dawnward Löwenheim-Skolem Theorem (Theorem 1.1.1) guarantees the existence of elementary subsystems in each cardinality strictly smaller than the cardinality of the given interpretation and greater than the cardinality of the language over K. A fortiori, infinite Henkin-Keisler models have elementray subsystems in each such cardinality. When K is countable, no infinite interpretation of type K can have finite elementary subsysyems. However, when K is uncountable, certain Henkin-Keisler models have elementary subsystems strictly smaller than the cardinality of the language over K.