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

  ##  [Model Completion](https://mathlogic.quantumdictionary.io/model-completion-0) 

  
 [![Mathematics & Logic Dictionary](/sites/default/files/styles/large/public/2026-01/Mathematics%20%26%20Logic.png.webp?itok=UhtTRPnp)](/topic-specific-dictionaries/natural-formal-sciences/mathematics-logic)

- Natural &amp; Formal Sciences -

**Mathematics &amp; Logic Dictionary**

 
Definition

A refinement of a first-order theory T producing a theory T* (when it exists) whose models are exactly the existentially closed models of T; T* is model-complete if every embedding between models of T* is elementary, equivalently every formula is equivalent to an existential formula in T*.

 
 ##  [Model Completion](https://puremath.quantumdictionary.io/model-completion-1) 

  
 [![Pure Mathematics Dictionary](/sites/default/files/styles/large/public/2026-01/Pure%20Mathematics.png.webp?itok=5pZnFQ59)](/topic-specific-dictionaries/mathematics-logic/pure-mathematics)

- Mathematics &amp; Logic -

**Pure Mathematics Dictionary**

 
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.

 
 ##  [Model Completion](https://algebra.quantumdictionary.io/model-completion-2) 

  
 [![Algebra](/sites/default/files/styles/large/public/2026-01/Algebra.png.webp?itok=3pHxBnUF)](/topic-specific-dictionaries/pure-mathematics/algebra)

- Pure Mathematics -

**Algebra Dictionary**

 
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.