Model complete theory

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title: "Model complete theory" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["mathematical-logic", "model-theory"] description: "Concept in model theory" topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Model_complete_theory" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Concept in model theory ::

In model theory, a first-order theory is called model complete if every embedding of its models is an elementary embedding. Equivalently, every first-order formula is equivalent to a universal formula. This notion was introduced by Abraham Robinson.

Model companion and model completion

A companion of a theory T is a theory T* such that every model of T can be embedded in a model of T* and vice versa.

A model companion of a theory T is a companion of T that is model complete. Robinson proved that a theory has at most one model companion. Not every theory is model-companionable, e.g. theory of groups. However if T is an \aleph_0-categorical theory, then it always has a model companion.

A model completion for a theory T is a model companion T* such that for any model M of T, the theory of T* together with the diagram of M is complete. Roughly speaking, this means every model of T is embeddable in a model of T* in a unique way.

If T* is a model companion of T then the following conditions are equivalent:

• T* is a model completion of T
• T has the amalgamation property.

If T also has universal axiomatization, both of the above are also equivalent to:

• T* has elimination of quantifiers

Examples

• Any theory with elimination of quantifiers is model complete.
• The theory of algebraically closed fields is the model completion of the theory of fields. It is model complete but not complete.
• The model completion of the theory of equivalence relations is the theory of equivalence relations with infinitely many equivalence classes, each containing an infinite number of elements.
• The theory of real closed fields, in the language of ordered rings, is a model completion of the theory of ordered fields (or even ordered domains).
• The theory of real closed fields, in the language of rings, is the model companion for the theory of formally real fields, but is not a model completion.

Non-examples

• The theory of dense linear orders with a first and last element is complete but not model complete.
• The theory of groups (in a language with symbols for the identity, product, and inverses) has the amalgamation property but does not have a model companion.

Sufficient condition for completeness of model-complete theories

If T is a model complete theory and there is a model of T that embeds into any model of T, then T is complete.

Notes

References

• {{cite journal |last=Saracino |first=D. |title=Model Companions for ℵ0-Categorical Theories |journal=Proceedings of the American Mathematical Society |volume=39 |issue=3 |date=August 1973 |pages=591–598

• {{cite journal |last=Simmons |first=H. |title=Large and Small Existentially Closed Structures |journal=Journal of Symbolic Logic |volume=41 |issue=2 |year=1976 |pages=379–390

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