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.

Solution

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

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A theory is model-complete when every embedding between its models is elementary. Every model-complete theory is inductive and hence admits a forall-exists axiomatization.


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