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 ℵ 0 {\displaystyle \aleph _{0}} -categorical theory, then it always has a model companion.12
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
- 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.4
Notes
- 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.
- Chang, Chen Chung; Keisler, H. Jerome (2012) [1990]. Model Theory. Dover Books on Mathematics (3rd ed.). Dover Publications. p. 672. ISBN 978-0-486-48821-9.
- 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.
- Marker, David (2002). Model Theory: An Introduction. Graduate Texts in Mathematics 217. New York: Springer-Verlag. ISBN 0-387-98760-6.
- Saracino, D. (August 1973). "Model Companions for ℵ0-Categorical Theories". Proceedings of the American Mathematical Society. 39 (3): 591–598.
- Simmons, H. (1976). "Large and Small Existentially Closed Structures". Journal of Symbolic Logic. 41 (2): 379–390.
References
Saracino 1973. - Saracino, D. (August 1973). "Model Companions for ℵ0-Categorical Theories". Proceedings of the American Mathematical Society. 39 (3): 591–598. ↩
Simmons 1976. - Simmons, H. (1976). "Large and Small Existentially Closed Structures". Journal of Symbolic Logic. 41 (2): 379–390. ↩
Chang & Keisler 2012. - Chang, Chen Chung; Keisler, H. Jerome (2012) [1990]. Model Theory. Dover Books on Mathematics (3rd ed.). Dover Publications. p. 672. ISBN 978-0-486-48821-9. ↩
Marker 2002. - Marker, David (2002). Model Theory: An Introduction. Graduate Texts in Mathematics 217. New York: Springer-Verlag. ISBN 0-387-98760-6. ↩