Examples in the theory of existential completeness

Abstract

Since A. Robinson introduced the classes of existentially complete and generic models, conditions which were interesting for elementary classes were considered for these classes. In [6] H. Simmons showed that with the natural definitions there are prime and saturated existentially complete models and these are very similar to their elementary counterparts which were introduced by Vaught [2, 2.3]. As Example 6 will show, there is a limit to the similarity—there are theories which have exactly two existentially complete models.In [6] H. Simmons considers the following list of properties, shows that each property implies the next one and asks whether any of them implies the previous one:1.1. T is ℵ0-categorical.1.2. T has an ℵ0-categorical model companion.1.3. ∣E∣ = 1.1.4. ∣E∣ < .1.5. T has a countable ∃-saturated model.1.6. T has a ∃-prime model.1.7. Each universal formula is implied by a ∃-atomic existential formula.[The reader is referred to [1], [3], [4] and [6] for the definitions and background.We only mention that T is always a countable theory. All the models under discussion are countable. Thus E is the class of countable existentially complete models and F and G, respectively, are the classes of countable finite and infinite generic models. For every class C,∣C∣ is the number of (countable) models in C.]