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).