Episode 1: The Continuum

Starting this week, I am going to try and do a “Math Post” every Friday. That is, I’m going to share with all of you something really cool that I’m learning about or that I have learned in the past. I think that this will be a great way for you to understand why I love math so much. Plus, you get to learn really cool things!


If you’re a math person and you’re reading this, then it will be a nice little refresher (or you can just stop reading). If you don’t consider yourself a math person, you’re in for a real treat (hopefully, if I don’t confuse you too much)! Enjoy!





Episode 1: The Continuum


So whether you want to believe it or not, everyone knows what a set is, and everyone has some sort of set. You have a set of socks, you have a set of shirts, and you probably have sets of silverware. This is how it works for numbers too. You take a group of numbers, and you basically have a set.

So for example, {1, 2, 3} is a set. We consider {1} to be a subset of {1, 2, 3}. We call the set which contains all possible subsets of a set the Power Set. So, the Power Set of {1, 2, 3} is

{ { }, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3} }

In fact, a set of n elements (in our case, n=3) has a Power Set of 2^n elements (in our case 8).

Cool, right? But what if our set looks something like the Natural Numbers,

{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 …}

Or, maybe even scarier, the Integers,

{ …, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, …}

Or, maybe even worse, the Rational Numbers,

{ 0, 1, 1/2, -1, 2, -1/2, 1/3, 1/4, -1/3, -2, 3, 2/3, -1/4, 1/5, ...}

How can we tell how big these sets are? They look like they go on forever (infinite). But is that it? Do we just label them as infinite sets and move on?

Actually, amazingly enough, all of these sets are smaller kinds of infinity. We call these sets Countably infinite. Meaning, we can make a list of everything in the set. Yes, all three of those sets are the same size of infinity. Doesn’t that seem strange?

Are you thinking it couldn’t get weirder?


Pshh, you haven’t seen anything yet :)

What about π? You know that number, right? The one everyone knows the first 3 digits to: three point one four something something? OR, what about √2?! Or √287?

These numbers, along with the Natural Numbers, the Integers, and the Rational Numbers, and a whole bunch of others (the Irrational Numbers), belong to what we call the Real Numbers.

All the Real Numbers live on the Real Line, which is called the continuum. Any number you can think of (as long as it isn’t imaginary) is a Real Number.

“A continuum is something every part of which has parts.”

Ok, I know, I know. I might be losing you, but don’t try and think about it too hard. The Real Numbers are uncountably infinite. Meaning, we cannot count the Real Numbers like we did with the other three sets. There is no place to start, and no order we can go in. It’s just crazy to even try to think about it.

BUT, now we are starting to get into the really cool stuff. There are sets of numbers which are BIGGER than the continuum. Crazy, right?!

Remember the Power Set? Imagine the Power Set of the continuum…

Whoa…

A SUBSET of this would be something like this:

{ {π, 1} {π, 2}, {π, 3}, {π, 4}, {π,5},... {π, 1,2}, {π, 1, 3}, {π, 1 , 4}, {π, 1, 5}, {π, 2 , 3}, {π, 2 , 4}, ... {π, 1, 2, 3}, {π, 1, 2, 4}, {π, 1, 2, 5}, ... }

..and that's just a SUBSET!!


What about the Power Set OF the Power Set of the continuum?

Eek!



Can you see now that there are infinitely many types of infinity?


"infinitely many types of infinity"







Doesn’t that just blow your mind?



Maybe a little?







Remember this from yesterday:

"Do you know how many there are? There's continuum many!...
That's not too many, you know!"

It makes more sense now, doesn’t it?




Isn’t math remarkable?