Ordinals

Ordinals are a class of numbers that extend the concept of ordering beyond the finite. While natural numbers describe quantity, ordinal numbers describe the position of an element within a well‑ordered set, that is, a set in which every non‑empty subset has a least element under the given order.

In set theory, each ordinal is identified with the set of all smaller ordinals, allowing ordinals to be constructed recursively: 0 is the empty set, and for any ordinal α, its successor α + 1 is α∪{α}. Limit ordinals, such as ω (the first infinite ordinal), are not successors but are defined as the union of all smaller ordinals. ω corresponds to the order type of the natural numbers, and larger ordinals like ω + 1, ω·2, and ω² represent increasingly complex well‑ordered structures.

Ordinal arithmetic differs from cardinal arithmetic. Addition and multiplication are defined by transfinite recursion and are not commutative: α + β is the order type of placing a copy of β after α, while β + α may yield a different order type. Exponentiation extends similarly, and the Cantor normal form expresses any ordinal uniquely as a finite sum of decreasing powers of ω with natural coefficients.

Ordinals have applications in many areas of mathematics. They provide a framework for transfinite induction and recursion, underpin the definition of rank in the cumulative hierarchy of sets, and appear in proof theory through large countable ordinals that measure the strength of formal systems. In topology, ordinals serve as prototypes for spaces with specific convergence properties, such as the ordinal interval [0, ω₁).