Infinity and Beyond

They say that you only really learn something after you’ve taught it for the first time. After trying to explain the concept of countable and uncountable infinity on Reddit, I decided to start this blog to try to explain concepts that I’ve learned a while ago to people that don’t necessarily like math. It will help me retain stuff and maybe show people a glimpse of what math is actually about beyond the boring high school curriculum that most people never see past.

The first time that I saw the proof for the concept that there are many infinities and not just one, my mind was figuratively blown. It’s a neat argument, so for now I’ll just provide some background information. An actual proof will have to come another day.

To understand this concept we begin by understanding sets, don’t worry sets are fairly easy.  A set can be a collection of any number of things. For example, the set of :a chair, a car, and a house. Here I’d like to throw in some notation that will come in handy later. When writing sets we write, {chair,car,house} for the above example. We can also give the entire set a single name. I can call the previous set A, by saying A = {chair,car,house}.

Math people prefer to start with numbers though so now I will move on to some less ridiculous examples. The set of numbers 1 through 10 is a set. It would be written, {1,2,3,4,5,6,7,8,9,10}, or a simpler notation is {1,2,…,10} with all of the numbers implied that are between 2 and 10. Keep in mind that we eventually want to talk about the size of sets, since our goal is to talk about the size of infinity.

It is simple to see that, the set A = {1,2,…,10} is larger than the set B= {1,2,…,5} since the first one has 10 things in it, and the second has 5. You’ve noticed I’m sure that the second set is actually inside of the first set so of course it is bigger! Although this is true, it is not the reason for the first set being larger because I can just put 5 things in the second set that aren’t numbers and it would still be of size 5 and therefore still smaller than the set A.

To deal with sets that don’t have comparable things inside, we bring in the concept of matching things up in two different sets. Lets go back to my first set and I’ll call it A = {chair,car,house} and I’ll call B = {1,2,3}. You can see right away that they have the same number of things, but to actually prove it we do the following.

Match up chair with 1, car with 2, house with 3. Since everything in the first set can be matched up precisely with one thing in the second set, the sets must be the same size. Using this concept we can move to infinity (and beyond).

The first time that people think about infinity is usually on the playground during an argument. “Well I have 10, well I have 100, well I have infinity, well I have infinity + 1.” So that is where we should begin also.

When we first learn to count we start at 1, then 2, then 3, and so on forever (if we had infinite time). I’ll call this set N = {1,2,3…}. It is called the set of “natural” numbers, mostly because it is a pretty natural set to think about. Just regular old fashioned numbers, no decimals or weird things. Clearly we cannot use any number to describe how big this set is because it is bigger than literally all of the numbers. It has size infinity.

Now, you probably are trying to think a few steps ahead of me and are thinking “aha I know of a bigger set, just put the number 0 in there.” Unfortunately, that is not the bigger infinity I am talking about and I’ll jump back to that in a second.

Instead I’ll first talk about the set of even numbers E = {2,4,6…}. Again you probably assume that this set is smaller than N. We all make that mistake, but to actually compare these two sets we have to go back to the concept of matching up things in the sets into pairs. If we can match everything up, then they are actually the same size. Luckily we can do this in a pretty simple way.

From = {1,2,3,…} I match everything up with it’s double in = {2,4,6,…}. So 1 matches up with 2 (from E), 2 (from N) matches up with 4 (from E) and so on. Since everything from both sets matches up to exactly one thing, the sets must be the same size. Yes, they are both what we call not just infinity, but “countable infinity.” The reason for this name is because we could, given enough time, count everything in the set. We can actually write everything in the set in a list.

With the same idea I just did, you can show that {0,1,2,3,…} is also the same size, that is countable infinity. Also the numbers = {…,-2,-1,0,1,2,…} are of the same size and so are the rational numbers = {all numbers that can be written as a fraction}. That last one takes a little more work to show, but it is a fun thing I might do next time.

I suppose this last part will be anticlimactic because I won’t go into as much detail on why it is true until next time. So what is the bigger infinity? I’ll start with ALL of the numbers between 0 and 1. What does this consist of? Well we already talked about the rational numbers, so how about the irrational numbers. Irrational numbers are numbers that do not have a fractional representation. They have infinite non repeating digits. One example is the number pi = 3.14…

So when I say ALL of the numbers between 0 and 1 I am talking about all of the rational numbers between 0 and 1 and all of the irrational numbers between 0 and 1. (It turns out you don’t even need the rational ones actually.) This set of numbers is clearly also infinity, but I claim that this set cannot be paired up in any way with the early countable infinity sized sets.  The proof of this is pretty nice, so I’ll work on that for next time, but know it is true for now and you can use this fact to blow minds.

How big is this set? Well actually it’s just as big as the set of ALL numbers, that is all rationals and irrationals and not just the ones between 0 and 1. It is counter intuitive, but you can match up all of the numbers between 0 and 1, to all of the real numbers. Remember, just because one set is inside the other set, doesn’t mean that you can’t do this. We matched up the even numbers with the natural numbers even though the even numbers are inside the natural numbers, so this should give you a reason to believe.

This infinity also has a name, it is called an “uncountable infinity.” This name comes from not being able to write every number in a list. It is actually the smallest uncountable infinity. To build higher infinities from this one you have to understand the idea of a “power set.” I’ll throw that in next time too I guess.

Alright, I’ll try to do the harder proofs for next time. I tried to write this so that non math people could follow along.

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