From nothing to something

This is a set of… nothing.

How do we get something from this? Why don't we take this empty set and put it inside another one? (After all, we have nothing else to work with).

There you go, we went from nothing to something. Thanks for reading this blog.

THE END.

…

Not so fast.

Let's explore more. We had $\{\}$ and now we have $\{\{\}\}$. What could the next member of the sequence be be? The phrase “next member” already hint at “order”, the idea that some elements come before or after some other elements.

To get to $\{\{\}\}$, we had to "cross" $\{\}$, and that's the thing we put inside an empty set. The member after $\{\{\}\}$, is also after $\{\}$. So to get to the member after $\{\{\}\}$, we cross both $\{\}$ and $\{\{\}\}$, so it seems natural that we put those two inside an empty set. So we get $\{\{\}, \{\{\}\}\}$.

In other words, to know if something comes after something else, we just check if that member is inside it.

Does $\{\}$ come before $\{\{\}, \{\{\}\}\}$? Well, is $\{\}$ inside $\{\{\}, \{\{\}\}\}$? Yep! So it comes before.

Does $\{\{\}\}$ come before $\{\{\}, \{\{\}\}\}$? Yes again since it's also inside $\{\{\}, \{\{\}\}\}$.

How convenient! We're not just nesting sets, we've made it so that each of them remembers how it was born. It's made from all the sets that came before.

Since we are talking about sets being inside other sets a lot, I'll introduce some notation so that it doesn't take too much space (and looks fancy).

Notation: To say “set A is inside a set B”, we write down “$A \in B$”. For example, $\{\} \in \{\{\}\}$.

Why this notation? It's just what people have mutually agreed upon to use to notate this.

Let's continue this and see what we get ourselves into.

After $\{\{\}, \{\{\}\}\}$, we get $\{\{\}, \{\{\}\}, \{\{\}, \{\{\}\}\}\}$ and after that, we get:

Woah. Things are getting out of hand, it's getting too big! It seems like it would be nice to represent each of these with a symbol so that we don't have to write all these long expressions down. We already get the point.

I'm feeling too lazy to invent a new symbol for each and every one of these, not least because there are infinitely many of them. Are there some other existing symbols that already represents this idea? That is, the notion of order, some things coming after something else. Well, I’ve got an idea, we could use the symbols that denote numbers! $1$ comes after $0$, and $2$ comes after $1$ and $0$, and there are infinitely many numbers. That's perfect.

and we could really just go on and on forever as we have infinite numbers whose symbols we could use.

Writing it a bit more cleanly:

So as you can see, to get what comes after a set $n$, you put $n$ and all its previous members (already inside $n$) into an empty set.

To write down this concept in less space, I'll introduce a new notation.

Notation: A set containing all the elements of $A$ and $B$ is denoted by $A \cup B$, and we call it “A union B”. For example, $\{0, 1\} \cup \{2\} = \{0, 1, 2\}$.

In other words, the successor of $2$ (that is, what comes after $2$) is a set that contains both the elements of $\{2\}$ and $2$ (as we know, $2$ is $\{1, 0\}$). If instead of “successor of $2$” we use the shorthand “$S(2)$”, then we get:

We can use the same logic for any of these sets $n$

The fact that we can use numbers to represent these sets makes me curious. Could we also do things with these sets that we can do with numbers, like addition? $S(n)$ is already like “adding” $1$ to $n$. Could we generalize this to “add” any two sets with each other just like how we can add numbers? Let's see what we can do with what we have.