Definable Set

A subset S of the domain of a structure M is definable (with parameters) if there exists a first-order formula φ(x, a) with parameters a from M such that S = { b in M : M ⊨ φ(b,a)}. If no parameters are used the set is definable without parameters (0‑definable).

A definable set in a structure M is a subset of M^n that equals the solution set of a first-order formula φ(x) possibly with parameters from M; that is, D = { a ∈ M^n : M ⊨ φ(a) } for some φ and parameter tuple from M.