$$\newcommand{\Nd}{\mathbb{N}} \newcommand{\Pc}{\mathcal{P}}$$

# Extraordinary ordinals

This (not-so-)short text is about ordinals. It may contain some maths, but don't worry, it won't hurt. Nonetheless, I think the following requirements should be fullfilled: you may want to know a little about what is a set, how we (usually) write them and what is an "abstract" function. Additionnaly, if you know the difference between a set and a (proper) class, that's good.

Do not hesitate to pause-and-ponder if you ever feel the need. It is a rather complicated subject and, depending on your knowledge and your intuition, it may be hard to grasp. Also, as I said, this is quite a long text, so take your time, you're not in a hurry.

Please note that this text is broken down into color-coded boxes:

• those with quartz border are the main text;
• those with red border are pictures and visual examples;
• those with green border are sidenotes;
• those with light cream border delve into harder topics, if you understand, that's good, if not, do not bother.

Also, mathematical statement are displayed like this: $e^{i\pi} + 1 = 0$ and sometimes you'll see this symbol:, hover or touch it to uncover some additionnal informations!

Finally, before we start, I would like to thank Grant Sanderson, who gave me the impetus to start this project, as a part of the Summer of Math Exposition v2 (SoME2)

### Orders!

When counting, one gives an order to things. There is the first thing, the second thing, the third thing, ... and sometimes, when there is nothing at all, there is a total of zero things. As you notice, counting in intrinsically bound to the idea of order.

Not any orders, in fact! When dealing with finite sets, only those which are total orders allows to count. By this, we mean orders on a set such that every two elements $a$ and $b$ are in relation: either $a < b$, $a > b$ or $a = b$. This may seem like an unnecessary remark, but the study of partial orders is particularly interesting (and we won't deal with them here).

One order is good, but two is better! Consider the usual ones on $A := \{0,1,2\}$ and $B := \{3,4,5\}$? They look different, but are pretty similar: they both have elements, both have a first, second and third element and both doesn't have a fourth element. In fact, they are said to be isomorphic, i.e. up to a change of label, they are identical. I will often use the symbol $a \mapsto b$ to tell "let's rename $a$ into $b$" and the symbol $A \simeq B$ to mean "$A$ and $B$ are isomorphic".

The idea of isomorphism is key in mathematics. It always means that some objects (set, well-orders, ring ...) are almost indistinguishable: they share the same structure, the only difference is the name of their elements. Here, in our example, we have the isomorphism $0 \mapsto 3$, $1 \mapsto 4$ and $2 \mapsto 5$.

Let's ask ourselves what we like when we count. If we are to generalize counting, we need to get rid of a few properties and, most importantly, to keep some. We first notice $\Nd$ has a least element: $0$. Also, when we count (downward) in the usual set $\Nd$ of the natural integer, we notice something: every strictly decreasing sequence of integer eventually comes down to $0$. So, we may want to go with that: every order we'll consider will have these properties: (1) having a least element and (2) every of decreasing sequence eventually comes down to it; and we will call them well-orders.

An equivalent property to being well-ordered for an ordered set $S$ is that every subset $T$ of $S$ has a least element (i.e. an element less than any other element in $T$). . Still using the $\Nd$ example, try and find a set of integers with no least element. If you can't, that's cool, because such a thing does not exists.

We may now try to find a classification of these order, and for each class, a quintessential order. In fact, we'll do better: first, some fidgeting with well-orders and then a classification.

### Numbers?

Let's look at a few particular well-ordered sets. The zeroth one is $A_0 := \emptyset = \{\}$, the first one $A_1 := \{\emptyset\}$, the second $A_2 := \{\emptyset, \{\emptyset\}\}$ and the $(n+1)$-th is the set containing all the previous ones, that is $A_{n+1} := \{A_1, A_2, \dots, A_n\} = A_n \cup \{A_n\}$. You first notice that if $n < m$, then $A_n \in A_m$.

Sidenote here: I will often use the notation $a := b$ to say: "let's define $a$ by it being equal to $b$", where $b$ is often complex-looking. This differs from the usual $a = b$ being only a statement: "I state that $a$ and $b$ are equal".

Let's do the converse: let's define an order on the family of the $A_n$ by saying that $x < y$ if and only if $x \in y$. This may seem overly complex (and complex it is), but bear with me and look at what we've just done. Starting with the empty set, we created a family of thing that behave just like the plain good old natural numbers. So, rather than calling this family $A_0$, $A_1$, $A_2$, $A_3$, ..., let's say that these are the natural numbers, the ones usually called $0$, $1$, $2$, $3$, ...

A closer look at the process we used to create our natural numbers shows that $n+1 := n \cup \{n\}$. That is, $n+1$ is the union of the set $n$ (so it contains every element that $n$ contains) and of the set containing only $n$. Imagine taking a big box an tossing in it the content of $n$ and then $n$ itself. That big box is now $n+1$.

So, why would we do that? Why would we call these things "numbers"? Because saying what a number precisely is is complicated (try it for yourself: how would you define a number?) and here, we just have a nice recipe! To be extra precise, we often call them "ordinal numbers", or simply "ordinals".

Here I come to warn you: from now on, things will get weirder and weirder. For instance, with this definition, a number is both an element and a set; $3$ is a set with three ordered elements, but also the greatest element of $4$.This may look like a weaknes or an overly-complex thing to do, but it will have its use.

### Comparisons

Imagine you have two well-orders $V$ and $W$ (whatever this means, we don't have much examples yet). What can you say about them? First, they could be equal, or, in a broader sense, isomorphic. But what if they are not ?

They could somehow still be related. What if a part (aka a subset) of one well-order (say $W$) is isomorphic to the other (say $V$)? That would mean $W$ is, in a certain way, bigger than $V$, since $W$ contains $V$. As an example, $A := \{ 0, 2 \}$, with $A \subset 4$ is isomorphic to the set $2$ (we have $A \simeq 2$), and so $4$ is bigger than $2$.

We now have a new mean of comparisons, not between elements in a well-orders, but among well-orders themselves. You can think of $2$ and $3$ as element of a bigger well-order (say, $4$), but also as well-orders and can be ordered without having to refer to a bigger set. In some weird way, $2 \preceq 3$ is an absolute statement, rather than one depending on the well order $4$, or $5$, ...

Here, I used the symbol $\preceq$ rather than $<$ in order to compare $2$ and $3$ as well-ordered sets. It is only to show explicitly that this is not the same thing, $<$ is to compare elements inside a well-order and $\preceq$ to compare two given well-orders. Again, if I put $W := \{a,b,c\}$ and $V := \{d,e\}$, ordered by $a < b < c$ and $d < e$, I can compare $W$ and $V$ through $\preceq$, but not through $<$.

Also, there is a little ambiguity, one you may not even have noticed: when using the symbol $<$, because it is relative to a well-order, what I state may not always be clear. Consider the sets $S := \{a,b,c\}$ and $T := \{a,b\}$. I could well-order $S$ with $a < b < c$ and $T$ with $b < a$ and by doing so, knowing wether $a$ is greater than $b$ depends on the well-order we put ourselves in, but we always have $S \preceq T$.

Before we move on, I shall just state one very important remark, although you may not feel like it is now: given two well-orders, one is always smaller than the other, or, in other words, embedded in the other. Note that I just said "smaller" and not "strictly smaller", because we always have $W \preceq W$ for every well-order $W$.

### To infinity!

We had some fun with well-orders, but still, the only one we know are finite well-orders. This means that for now, all we've done is a formal definition of what is a natural integer, but we did't go beyond infinity. Every set we know is finite.

And here comes the deus ex machina, the axiom saving us: the collection of all natural numbers is a set. Let's call it $\omega$. Why does its existence matters ? Because now, we have a new well-order! One that we did not know previously existed, an infinite one!

Before we go further, we should notice that what I called $\omega$ may look like what is usually called $\Nd$. Why naming it differently? Because I swiped a few things under the rug, some weird technicalities, like that we cannot be sure that $\omega$ and $\Nd$ are exaclty the same object, that $\Nd$ do is a set and some other odd stuff. You shouldn't care too much about this, because a teacher once told us that "it is a matter of faith".

What does $\omega$ looks like? It looks like a ladder, like all the finite numbers we just (re-)created, but an infinite one. It never stops an goes on and on and on forever! Nonetheless, it goes like that in only one direction, only upward; it is solidely rooted in the ground at the rung $0$.

### The idea of doing this came to me thanks to ...

• Grant Sanderson, author of the YouTube channel 3Blue1Brown;
• some of my teachers whom I won't name but did a really great job;
• my love of HTML, CSS and mathematical logic.

### The images come from ...

• Jonathan Hoefler, Public domain, via Wikimedia Commons, for the Wikipedia logo;
• Reidab, Grunt, 3247, Pumbaa, Public domain, via Wikimedia Commons, for the Wikimedia logo;
• and me: the remaining ones all are homemade, handmade, free-to-use, public domain SVG files.

### The fonts are ...

• Ovo, by Nicole Fally, used for all the content texts;
• Alegreya Sans, by Juan Pablo del Peral, for all the titles.