In mathematical logic and set theory, a **definable set** refers to a set whose properties can be precisely described using a formal language or a logical formula. More specifically, a set \( S \) is considered definable in a mathematical structure if there exists a formula in the language of that structure such that the set \( S \) consists exactly of the elements that satisfy the formula. ### Types of Definability 1.
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A definable set in a first-order structure is the set of tuples satisfying one first-order formula, possibly with a specified finite tuple of parameters from the structure. A set definable over uses parameters from . Parameter-free definability uses no extra named elements. The distinction matters: any singleton becomes definable if its own element is allowed as a parameter, while many singletons are not definable without parameters. Finite Boolean combinations of definable sets are definable, using logical conjunction, logical disjunction and logical negation.