model complete theory
{{Short description|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 -categorical theory, then it always has a model companion.{{sfn|Saracino|1973}}{{sfn|Simmons|1976}}
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:{{sfn|Chang|Keisler|2012}}
- 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.{{sfn| Marker|2002}}
Notes
{{reflist}}
References
- {{Cite book|last1=Chang|first1=Chen Chung|author1-link=Chen Chung Chang|last2=Keisler|first2=H. Jerome|author2-link=Howard Jerome Keisler|title=Model Theory|year=1990|orig-year=1973|publisher=Elsevier|edition=3rd|series=Studies in Logic and the Foundations of Mathematics|isbn=978-0-444-88054-3}}
- {{Cite book|last1=Chang|first1=Chen Chung|author1-link=Chen Chung Chang|last2=Keisler|first2=H. Jerome|author2-link=Howard Jerome Keisler|title=Model Theory|year=2012|orig-year=1990|publisher=Dover Publications|edition=3rd|series=Dover Books on Mathematics|pages=672|isbn=978-0-486-48821-9}}
- {{cite book|last=Hirschfeld|first=Joram|last2=Wheeler|first2=William H.|chapter=Model-completions and model-companions|title=Forcing, Arithmetic, Division Rings|series=Lecture Notes in Mathematics|publisher=Springer|volume=454|pages=44–54|year=1975|isbn=978-3-540-07157-0|mr=0389581|doi=10.1007/BFb0064085}}
- {{cite book | last=Marker | first=David | title= Model Theory: An Introduction | publisher=Springer-Verlag|location= New York| year=2002 | isbn=0-387-98760-6| series=Graduate Texts in Mathematics 217}}
- {{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
}}
{{Mathematical logic}}
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