Monthly Archives: August 2021

The Unabashed Expanse of Ordinal numbers

Let me take you on a magical journey to explore one of the most fundamental and elegant, yet underappreciated, concept in mathematics – the ordinal numbers.

This cannot be done without a few words on what they are and and why we need them, but my focus will be on trying to instill a sense of awe at just how many of them there are – something that I have always found beautiful.

A language primer

The concepts of ordinal numbers – and their close cousins, the cardinal numbers – were not invented by mathematicians. They were originally a language construct, which we are all familiar with in our spoken language of choice, be it English, Hebrew or whatever.

A cardinal numeral is a way to express how many things there are. “One”, “two”, “three”, “four”, “fourty-two”, “seventy-eight hundred and twenty-five” are all cardinal numerals.

An ordinal numeral describes the position of something in a list of ordered items. “First”, “second”, “third” and “fourth” are ordinal numerals.

Back to math

In mathematics, cardinal numbers are basically the same as their natural language counterpart – they describe the size of a set, how many elements there are. 0, 1, 2, 3, etc. – known as “natural numbers” or “non-negative integers” – are cardinal numbers. Where this gets interesting – and where they truly diverge from the linguistic concept – is with infinite sets. Their size is measured with infinite cardinal numbers, which have names such as \aleph_0 or \beth_1.

Ordinal numbers are silghtly more complicated. They represent the different fundamental ways in which the elements of a set can be ordered, known also as “order types”. I’ll show you what I mean, but I should warn you that they are not terribly interesting in the realm of finite sets. For every given finite size for a set, there is exactly one fundamental way to order it – so each finite cardinal number corresonds to exactly one finite ordinal number. Because finite cardinal and ordinal numbers are basically the same, it may not be clear why we even introduced ordinal numbers as a new concept. This should become clearer when we explore infinite numbers.

So let’s take a set with, say, 4 elements – \{A,B,C,D\}. There are many ways to order it (24, to be exact, if you remember your combinatorics) – for example, A<C<B<D, or C<D<B<A, or D<A<B<C. But all these orderings are fundamentally the same. They all follow the pattern – “One item, then another, then another, then another”. Different orderings have different items in each position, but they all follow the same basic structure. This structure is represented by the ordinal number 4 (so called because it is the only way to order sets of size 4).

Continue reading