Ordinale

Ordinale is a term used in linguistics, mathematics, and related fields to describe things that express or concern order or position in a sequence. In grammar, ordinal numbers serve as adjectives or determiners that denote a position in a series, such as first, second, and third. They typically agree with the noun in gender and number in many languages, and they contrast with cardinal numbers, which express quantity.

In mathematics, the concept of ordinals generalizes the idea of position beyond simple counting. Finite ordinals correspond to natural numbers, while infinite ordinals describe positions in well-ordered sequences that continue indefinitely. A foundational construction is the von Neumann ordinal, where each ordinal n is identified with the set of all smaller ordinals. Notable infinite ordinals include symbols like omega (ω), representing the order type of the natural numbers, and more complex ordinals built from it. Ordinals are central to set theory and logic, underpinning notions of well-ordering, transfinite induction, and recursion.

Distinction from cardinal numbers is a standard feature of the concept: cardinals measure the size of a set (how many elements), whereas ordinals measure order (the position of elements within a well-ordered sequence). Finite ordinals align with finite cardinals, but they diverge for infinite sets, where many different ordinals can share the same cardinality.

Applications of ordinals appear across disciplines. In language, ordinals help specify order in lists and processes. In computing and mathematics, they are used to index data, reason about sequences, and formalize proofs and algorithms that proceed in a well-defined order. Etymologically, the term derives from Latin ordo, meaning order.