Ordinal numbers let you keep counting after infinity
Count first, second, third and so on through every natural number. Georg Cantor asked: what comes next? His answer was omega, the first position beyond all the finite ones, followed by omega plus one, omega plus two and onward. These ordinal numbers, introduced in 1883, turned counting into a tool that works on infinite collections.
A natural number can do two jobs. It can say how many things a set holds, or it can say where an item sits in a line. For finite sets those jobs coincide, since counting labels tells you the size. In the infinite case they split apart. Size becomes the theory of cardinal numbers, while position becomes the theory of ordinals, and different infinite ordinals can describe arrangements of sets that are exactly the same size.
The key requirement is a well-order: an ordering in which every non-empty collection has a smallest member, so there is always a definite next unused label. The ordinary real numbers fail this test, since an open interval has no least element. Ordinals are defined as the standard patterns of well-ordered sets, and any two well-orders can be compared, with one matching the start of the other. Assuming the axiom of choice, every set can be well-ordered.
Well-ordering powers transfinite induction, an extension of the familiar proof technique. If some statement failed for any ordinal, there would have to be a smallest ordinal where it fails; show no such smallest failure exists and the statement holds everywhere. Counting through omega times two works like a nested loop in programming, running through one full copy of the natural numbers and then another.
Ordinals behave strangely. They can be added, multiplied and raised to powers, but order matters in all three operations, unlike ordinary arithmetic. There is also no largest ordinal: for any set of them, set theory supplies a bigger one. The Burali-Forti paradox asks about the collection of all ordinals, and the resolution is that it is too big to be a set at all. Cantor had arrived here from a study of trigonometric series begun in 1872.
Source: Ordinal number
