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.

## Contents

* 1 Model companion and model completion
* 2 Examples
* 3 Non-examples
* 4 Sufficient condition for completeness of model-complete theories
* 5 Notes
* 6 References

## 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 ℵ0-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

* 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

1. ↑ Saracino 1973.
2. ↑ Simmons 1976.
3. ↑ Chang & Keisler 2012.
4. ↑ Marker 2002.

## References

* Chang, Chen Chung; Keisler, H. Jerome (1990). 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). Model Theory. Dover Books on Mathematics (3rd ed.). Dover Publications. pp. 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. 454. Springer. pp. 44–54. doi:10.1007/BFb0064085. ISBN 978-3-540-07157-0.

* 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.

* v
* t
* e

Mathematical logic

General|

* Formal language
* Formation rule
* Formal proof
* Formal semantics
* Well-formed formula
* Set
* Element
* Class
* Classical logic
* Axiom
* Rule of inference
* Relation
* Theorem
* Logical consequence
* Type theory
* Symbol
* Syntax
* Theory

Systems|

* Formal system
* Deductive system
* Axiomatic system
* Hilbert style systems
* Natural deduction
* Sequent calculus

Traditional logic|

* Proposition
* Inference
* Argument
* Validity
* Cogency
* Syllogism
* Square of opposition
* Venn diagram

Propositional calculus
and Boolean logic

* Boolean functions
* Propositional calculus
* Propositional formula
* Logical connectives
* Truth tables
* Many-valued logic

Predicate logic|

* First-order
* Quantifiers
* Predicate
* Second-order
* Monadic predicate calculus

Naive set theory|

* Set
* Empty set
* Element
* Enumeration
* Extensionality
* Finite set
* Infinite set
* Subset
* Power set
* Countable set
* Uncountable set
* Recursive set
* Domain
* Codomain
* Image
* Map
* Function
* Relation
* Ordered pair

Set theory|

* Foundations of mathematics
* Zermelo–Fraenkel set theory
* Axiom of choice
* General set theory
* Kripke–Platek set theory
* Von Neumann–Bernays–Gödel set theory
* Morse–Kelley set theory
* Tarski–Grothendieck set theory

Model theory|

* Model
* Interpretation
* Non-standard model
* Finite model theory
* Truth value
* Validity

Proof theory|

* Formal proof
* Deductive system
* Formal system
* Theorem
* Logical consequence
* Rule of inference
* Syntax

Computability
theory|

* Recursion
* Recursive set
* Recursively enumerable set
* Decision problem
* Church–Turing thesis
* Computable function
* Primitive recursive function

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Model complete theory. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Model_complete_theory&oldid=4151163"
*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template