definibility

Definibility, or definability, is the property of being definable: the ability to specify or characterize something using a formal or natural language description within a given framework. In mathematics and logic, definability is usually studied inside a structure, such as a mathematical object equipped with a language of operations, relations, and constants.

In model theory, a subset A of M^n is definable in a structure M if there exists a first-order formula φ(x1,...,xn) in the language of M such that A = { (a1,...,an) ∈ M^n : M ⊨ φ(a1,...,an) }. If parameters from M are allowed, A is definable with parameters; if not, it is definable without parameters. An element a ∈ M is definable over a subset C ⊆ M if there is a formula φ(x,c) with c from C such that M ⊨ φ(a,c) and any b with M ⊨ φ(b,c) must equal a. The set of all elements definable over C is the definable closure of C, and automorphisms of M fixing C pointwise must fix any element definable over C.

Examples in simple structures are straightforward: the singleton {e} in a group (G,·) is definable by x = e. In finite structures, every subset is definable because one can specify it by a finite disjunction of equalities.

Definability is language- and structure-dependent; a concept may be definable in one framework but not in another. It contrasts with mere expressibility in natural language and with notions of definability in philosophy, where reference and description play central roles.