Definable

Definable is a term used in model theory to describe a subset of a mathematical structure that can be described by a first-order formula. Let M be a structure for a given language L. A subset S of M^n is called definable if there exists an L-formula φ(x1, ..., xn) such that S = {a in M^n : M ⊨ φ(a)}. More generally, S is definable over a subset A ⊆ M if φ may include parameters from A, written as φ(x; a) with a in A. If no parameters are used, S is definable over the empty set.

Definability abstracts the idea of describing a set by a logical condition rather than by explicit construction. It is central to comparing structures and studying their internal geometry, as definable sets behave well under logical operations and model-theoretic techniques. A subset is locally definable if it can be described by a family of formulas with parameters from a set.

Examples illustrate the concept. In any group G, a subgroup H is definable if there exists a formula φ(x) such that H = {g in G : G ⊨ φ(g)}. In real closed fields, definable subsets of the field are precisely the semialgebraic sets (constructible using polynomial equations and inequalities, possibly with quantifiers). This aligns with quantifier elimination, which describes how logical formulas can be reduced to explicit algebraic descriptions in that theory. Definable maps are functions whose graphs are definable sets; equivalently, a function f: D → M^m is definable if its graph is a definable subset of D × M^m.

Related notions include the definable closure and types. The definable closure of a set A in M consists of those elements uniquely determined by a formula over A. Definability provides a unifying framework for understanding algebraic, geometric, and logical structure across theories.