In the last post we prepared the ground by defining **Well-ordered** sets,–**transitive **sets, and talked about the order .

## What is it?

It is now finally time to define what ordinals are: A set is called an **ordinal** if it is:

- -transitive
- well ordered by the order ‘‘ ()

Ok, in first sight, this definition doesn’t look that interesting or intuitive. The best way to feel what ordinals are is by examples:

## Who are the ordinals?

The most basic ordinal I can think of is the , which is a trivial example, What about the set ? It is transitive since there are no elements in the only element of the set (which is the empty set), and it is well order since the only susbet of is itself (excluding the empty set) which is well ordered, thus, it is an ordinal.

Let’s proceed, what about the set: . This set is indeed transitive ( and ) and well ordered, thus, it is an ordinal.

Well, I think I got it, from the same logic I **think** that the set **should** also be an ordinal, let’s check: in this set we have:

\emptyset\in \{\emptyset\}\in\{\{\emptyset\}\}

Hold up for a moment, , then . Thus, ‘‘ is not even transitive, therefore, this is not even a valid order! and certainly not an ordinal! As it turns out, my intuition betrayed me! However, Now I know that I need to be carefull, and create a set where is indeed an order, Let’s try this one:

O=\{\emptyset, \{\emptyset\}\ ,\{\emptyset, \{\emptyset\}\}\}

That one is going to work, trust me, I checked! This leads me to the next logic, We start with some ordinal , and create another ordinal by taking and add itself to it. i.e.

\begin{array}{c} \emptyset\\ \{\emptyset\}\\ \{\emptyset,\{\emptyset\}\}\\ \{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\}\}\\ \{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\},\{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\}\}\}\\ \{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\},\{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\}\},\{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\},\{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\}\}\}\} \end{array}

Those are the first 6 ordinals according to this logic. However, I still need to justify this logic formally.

Before I’ll prove it I want you to notice that: If is an ordinal, then , why?

Suppose that , since the order on is we have which is a contradiction to the order’s first axiom : anti-reflexive.

## The ordinals genarator

so let’s prove my ‘conjecture’: If is an ordinal, then is also an ordinal. In order to prove that we need to show few things:

is -transitive : I proved in the last post that if is transitive, then is transitive as well.

is a valid well-order:

**Anti-reflexive**

Suppose that . Then we have two options:

- : Then since is ordered with respect to , we get that .

- :Then and I just proved that since is an ordinal, then .

**Transitive**

Suppose that such that . Again, we have two options:

If : thus .:

Since is transitive and we get that . We know now that , thus .

Therefore . Recall that is an ordinal, then, from the transitivity of ‘‘ we get that .

On the other hand, if : Then . Since is transitive we know that , therefore .

**Well ordered**

Let’s pick an arbitrary subset . And once again, we are facing two options:

If : Then is the first element in .

Else, . Since and is well ordered, then has a first element .

In fact, , then even if , is ‘still’ the first element of .

**Proof review**

And that’s it, we’ve proved that is indeed an ordinal. I think this proof forced me to ‘get my hand dirty’ and play with the definitions carefully. A good practice for you would be to try and prove it again by yourself, it’s a really good test for yourself that checks if you understand the definitions and know how to use them.

This theorem allows me to define a function which is kind of an ‘ordinal generator’. It’s input is some ordinal and it’s output is , which is also an ordinal.

S(A)=A\cup\{A\}

I’ll come back to this function later.

## Comparing ordinals

In this post I have one main goal: I want to prove a **super strong** property of ordinals: If are two ordinals, then or or .

That’s it, those are all the options! this theorem shows that you can ‘**compare**‘ two ordinals, **any** two ordinals! This property will be extremely useful for us, and I think you can already imagine why such a property is useful.

However, proving this theorem **won’t** be a walk in the park, I have to prove some more theorems first in order to prove the desired property.

## Some properties of ordinals

#### Transitive subset

Let’s pick some ordinal and a **transitive** subset . If you recall, I have proved in the last post that a subset of a well-ordered set, is also well-ordered, thus is well-ordered, then we found that is also an ordinal.

However, we can take this one step forward: obviously, it is possible that will be itself, but what if it doesn’t?

That means the subset is non-empty and since is well ordered, it has a first element which I’ll denote by .

Notice that (since ).

Let’s pick now some element . Since we get from transitivity that , and recall that by definition, is the ‘smallest’ element which is not in , thus , and we just proved that .

Now, we can pick another , we know that , and we also know that . Since is well-ordered, it is in particular linearly ordered as well. This gives us options:

- which is impossible since .
- , and from the transitivity of we get that which is, again, impossible.

Thus and we proved that . Combinig the to results to get that , and , thus .

I just proved that a transitive subset of an ordinal is also an ordinal and even more than that: I proved that it has to be the whole set (which is the original ordinal) or an **element** of the original ordinal – And if that is the case, the element is the smallest one which doesn’t belong to the subset!

To summarize it shortly – we found that a subset of an ordinal , is an ordinal **itself**, and it is also an **element** of .

#### Element of an ordinal

Let’s try something a little different now, again we are starting with some ordinal , however, we are picking an **element** . Let’s see what we can learn about such an element:

Suppose that since , we can conclude (as we did here in the first case of ‘transitive’) that . Since ‘‘ is an order on , from transitivity, we get that , thus, is a transitive set.

Moreover, since is transitive and , we know that if then as well, thus, , and we can now use our latest result to get that is an ordinal as well.

This allows us to conclude something even stronger: If is an ordinal, then **proper subsets** (not the whole set) of and **elements** of are the **same thing!**

## Set of ordinals

I need to prove just one more theorem to reach my goal for this post: Suppose that is a set of ordinals, then is an ordinal

In addition, it is the first element in with respect to the order .

moreover, it is an element of all the elements of ( for every ).

I’ll prove it in three steps:

##### Step 1: ** is an ordinal**

Iv’e proved that intersection of transitive sets is also transitive in the last post. Thus is transitive.

Now pick some . We know that

Moreover, we’ve proved that must also be an ordinal as a transitive subset of one, while also being an element of or itself!

##### Step 2:

As I just said, we know that for every , we have two options:

- .

Suppose that for every .

Then it is also an element of their intersection, thus . However, since is an ordinal, we already know that such a thing is impossible.

Thus, for some , which proves that .

##### Step 3: ** is the first element of the set**

Since for any (such that ), we’ve seen that .

Thus, it is indeed the first element of , as desired.

## Finally reached my goal!

That’s it, we are now ready to prove that if are ordinals then or or :

Consider the set and the intersection . using the last theorem, we know that is an ordinal and it is the first element of the set . Thus, we have two options:

- , then is the first element and we get that or .
- , then is the first element and we get that or .

Therefore, the only three options are or or , and that’s what we wanted to prove!

## Summary

We’ve reached and proved the main goal in this post, and I think we’ve got some more sense about ordinals.

For me, they give me a sense of, well…. **order**, and order is everything I’ve been talking so far. In fact, I also feel like they behave really well, like they are super well defined or something like that… And if you know how to work with them properly and obey their strict rules, they won’t let you down.

In the next post, we will start reaping the benefits of ordinals and we will get to know the strength ordinals have even more.