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

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

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

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

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

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

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

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

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.