Model complete theory

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.^[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]

References

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.

Mathematical logic

General

Theorems (list)
and paradoxes

Gödel's completeness and incompleteness theorems
Tarski's undefinability
Banach–Tarski paradox
Cantor's theorem, paradox and diagonal argument
Compactness
Halting problem
Lindström's
Löwenheim–Skolem
Russell's paradox

Logics

Traditional	Classical logic Logical truth Tautology Proposition Inference Logical equivalence Consistency Equiconsistency Argument Soundness Validity Syllogism Square of opposition Venn diagram
Propositional	Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued logic 3 finite ∞
Predicate	First-order list Second-order Monadic Higher-order Fixed-point Free Quantifiers Predicate Monadic predicate calculus

Set theory

Set hereditary Class (Ur-)Element Ordinal number Extensionality Forcing Relation equivalence partition Set operations: intersection union complement Cartesian product power set identities
Types of sets	Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter Recursive Fuzzy Universal Universe constructible Grothendieck Von Neumann
Maps and cardinality	Function/Map domain codomain image In/Sur/Bi-jection Schröder–Bernstein theorem Isomorphism Gödel numbering Enumeration Large cardinal inaccessible Aleph number Operation binary
Set theories	Zermelo–Fraenkel axiom of choice continuum hypothesis General Kripke–Platek Morse–Kelley Naive New Foundations Tarski–Grothendieck Von Neumann–Bernays–Gödel Ackermann Constructive

Formal systems (list),
language and syntax

Alphabet Arity Automata Axiom schema Expression ground Extension by definition conservative Relation Formation rule Grammar Formula atomic closed ground open Free/bound variable Language Metalanguage Logical connective ¬ ∨ ∧ → ↔ = Predicate functional variable propositional variable Proof Quantifier ∃ ! ∀ rank Sentence atomic spectrum Signature String Substitution Symbol function logical/constant non-logical variable Term Theory list
Example axiomatic systems (list)	of arithmetic: Peano second-order elementary function primitive recursive Robinson Skolem of the real numbers Tarski's axiomatization of Boolean algebras canonical minimal axioms of geometry: Euclidean: Elements Hilbert's Tarski's non-Euclidean Principia Mathematica