Ordinal number

An ordinal number specifies the position of an item in an ordered sequence. In mathematics, ordinals also describe the order types of well-ordered sets, extending beyond finite positions.

Natural number: A number used for counting or ordering, commonly beginning with zero or one. Finite ordinal positions are represented by natural numbers.
Ordinal successor: The ordinal obtained by placing one new element after every element of a given ordinal. Successor operation gives the next position after any ordinal.
Georg Cantor: A German mathematician who founded set theory and developed the theory of transfinite numbers. Cantor introduced ordinal numbers to distinguish different kinds of infinite order.
Transfinite recursion: A method for defining objects in stages indexed by ordinal numbers. Ordinal stages let definitions continue through limit points as well as successors.
Ordinal numeral: A linguistic form such as “first” or “third” that expresses rank or position. English ordinal numerals express the everyday positional sense of ordinal number.
Sequence: An ordered list of elements indexed by positions. An ordinal identifies a position within an ordered list.
Limit ordinal: A nonzero ordinal that is not the successor of any ordinal. Limit ordinals mark stages with no immediately preceding position.
Set theory: The mathematical study of sets, their elements, and the relations among them. Modern ordinal numbers are defined and studied within set theory.
Ordinal analysis: A proof-theoretic method that assigns ordinals to formal theories to measure their strength. The assigned ordinal bounds the transfinite induction a theory can justify.
Ordinal indicator: A symbol or suffix, such as ‑th, that marks an ordinal numeral in writing. The indicator marks written forms like 4th without itself being an ordinal.

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