In
model theory
In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the s ...
, a
first-order
In mathematics and other formal sciences, first-order or first order most often means either:
* "linear" (a polynomial of degree at most one), as in first-order approximation and other calculus uses, where it is contrasted with "polynomials of high ...
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.
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
A diagram is a symbolic representation of information using visualization techniques. Diagrams have been used since prehistoric times on walls of caves, but became more prevalent during the Enlightenment. Sometimes, the technique uses a three- ...
of ''M'' is
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
. 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:
* ''T''* is a model completion of ''T''
* ''T'' has the
amalgamation property
In the mathematical field of model theory, the amalgamation property is a property of collections of structures that guarantees, under certain conditions, that two structures in the collection can be regarded as substructures of a larger one.
This ...
.
If ''T'' also has universal axiomatization, both of the above are also equivalent to:
* ''T''* has
elimination of quantifiers Quantifier elimination is a concept of simplification used in mathematical logic, model theory, and theoretical computer science. Informally, a quantified statement "\exists x such that \ldots" can be viewed as a question "When is there an x such ...
Examples
*Any theory with
elimination of quantifiers Quantifier elimination is a concept of simplification used in mathematical logic, model theory, and theoretical computer science. Informally, a quantified statement "\exists x such that \ldots" can be viewed as a question "When is there an x such ...
is model complete.
*The theory of
algebraically closed field
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in .
Examples
As an example, the field of real numbers is not algebraically closed, because ...
s is the model completion of the theory of fields. It is model complete but not complete.
*The model completion of the theory of
equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relation ...
s 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
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
* Do ...
s).
*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.
[David Marker (2002). ''Model Theory: An Introduction''. Springer-Verlag New York.]
Notes
References
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{{Authority control
Mathematical logic
Model theory