Ordinal number

Ordinal numbers are a special kind of number used in mathematics to show the position of things in a line-up. For example, we use ordinal numbers when we say "first," "second," or "third." They help us know the order of things, like who is first, second, or third in a race.

In more advanced math, ordinal numbers help us work with very big groups, even ones that go on forever. They go past the usual counting numbers to include special numbers like ω (omega), which comes after all the regular counting numbers. This helps mathematicians study and compare different kinds of endless groups.

Ordinal numbers were introduced by Georg Cantor in 1883. He made them to better understand endless lines and to sort out special groups he had studied before. Unlike regular counting numbers, ordinal numbers help us talk about the order of things, even when there are endless numbers of them.

Motivation

A natural number can tell us two things: how many things are in a group, or the position of something in a list. When we think about very large, even endless groups, we use special numbers called ordinal numbers to talk about positions.

Counting can be like a step-by-step process. For normal numbers, this is easy: if something is true for the number 0, and if it being true for a number n means it is true for n + 1, then it is true for all natural numbers. This idea helps us understand the first infinite ordinal, written as ⁠ ω {\displaystyle \omega } !{\displaystyle \omega } ⁠.

Sometimes we need to count beyond just one endless group. For example, we might count through all the natural numbers (0, 1, 2, …), then start again with another group (ω + 1, ω + 2, …). Ordinals help us describe very complex counting processes.

Definitions

Well-ordering

When we label things in order, like first, second, third, and so on, we are using a special kind of order called "well-ordering." This means that in any group of these labels, there is always a smallest one. For example, if you have the numbers 1, 2, and 3, the smallest number is 1.

Well-ordering is different from just any order. For instance, if you think about all the numbers between 0 and 1, there isn’t really a smallest one because you can always find a number in between. Well-ordering helps us make sure we can always find a starting point when we count or arrange things.

Order types

Every time we arrange a set of things in a well-ordered way, like lining up toys from smallest to largest, there is a special number called an ordinal that matches exactly how they are ordered. This ordinal is unique for that arrangement.

For example, if you have three toys and line them up as toy A, toy B, then toy C, the ordinal for this arrangement is unique to that order. No other way of arranging these toys will give the same ordinal.

Transfinite induction

Transfinite induction is a way to prove something is true for all ordinals. If a statement is true for all smaller ordinals, then it is true for the next one too. This helps us understand patterns that continue forever, even beyond the normal counting numbers.

Ordinal arithmetic

We can add, multiply, and raise ordinals to powers, just like we do with regular numbers. These operations help us understand how ordinals behave when we combine them. There are special ways to write ordinals using a method called Cantor normal form, which uses the Greek letter ω to show very large numbers.

Ordinals are also a type of number called surreal numbers, and they can be used in games like Nim, where they follow special rules for adding and multiplying.

Ordinals and cardinals

Ordinals help us describe places in a line, like first, second, or third. They can also be used for very big groups, even ones that never end.

For a small group of things, we can name them by giving each one a number in order. This idea can also work for bigger groups, using special symbols like Greek letters to help keep track.

Some "large" countable ordinals

Further information: Large countable ordinal

Some very big numbers, called ordinals, follow special rules. For example, there is a number called ε₀ that comes after all the numbers you can make with certain steps.

No matter how clever you are at making big numbers, there will always be an even bigger one you cannot reach with your rules. One of these hard-to-reach numbers is called the Church–Kleene ordinal. Even with its complex name, it is still a countable number, meaning it can be listed in order, even if it is very large.

Topology and ordinals

Further information: Order topology

We can think of ordinal numbers as special spaces by giving them a special arrangement called order topology. This arrangement is simple, like separate points, only for small ordinal numbers — specifically, when they are less than or equal to ω (a special infinite number). For larger arrangements, a part of ω + 1 is considered "open" only in two cases: if it has almost all its points or if it does not include ω itself.

See the Topology and ordinals section of the "Order topology" article.

History

The idea of transfinite ordinal numbers began in 1883 with work by Cantor on special sets of numbers. Cantor studied sets of real numbers and made new sets by taking away some points. He used these ideas to build a very long list of sets that could continue without end.

Cantor showed that these sets could be organized using special numbers called ordinal numbers. He used these numbers to prove important facts about how these sets work. His work helped create new ways to understand very large and endless sets.