formaalset

Formaalset is a term used in Estonian mathematical and logical literature to denote a set defined within a formal system by a precise, rule-based description. It reflects the idea of separable collections that can be characterized by a formal predicate rather than by empirical or intuitive means. In English-language mathematics the corresponding notion is usually referred to as a definable or formula-definable set.

In model-theory terms, let L be a formal language and M an L-structure. A set S is a formaalset if there exists an L-formula φ(x) with parameters from M such that S = { a in M^n | M ⊨ φ(a) }. The definability depends on the chosen language and structure; the same set may or may not be definable in a different theory.

In computational contexts, a formaalset may also be described by an algorithm or decision procedure; when such a description exists and halting on all inputs is guaranteed, the set is computable (recursive). If the description only allows for semi-decision procedures, the set may be recursively enumerable.

Examples include the set of natural numbers satisfying x mod 3 = 0, definable in arithmetic by a formula, or the set of strings over an alphabet that are palindromes, definable in the language of concatenation and equality.

Formaalset is primarily a term within Estonian-language texts; in precise mathematical writing it is common to use definable set or, for computational aspects, computable/recursively enumerable. The concept plays a role in discussions of definability, decidability, and the expressive power of formal languages.