®

I’ve always been very interested in mathematics. Since high school I’ve been kinda obsessed with set theory. Set theory has its roots in ancient greek philosophy, and is the most basic form of mathematics; it’s also the most difficult to understand.  At first it’s very simple and straight forward, but as you delve into it, you’ll quickly realize that it has the potential to drive you completely crazy.

A set is a collection of objects. That’s it. From that very basic concept, ALL of the modern mathematics currently known to man can be derived. Crazy right? That’s nothing. 

A set is specified by its contents. That is, every set is defined by what it contains. If you can’t define the contents of a set, it’s impossible to know what does and what doesn’t belong to that set. This means that the set is “undefined.”

We use curly brackets to denote sets, like this:

A set of my favorite brands = {vans, alife, supreme, levi’s, apple}

But what about a set that doesn’t contain anything? How is that described?

That’s the empty set… My favorite set. Not only because its the most famous set, but because it can be used to define numbers.

We represent the empty set by curly brackets:

The empty set = {}

(You may be asking yourself “why is there a need to define numbers?” I understand that it may seem crazy to even care about this, but that’s what mathematicians do. We obsess about the most subtle facts of nature.)

Check it out:

0 = the empty set = {}
1 = a set that contains the empty set = {{}}
2 = a set that contains a set that contains the empty set = {{{}}}
3 = a set that contains….                                            

You get the idea right? So numbers are actually just representations of certain sets. Imagine what it would look like if we had to write out the set notation for 1,094,889,648,946,149,361,847,291,748,937.

The empty set only helps us describe whole, positve numbers. How can we define numbers like -1, or 12.9898, or numbers that never end, like pi? That’s another topic altogether. 

And what about sets that contain an infinite amount of objects? Are they all the same set? NO. They’re not. This means that there exist different types of infinity. Crazy right? That’s nothing.

If you’re still reading this you’re probably interested in set theory by now. Here’s some more info on the topic.