Formuladefinable

Formuladefinable is a term used in mathematical logic and model theory to describe a subset, relation, or property that can be described by a logical formula within a given formal structure. In this setting, a formula is an expression built from symbols, variables, logical connectives, and potentially parameters from the structure's domain.

A subset D of the domain M is said to be definable (formuladefinable) in M if there exists a first-order formula φ(x) with one free variable such that D = {a in M : M satisfies φ(a)}. If parameters from M are allowed, the formula may include those fixed elements; then D is definable over a set A, written D = {a in M : M |= φ(a, a1, ..., an)} with a ⊆ A.

Definable sets and relations are central in model theory because they connect logical syntax with semantic structure. They help describe which subsets can be uniquely specified by a formula, independently of particular representations. Concepts such as definable closure, imaginaries, and definable groups use definability to study algebraic and geometric properties inside models.

Examples illustrate the idea. In the real field R, the set of nonnegative numbers is definable by φ(x) = (x ≥ 0). The set of squares is definable by ∃y (x = y^2). In a group, the centralizer of an element g is definable by φ(x) = xg = gx.

Definability is a flexible notion across areas such as algebraic geometry, o-minimal structures, and stability theory. It provides a precise way to express which subsets or relations are describable using the language of a given structure.