Alef is Infinite

Infinity, the never ending. One of the most interesting ideas the human mind can conceive. Something that goes on and on and on and never stops. Imagine, for an instant, that this text contained infinitely many words. In that case, even if you could read any number of words per second, still you would never finish it. This is more absurd than it sounds.

For example, the Pacific Ocean has so much less water than infinite water. The comparison is as good as comparing $1$ to infinity. The amount in the Pacific Ocean is not closer to infinity than a single droplet is. In order to convince yourself of this fact, think about spliting the Ocean in droplets and numbering them from droplet $1$ to droplet one gazillion or whatever. Still, the last one would be a finite number. Now think about what it means when we find the difference between two quantities: $10-4 = 6$ because $6$ is what you add to $4$ in order to get $10$. Very well. Now let us analyze whether or not droplet $2$ is closer to inifinity than droplet $1$: the number of droplets you would have to count to reach infinity from droplet $1$ is, well, infinitely many. If you literaly spent forever counting $1$ droplet per second, then you would be able to finish. Same thing if you start from droplet $2$, $3$, $1000000$ or from any finite number for that matter.

If we use the symbol $\infty$ to represent infinity for now, we can convey what we said above in math notation the following way: $\infty - n = \infty,$ for any finite $n$. But there is a problem: is $\infty$ really a number in the sense of us being able to do arithmetic with it? Or is it simply a symbol representing the idea of never ending and, in that sense, not a number, so no arithmetic? Well, both.

1 Enumeration

When we talk about infinity, we in general mean one of two things. More commonly, we mean something that is never ending. And it is not hard to formalize this idea:

The process of assigning these numbers is generally referred to as counting. For example, someone could open a packet of pasta an count how many individual pieces there are. Obviously, eventually you would run out of pasta as you reached some positive integer $n$ in the counting process. Suppose, for instance, that you go like “$1, 2, 3, \dots, 467, 468$”. Such $n$ ($468$, in this case) is defined to be the amount of pieces, or the cardinality of the set.

But how about the positive even numbers? Could we count them? In other words, could we assign each even number a positive integer $1, 2, \dots$ such that each even would be assigned something and no two evens would be assigned the same thing?

Well, yes. Below, you can see the even numbers to the left in a never ending column; to the right, you will see the numbers $1, 2, 3, \dots$ The arrow $\longmapsto$ means “is mapped to”, or “is assigned to”:

$$\begin{align*} 2 &\longmapsto 1 \\ 4 &\longmapsto 2 \\ 6 &\longmapsto 3\\ &\vdots\\ 2n &\longmapsto n\\ &\vdots \end{align*}$$

So, what I mean above is that we will assign to each even number of the form $2n$ the number $n$. For example, $10 = 2 \cdot 5$, so $10$ is assigned to $5$.

With that, it is very clear that all even numbers get assigned a number and that the numbers two different evens get assigned are always different.

As a double check to yourself, try to make sense that a similar thing can be done to the odd numbers.

All of this being clear, we shall give this type of infinity a name.

There are some other ways to affirm that a set is enumerable, like saying that it is countably infinite.

And now we can explore some classic examples of proving sets enumerable - or not!

2 Cantor’s Diagonal