In
model theory
In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (mat ...
, a branch of
mathematical logic
Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...
, an elementary class (or axiomatizable class) is a
class
Class, Classes, or The Class may refer to:
Common uses not otherwise categorized
* Class (biology), a taxonomic rank
* Class (knowledge representation), a collection of individuals or objects
* Class (philosophy), an analytical concept used d ...
consisting of all
structures
A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such as ...
satisfying a fixed
first-order theory
A theory is a systematic and rational form of abstract thinking about a phenomenon, or the conclusions derived from such thinking. It involves contemplative and logical reasoning, often supported by processes such as observation, experimentation, ...
.
Definition
A
class
Class, Classes, or The Class may refer to:
Common uses not otherwise categorized
* Class (biology), a taxonomic rank
* Class (knowledge representation), a collection of individuals or objects
* Class (philosophy), an analytical concept used d ...
''K'' of
structures
A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such as ...
of a
signature
A signature (; from , "to sign") is a depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. Signatures are often, but not always, Handwriting, handwritt ...
σ is called an elementary class if there is a
first-order theory
A theory is a systematic and rational form of abstract thinking about a phenomenon, or the conclusions derived from such thinking. It involves contemplative and logical reasoning, often supported by processes such as observation, experimentation, ...
''T'' of signature σ, such that ''K'' consists of all models of ''T'', i.e., of all σ-structures that satisfy ''T''. If ''T'' can be chosen as a theory consisting of a single first-order sentence, then ''K'' is called a basic elementary class.
More generally, ''K'' is a
pseudo-elementary class if there is a first-order theory ''T'' of a signature that extends σ, such that ''K'' consists of all σ-structures that are
reducts to σ of models of ''T''. In other words, a class ''K'' of σ-structures is pseudo-elementary if and only if there is an elementary class ''K
''' such that ''K'' consists of precisely the reducts to σ of the structures in ''K
'''.
For obvious reasons, elementary classes are also called axiomatizable in first-order logic, and basic elementary classes are called finitely axiomatizable in first-order logic. These definitions extend to other logics in the obvious way, but since the first-order case is by far the most important, axiomatizable implicitly refers to this case when no other logic is specified.
Conflicting and alternative terminology
While the above is nowadays standard terminology in
"infinite" model theory, the slightly different earlier definitions are still in use in
finite model theory
Finite model theory is a subarea of model theory. Model theory is the branch of logic which deals with the relation between a formal language (syntax) and its interpretations (semantics). Finite model theory is a restriction of model theory to inte ...
, where an elementary class may be called a Δ-elementary class, and the terms elementary class and first-order axiomatizable class are reserved for basic elementary classes (Ebbinghaus et al. 1994, Ebbinghaus and Flum 2005). Hodges calls elementary classes axiomatizable classes, and he refers to basic elementary classes as definable classes. He also uses the respective synonyms EC
class and EC class (Hodges, 1993).
There are good reasons for this diverging terminology. The
signature
A signature (; from , "to sign") is a depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. Signatures are often, but not always, Handwriting, handwritt ...
s that are considered in general model theory are often infinite, while a single
first-order sentence contains only finitely many symbols. Therefore, basic elementary classes are atypical in infinite model theory. Finite model theory, on the other hand, deals almost exclusively with finite signatures. It is easy to see that for every finite signature σ and for every class ''K'' of σ-structures closed under isomorphism there is an elementary class
of σ-structures such that ''K'' and
contain precisely the same finite structures. Hence, elementary classes are not very interesting for finite model theorists.
Easy relations between the notions
Clearly every basic elementary class is an elementary class, and every elementary class is a pseudo-elementary class. Moreover, as an easy consequence of the
compactness theorem
In mathematical logic, the compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model. This theorem is an important tool in model theory, as it provides a useful (but generall ...
, a class of σ-structures is basic elementary if and only if it is elementary and its complement is also elementary.
Examples
A basic elementary class
Let σ be a signature consisting only of a
unary function
In mathematics, a unary function is a function that takes one argument. A unary operator belongs to a subset of unary functions, in that its codomain coincides with its domain. In contrast, a unary function's domain need not coincide with its r ...
symbol ''f''. The class ''K'' of σ-structures in which ''f'' is
one-to-one is a basic elementary class. This is witnessed by the theory ''T'', which consists only of the single sentence
:
.
An elementary, basic pseudoelementary class that is not basic elementary
Let σ be an arbitrary signature. The class ''K'' of all infinite σ-structures is elementary. To see this, consider the sentences
:
"
",
:
"
",
and so on. (So the sentence
says that there are at least ''n'' elements.) The infinite σ-structures are precisely the models of the theory
:
.
But ''K'' is not a basic elementary class. Otherwise the infinite σ-structures would be precisely those that satisfy a certain first-order sentence τ. But then the set
would be inconsistent. By the
compactness theorem
In mathematical logic, the compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model. This theorem is an important tool in model theory, as it provides a useful (but generall ...
, for some natural number ''n'' the set
would be inconsistent. But this is absurd, because this theory is satisfied by any finite σ-structure with
or more elements.
However, there is a basic elementary class ''K
''' in the signature σ' = σ
, where ''f'' is a unary function symbol, such that ''K'' consists exactly of the reducts to σ of σ'-structures in ''K
'''. ''K
''' is axiomatised by the single sentence
, which expresses that ''f'' is injective but not surjective. Therefore, ''K'' is elementary and what could be called basic pseudo-elementary, but not basic elementary.
Pseudo-elementary class that is non-elementary
Finally, consider the signature σ consisting of a single unary relation symbol ''P''. Every σ-structure is
partitioned into two subsets: Those elements for which ''P'' holds, and the rest. Let ''K'' be the class of all σ-structures for which these two subsets have the same
cardinality
The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thum ...
, i.e., there is a bijection between them. This class is not elementary, because a σ-structure in which both the set of realisations of ''P'' and its complement are countably infinite satisfies precisely the same first-order sentences as a σ-structure in which one of the sets is countably infinite and the other is uncountable.
Now consider the signature
, which consists of ''P'' along with a unary function symbol ''f''. Let
be the class of all
-structures such that ''f'' is a bijection and ''P'' holds for ''x''
iff
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either both ...
''P'' does not hold for ''f(x)''.
is clearly an elementary class, and therefore ''K'' is an example of a pseudo-elementary class that is not elementary.
Non-pseudo-elementary class
Let σ be an arbitrary signature. The class ''K'' of all finite σ-structures is not elementary, because (as shown above) its complement is elementary but not basic elementary. Since this is also true for every signature extending σ, ''K'' is not even a pseudo-elementary class.
This example demonstrates the limits of expressive power inherent in
first-order logic
First-order logic, also called predicate logic, predicate calculus, or quantificational logic, is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over ...
as opposed to the far more expressive
second-order logic
In logic and mathematics, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory.
First-order logic quantifies on ...
. Second-order logic, however, fails to retain many desirable properties of first-order logic, such as the
completeness and
compactness
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it ...
theorems.
References
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{{DEFAULTSORT:Elementary Class
Model theory