TITLE: Definition of Ordinals in Set Theory in Layman Terms
QUESTION [5 upvotes]: I've taken a huge interest in the mathematical concept of infinity and often been contemplating the same over years. But the fundamental concept of set theory ordinals continues to evade my understanding. The questions below comprise (more or less) the gaps in my comprehension of the mathematical infinite:

Ordinal numbers in general (1st, 2nd, 3rd, 4th...) are entirely different from ordinal numbers in set theory, correct?
I understand that set theory ordinals are basically sets that contain a least element by definition. But, is it necessary for the elements of an ordinal to be strictly in order? For example, must the ordinal 4 be represented as {∅, {∅}, {∅,{∅}}, {∅, {∅}, {∅,{∅}}}....} and not as {{∅,{∅}}, ∅, {∅}, {∅, {∅}, {∅,{∅}}}....}?
The cardinality of ω is א‎0 (please correct me if I'm wrong), but where exactly is the position of ω along the number line. Is it א‎0th position (so to speak)?

I apologize for the naivety of the questions above (honestly, I really don't find a layman explanation of ordinals anywhere on the web. I saw a very good YouTube video though). The objective is to understand the core concept of set theory ordinals (well enough to be able to explain the same to a layman) rather than memorizing formal, mathematical definitions with little to no true comprehension of the same. Thanks in advance!

REPLY [3 votes]: To understand ordinals you first need to understand what is a well-order.
Well-orders are a type of strict partial orders which satisfy the following axiom: Every non-empty set has a minimum. This implies that well-orders are linear orders, and that they look kinda like the natural numbers. The most important property of well-orders is that we can use them for inductive constructions and proofs, much like the natural numbers.
Now, when we want to do something by induction and use some well-ordered set, the specific set does not matter. What matters is how does the set "looks like". In mathematical terms we care about its order type, rather than its actual elements. Two partially ordered sets have the same order type if there is an order isomorphism between them: a bijection between the two sets which preserves the order attached to each set.
Let's consider a quick example: $\Bbb N$ and $\Bbb N\cup\{-42\}$, both ordered by the usual order of the integers, are isomorphic, they have the same order type. The isomorphism can be even given explicitly, $$f(x)=\begin{cases}0 & x=-42\\ x+1 & x\in\Bbb N\end{cases}$$
I will let you check and see that this is an isomorphism.
Two important theorems about well-ordered sets are these:

If $A$ and $B$ are two well-ordered sets, and there is an order isomorphism from $A$ into some $B'\subseteq B$ and from $B$ into some $A'\subseteq A$, then in fact there is an isomorphism between $A$ and $B$. Moreover, this isomorphism is unique.

This is a non-trivial fact. Take a look at $\Bbb Q$ and $\Bbb Q\cap[0,1]$. We can embed each into the other, but they are not isomorphic: one of them has a minimum and maximum and the other does not. And even if we consider a simpler example, $\Bbb Z$ and $\Bbb Z$ itself, there are many ways to move the elements around while preserving the order.

If $A$ and $B$ are two well-ordered sets, then either $A$ is isomorphic to an initial segment of $B$ or $B$ is isomorphic to an initial segment of $A$. (And this isomorphism is unique, as a consequence of the previous theorem.)

Again, this is nontrivial. Consider $\Bbb Z$ and $\Bbb Q$. Neither one is isomorphic to the other, and neither one is isomorphic to an initial segment of the other.
So what do these two theorems tell us? They us that well-orders are very robust, they have great comparability properties, and so if we only consider equivalence classes of well-orders under isomorphism, this induces a natural ordering between the classes, which itself happens to be a well-ordering. That's great!
But now we have a mild problem. Other than the empty order, there is a proper class of well-ordered sets from each order type. Just note that for the order type of "one element", every singleton can be a candidate for the well-ordered set, and there is a proper class of those. This is not a huge problem, but wouldn't it be nice if we could replace these proper classes by sets?
At some point in the early years of set theory, von Neumann suggested that instead of thinking in terms of abstract well-ordering, since they are so nice, we pick canonical representatives from each isomorphism class. And he proved that you can find a unique set which is transitive and well-ordered by $\in$ inside each class.
(Here we say that $A$ is a transitive set if whenever $a\in A$, it follows that $a\subseteq A$.)
And this is the von Neumann ordinal assignment. It picks from each order type, the unique transitive set which is well-ordered by $\in$. Then, a magical thing happens, it turns out that $0=\varnothing$, and the successor of $\alpha$ is $\alpha\cup\{\alpha\}$, and that if $\{\alpha_i\mid i\in I\}$ were defined, and $I$ is a well-ordering without a maximal element, then $\bigcup\{\alpha_i\mid i\in I\}$ is the limit of the $\alpha_i$'s.
And that gives us a very robust structure for the von Neumann ordinals: $\alpha=\{\beta\mid\beta<\alpha\}$, where $<$ here indicates "isomorphic to a proper initial segment".
Let me digress now, and since you brought $\aleph_0$ into the discussion, talk about cardinals for a moment. Cardinals are also isomorphism classes, like order type, only here we use just plain ol' bijections without requiring they preserve any structure. In the days of yore cardinals were also proper classes (except that of the empty set). We would say that $A$ has cardinality $\aleph_0$ if it has a bijection with the natural numbers, for example. 
As set theory grew, we realized that if $A$ can be well-ordered, then there is a smallest order type of such a well-ordering (because the order types of well-ordered sets have a well-ordering themselves). This meant that we can assign to a set that can be well-ordered the least ordinal that can be put in bijection with that set. And after you have the von Neumann ordinals, this points at a specific set, too. Of course, assuming the axiom of choice every set can be well-ordered and that's fine, but not assuming choice, we need to find a better solution, and I won't discuss it here.
We have since come to confuse cardinals with ordinals on a regular basis. This is fine if all parties involved are well-aware of the differences and the implicit contexts and types. But I would generally advise against it (and I try my best to discern in my work between the cardinal and the ordinal, whenever possible). The two systems are equipped with addition, multiplication and exponentiation. And they often disagree. $\aleph_0+\aleph_0=\aleph_0$ as cardinal addition, but $\omega+\omega\neq\omega$ as ordinal addition.
So talking about $\aleph_0$ as if it was an ordinal is technically not a mistake, but it can lead others to mistaken conclusions (I recall several occasions that I had struggled with such notation; and historically we can find examples of people who didn't understand implicit contexts and made mistakes.)
Now, you asked whether or not ordinals denote "first" and "second" and "$\alpha$th". And the answer is yes. An ordinal is the answer to "how long is the queue to the bathroom". Cardinals, however, are the question "how many people are in the queue to the bathroom". As long as the queue is finite, the answers are the same. But if the queue has order type $\omega$, then every person has to wait only a finite time before using the facilities; but if you join at the end of the queue, we didn't increase the "how many" answer, but you clearly have to wait an infinite time before you can get to the bathroom.
So using $\aleph_0$ to denote a location in the queue is not ideal, as I remarked. Sure, you can insist that $\aleph_0$ and $\omega$ are both designated by the same set, therefore you can talk about the $\aleph_0$th ordinal being $\omega$. But it is much better to separate in your mind the two notions of cardinals and ordinals, at least until you have a better grasp of the two.