Theories with models of prescribed cardinalities

The Löwenheim–Skolem theorem states that if a theory has an infinite model it has models of all cardinalities greater than or equal to the cardinality of the language in which the theory is defined. A natural question is what happens if there is a model whose cardinality is less than that of the language. If κ is an infinite cardinal less than the first measurable cardinal and κ < κω, the Rabin–Keiler theorem [1, p. 139] gives an example of a theory which has a model of cardinality κ in which every element is the interpretation of a constant and all other models have cardinality μ ≥ κω. Keisler has also shown that if a theory has a model of cardinality κ it has models of all cardinalities μ ≥ κω. We will show that within the bounds of the above theorems anything can happen. The main result is as follows.