##  [Model Completion](/model-completion-2) 

 Definition

A model completion of a theory T is a model companion T* with the stronger property that every model of T embeds as a substructure (not merely embeds) into a model of T*, equivalently T* is model‑complete and every model of T has an existentially closed extension that is a superstructure in which the embedding is an inclusion.

 

 

 

 

 

 





## Principle

Principle

Model completion upgrades companionship by requiring that existential closure be realized by actual substructure inclusion: the completion supplies canonical completions of models of T within which first‑order types are resolved as much as possible consistent with T.

 

 

 

 

 





## Demonstration

Demonstration

The theory of algebraically closed fields (ACF) is the model completion of the theory of fields: every field embeds as a subfield of an algebraically closed field, and ACF is model‑complete, so ACF is a model completion of the theory of fields.

 

 

 

 

## Misapplication

Misapplication

Asserting that a model companion is automatically a model completion; some companions exist that do not realize embeddings as substructures for every model of the base theory, so one must check the stronger embedding condition rather than assume it.

 

 

 

 

 





## Consequence

Consequence

When a model completion exists, it often yields strong structural control such as quantifier elimination (in favorable languages), a clear description of existentially closed models, and canonical ways to extend models; this facilitates classification and decidability analyses.

 

 

 

 

## Reversal

Reversal

A theory may have a model companion that is not a model completion: embeddings of models of T into models of the companion may not be realized as substructure inclusions, so the companion does not provide canonical substructure completions.

 

 

 

 

 





## Boundary

Boundary

Existence of a model completion is not guaranteed and typically depends on syntactic and semantic properties of T; the notion presupposes a fixed language and Tarski semantics and excludes weaker companions that lack the substructure embedding property.

 

 

 

 

 





## Semantic Tension

Semantic Tension

The critical tension is between coronation by model‑completeness alone and the extra geometric/algebraic demand that completions produce substructure inclusions; authors sometimes blur the two, so clarity about the embedding form is essential.

 

 

 

 

 





## Synthesis

Synthesis

A model completion is the strongest form of model companion: a model‑complete theory that realizes every model of the original theory as a substructure of some model of the completion, thereby providing canonical, existentially closed completions of models of T.