Logic Colloquium 2004, Tutorial on Geometric Stability Theory, Model Completion

Model Completions

Let M be some class of models. We say the the model M ∈ M is existentially closed in M if whenever M ⊆ N ∈ M is an extension of M in M and φ(x) is a formula with parameters from M for which N satisfies (∃ x) φ(x), then so does M

We say that M is existentially closed (without qualification) if it is existentially closed in Mod(Th(M)), the class of models elementarily equivalent to M.

The theory T is model complete if every model of T is existentially closed.

We say that the theory T' is the model companion of the theory T if

T' is model complete,
every model of T embeds as a substructure of a model of T', and
vice versa: every model of T' embeds as a substructure of a model of T.

If, in addition, T' eliminates quantifiers, namely, every formula is equivalent modulo T' to a quantifier-free formula, then we say that T' is the model completion of T.