Ordinal number

Ordinal numbers are a special kind of number used in mathematics to describe the position of things in a sequence. For example, we use ordinal numbers when we say "first," "second," or "third." These help us understand order, like knowing who comes first, second, or third in a race.

In more advanced math, ordinal numbers help us work with very large collections, even ones that are infinite. They go beyond the usual counting numbers to include special numbers like ω (omega), which comes after all the natural numbers. This helps mathematicians study and compare different kinds of infinite collections.

Ordinal numbers were introduced by Georg Cantor in 1883. He created them to better understand infinite sequences and to classify special sets he had studied earlier. Unlike regular counting numbers, ordinal numbers help us talk about the order of things, even when there are infinitely many 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, …). This is like putting one counting process inside another. Ordinals help us describe very complex counting processes, such as counting through many layers of nested groups.

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 positions in sequences, like first, second, or third. They can also apply to very large, even endless sets.

For a small group of items, we can list them by giving each one a number in order. This idea can be stretched to work with bigger and even infinite collections, using special symbols like Greek letters to keep track.

Some "large" countable ordinals

Further information: Large countable ordinal

Some very big counting numbers, called ordinals, can be made by following special rules. For example, there is a number called ε₀ that comes after all the numbers you can make by repeating certain steps.

No matter how cleverly you make new big numbers, there will always be an even bigger one that you cannot reach with your rules. One of the most important of these hard-to-reach numbers is called the Church–Kleene ordinal. Even though it has a 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 kind of arrangement called order topology. This arrangement is very simple, like separate points, only when the ordinal number is small enough — specifically, when it is less than or equal to ω (a special infinite number). For larger arrangements, a part of ω + 1 is considered "open" only if it either has almost all its points or does not include ω itself.

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

History

The idea of transfinite ordinal numbers started in 1883 with work by Cantor on special sets of numbers. Cantor looked at sets of real numbers and created new sets by removing certain points. He used these ideas to build a long list of sets that could go on forever.

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