##  [Model Completion](/model-completion-1) 

 Definition

A model completion of a first-order theory T is a theory T* that is a model companion of T, is model-complete (every embedding between models of T* is elementary), and whose models are precisely the existentially closed models of T.

 

 

 

 

 

 





## Principle

Principle

Extend T to a theory whose models satisfy all existential consequences consistent with T so that embeddings preserve all first-order formulas, reducing reasoning to the study of existential closure.

 

 

 

 

 





## Demonstration

Demonstration

For the theory of fields, the theory of algebraically closed fields (ACF) is the model completion: ACF is model-complete and its models are exactly the fields that are existentially closed among all fields.

 

 

 

 

## Misapplication

Misapplication

Treating any conservative or complete extension of T as a model completion; existence of a model completion is nontrivial and may fail for many theories.

 

 

 

 

 





## Consequence

Consequence

When a model completion exists it yields uniform control of definable sets (often quantifier elimination) and uniqueness up to logical equivalence, simplifying classification of models via existential closure.

 

 

 

 

## Reversal

Reversal

The opposite notion is a theory T with many proper extensions that do not make embeddings elementary; there is no single companion capturing existential closure and embeddings can change truth of formulas.

 

 

 

 

 





## Boundary

Boundary

Applies to first-order theories; not every theory has a model completion. Existence typically requires stability of certain embedding and amalgamation properties and is sensitive to signature and axiomatization.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Competes with the idea of a model companion or conservative extension: a model companion need not be model-complete, and insisting on model-completeness tightens existence conditions and semantic scope.

 

 

 

 

 





## Synthesis

Synthesis

A model completion is the strongest companion of T that makes existential closure into an intrinsic property of models by ensuring every embedding is elementary, thereby converting external existential constraints into internal axioms.