Vaught's Test

Let be an -Theory with no finite Models. Suppose there is some such that: any two models of with cardinality are Elementary Equivalent Then is a Complete Theory.

Proof

Assume for contradiction that is not complete. Then there is a sentence such that and are consistent. Because has no finite models, there are infinite models and of and respectively. By The Löwenheim-Skolem Theorems, these theories have models of size . But then they are Elementary Equivalent which is a contradiction.