# Model complete 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[edit]

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 -categorical theory, then it always has a model companion.^{[1]}^{[2]}

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:^{[3]}

*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[edit]

- 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[edit]

- 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[edit]

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.^{[4]}

## Notes[edit]

**^**D. Saracino.*Model Companions for*ℵ_{0}-*Categorical Theories*. Proceedings of the American Mathematical Society Vol. 39, No. 3 (Aug., 1973), pp. 591–598**^**H. Simmons.*Large and Small Existentially Closed Structures*. J. Symb. Log. 41 (2): 379–390 (1976)**^**Chang, C. C.; Keisler, H. Jerome (2012).*Model Theory*(Third edition ed.). Dover Publications. pp. 672 pages.**^**David Marker (2002).*Model Theory: An Introduction*. Springer-Verlag New York.

## References[edit]

- Chang, Chen Chung; Keisler, H. Jerome (1990) [1973],
*Model Theory*, Studies in Logic and the Foundations of Mathematics (3rd ed.), Elsevier, ISBN 978-0-444-88054-3 - Hirschfeld, Joram; Wheeler, William H. (1975), "Model-completions and model-companions",
*Forcing, Arithmetic, Division Rings*, Lecture Notes in Mathematics, vol. 454, Springer, pp. 44–54, doi:10.1007/BFb0064085, ISBN 978-3-540-07157-0, MR 0389581