Ordinal Number

In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals. Like other kinds of numbers, ordinals can be added, multiplied, and exponentiated.

Ordinals were introduced by Georg Cantor in 1883 to accommodate infinite sequences and to classify sets with certain kinds of order structures on them. He derived them by accident while working on a problem concerning trigonometric series—see Georg Cantor.

Two sets S and S' have the same cardinality if there is a bijection (i.e. one-to-one onto function) f which maps each element x of S to a unique element y = f(x) of S' and each element y of S' comes from exactly one such element x of S. S and S' are order isomorphic if there is a partial ordering < defined on S and a partial ordering <' defined on S', such that the function f preserves the ordering. That is, f(a) <' f(b) if and only if a < b. Every well ordered set S is order isomorphic to exactly one ordinal number.

The finite ordinals (and the finite cardinals) are the natural numbers: 0, 1, 2, …, since any two total orderings of a finite set are order isomorphic. The least infinite ordinal is ω, which is identified with the cardinal number . However in the transfinite case, beyond ω, ordinals draw a finer distinction than cardinals on account of their order information. Whereas there is only one countably infinite cardinal, namely itself, there are uncountably many countably infinite ordinals, namely

ω, ω + 1, ω + 2, …, ω·2, ω·2 + 1, …, ω2, …, ω3, …, ωω, …, ωωω, …, ε₀, ….

Here addition and multiplication are not commutative: in particular 1 + ω is ω rather than ω + 1 and likewise, 2·ω is ω rather than ω·2. The set of all countable ordinals constitutes the first uncountable ordinal ω₁, which is identified with the cardinal (next cardinal after ). Well-ordered cardinals are identified with their initial ordinals, i.e. the smallest ordinal of that cardinality. The cardinality of an ordinal defines a many to one association from ordinals to cardinals.

In general, each ordinal α, is the order type of the set of ordinals strictly less than the ordinal, α itself. This property permits every ordinal to be represented as the set of all ordinals less than it. Ordinals may be categorized as: zero, successor ordinals, and limit ordinals (of various cofinalities). Given a class of ordinals, one can identify the α-th member of that class, i.e. one can index (count) them. Such a class is closed and unbounded if its indexing function is continuous and never stops. The Cantor normal form uniquely represents each ordinal as a finite sum of ordinal powers of ω. However, this cannot form the basis of a universal ordinal notation due to such self-referential representations as ε₀ = ωε₀. Larger and larger ordinals can be defined, but they become more and more difficult to describe. Any ordinal number can be made into a topological space by endowing it with the order topology; this topology is discrete if and only if the ordinal is a countable cardinal, i.e. at most ω. A subset of ω + 1 is open in the order topology if and only if either it is cofinite or it does not contain ω as an element.