Ordinal number

In set theory, ordinal, ordinal number, and transfinite ordinal number refer to a type of number introduced by Georg Cantor in 1897, to accommodate infinite sequences and to classify sets with certain kinds of order structures on them. Ordinals are an extension of the natural numbers different from integers and from cardinals.

Well-ordering is total ordering with transfinite induction, where transfinite induction extends mathematical induction beyond the finite. Ordinals represent equivalence classes of well orderings with order-isomorphism being the equivalence relationship. Each ordinal is taken to be the set of smaller ordinals. 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. A class is closed and unbounded if its indexing function is continuous and never stops. One can define addition, multiplication, and exponentiation on ordinals, but not subtraction or division. The Cantor normal form is a standardized way of writing down ordinals. There is a many to one association from ordinals to cardinals. 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.