äärellisen

äärellisen is a Finnish adjective meaning finite or bounded. It is used mainly in mathematical, logical, and philosophical texts to describe objects or concepts that have a finite extent, size, value, or range. In ordinary language, the more common synonym is rajallinen, but äärellinen appears in technical writing to emphasize the idea of being limited by explicit bounds rather than by everyday common sense.

Etymology and form. The word is formed with the suffix -llinen from a stem related to äär, meaning edge or limit. As an adjective, äärellinen inflects like other Finnish adjectives and appears in various cases and numbers; the genitive singular form is äärellisen, which is used when the adjective modifies a noun in the genitive. For example, äärellisen luvun arvo means the value of a finite number, and äärellisen järjestelmän ominaisuudet refers to the properties of a finite system.

Usage. In mathematics, äärellinen is used to describe finite sets, sequences, or intervals, and to distinguish them from infinite or unbounded cases. It also appears in discussions of boundedness in analysis and in formal logic when finite structures or finite models are considered. Outside technical contexts, builders of mathematical explanations may prefer rajallinen, but äärellinen remains correct and precise in specialized literature.

See also. Rajallinen, finite set, boundedness, finite model theory.