Past exam of the model theory course of the University of Oxford / 2011 / 2 / a

A complete theory is a consistent first-order theory deciding every sentence in its language:Equivalently, every two first-order models of $T$ are elementarily equivalent. Consistency excludes the vacuous situation in which a first-order theory has no first-order models and entails both alternatives.

A first-order theory has quantifier elimination if every first-order formula is equivalent over the first-order theory to a quantifier-free formula with the same free variables:Logical truth and falsity may be treated as the empty Boolean combinations when considering sentences in a language without closed terms. The assertion concerns first-order formulas with free variables, not merely sentences.

A model-complete theory is a first-order theory for which every structure embedding between two of its first-order models is an elementary embedding. Thus, if $f:\mathcal{A}\hookrightarrow \mathcal{B}$ is such an embedding, then for every first-order formula $\phi (\mathbf{x})$ and tuple $\mathbf{a}$ from $A$,first-order model completeness does not assert that every two first-order models are elementarily equivalent: it constrains embeddings that exist between first-order models, whereas completeness constrains all first-order models.