Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 120 / 1 / d / Solution

A model-complete theory is one for which every embedding between models is elementary. If $(M_i)$ is an embedding chain of models, all transition embeddings are therefore elementary. The elementary chain theorem shows that the union is a model elementarily extending every $M_i$, so the theory is preserved under unions of embedding chains. Part (c) then gives an axiomatization by forall-exists sentences.