Model (in logic)
An interpretation of a formal language satisfying certain axioms (cf. Axiom). The basic formal language is the first-order language of a given signature
including predicate symbols
,
, function symbols
,
, and constants
,
. A model of the language
is an algebraic system of signature
.
Let be a set of closed formulas in
. A model for
is a model for
in which all formulas from
are true. A set
is called consistent if it has at least one model. The class of all models of
is denoted by
. Consistency of a set
means that
.
A class of models of a language
is called axiomatizable if there is a set
of closed formulas of
such that
. The set
of all closed formulas of
that are true in each model of a given class
of models of
is called the elementary theory of
. Thus, a class
of models of
is axiomatizable if and only if
. If a class
consists of models isomorphic to a given model, then its elementary theory is called the elementary theory of this model.
Let be a model of
having universe
. One may associate to each element
a constant
and consider the first-order language
of signature
which is obtained from
by adding the constants
,
.
is called the diagram language of the model
. The set
of all closed formulas of
which are true in
on replacing each constant
by the corresponding element
is called the description (or elementary diagram) of
. The set
of those formulas from
which are atomic or negations of atomic formulas is called the diagram of
.
Along with models of first-order languages, models of other types (infinitary logic, intuitionistic logic, many-sorted logic, second-order logic, many-valued logic, and modal logic) have also been considered.
For references see Model theory.
Comments
English usage prefers the word "structure" where Russian speaks of a "model of a language" or an "algebraic system" ; "model" is reserved for structures satisfying a given theory (set of closed formulas).
Model (in logic). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Model_(in_logic)&oldid=15173