How large is infinity?

If you are like most people, then you have probably heard children arguing over some kid thing. Have you ever been astonished how easily they resort to infinity? It's like these kids don't even know how large it really is!

But how large is infinity? Let's start looking into it with some actual numbers that you can write down in some form. The first really big and easy to understand number is googol. You can google it, googol is a real thing, and Google is actually a misspelling of googol. (Fun fact: the study of very large numbers is called googology.)

Googol is simply \(10^{100}\). That's 1 followed by a hundred zeros, so yeah, it's quite a large number. One way to illustrate the size of googol is to compare the mass of the electron, which is \(10^{-31}\) kg, to the estimated mass of the universe, about \(10^{50}\) or \(10^{60}\) kg.

If you take one googol of electrons, their mass will be up to \(10^{19}\) times larger than the mass of the whole universe. That's about the same ratio as between a medium sized cargo ship and the Earth. So googol really is quite large, right? Let's take it up a notch.

Humans really like prime numbers. We are basically obsessed with them. Prime numbers are a whole branch of study in mathematics, and it gets a lot of attention. Primes are just numbers that are evenly divisible with themselves and one, big whoop.

Well, the big whoop is that they can be used for encryption purposes. If you take two really large prime numbers and do some mathematical tricks with them, you can make almost an unbreakable encryption. There are some really big primes that are illegal to own for that reason.

But owning them would be rather cumbersome. You see, the largest currently known prime is \(2^{77,232,917}-1\), which has 23 249 425 digits. For comparison, the googol has only 101 digits.

If you wanted to print out the largest known prime, it would require about 10 000 A4-pages (give or take a few thousand). Take the biggest book you know and look at how many pages it has. I can guarantee you, it does not have 10 000 pages.

So some primes are unbelievably huge, surely it doesn't get much bigger than this? Don't worry, it does. A lot bigger.

Graham's number is a good example of a ridiculously large number that has been used in a mathematical proof. It's definition is a bit technical but let's take a crack at it. First you start from the number three. Then you raise it to the power of three and then raise also it's exponent to the power of three, like this \(3^{3^{3}}\), which yields 7 625 597 484 987.

Then you take another 3 and raise it and it's powers to the power of three \(3^{3^{3}}-1\) times, so that you basically get a huge tower of powers of three like this \(3^{3^{.^{.^{.^{3}}}}}\). I think I'll be lazy and not write that out, so the new number is just \(x\).

After that, you again take a three and raise it and it's powers to the power of three \(x-1\) times. So you raise a three to the power of three as many times as the last product minus one. Then you repeat this process a total of \(3^{3^{3}}\) times.

It's nuts, right? This has got to be the biggest number ever in the history of mankind! Actually, this process yields only the first iteration cycle of Graham's number, which is usually denoted as \(g_1\).

To get to the next step, \(g_2\), you need a similar iteration cycle, where you first raise a three to the power of three \(g_1\) times, which yields some number \(y\). Then you do the cycle \(y\) times and you get the \(g_2\).

As you can imagine, the number we get from this iteration increases very rapidly. And we do this a total of \(64\) times, to finally get the Graham's number, or in other words \(g_{64}\), which is an unimaginably large number.

In fact, it's so large that you can't print it out, express it numerically or even store it. It's just physically impossible, there aren't enough atoms or space in the observable universe to do that. That's why we have to express the Graham's number with this iterative cycle, it's the only way to contain it!

Now this must be the biggest number there is, right? Sorry, but it's not. There are several larger numbers than the Graham's number, for example TREE(3), which is such a large number that \(g_{64}\) seems like an insignificant speck of dust when compared to it.

The definition of TREE(3) is even more technical than the definition for Graham's number, so I'm just gonna leave here a Numberphile video explaining what it is in simple terms

And you know what? As humongous as TREE(3) is, it is completely insignificant when compared to Loader's number, which in turn is only a tiny fraction of BIG FOOT, whereas Utter Oblivion is still much larger than that!

Actually, Utter Oblivion is probably the largest (although not that well defined) number known at the moment. The thing is, there is no limit to how large numbers we can define, and Utter Oblivion will probably become insignificant in a few years.

Is your brain hurting yet? Well here comes the grande finale: any number, even the unimaginably ridiculously huge Utter Oblivion (or any number that is larger than it) is closer to zero than to infinity. That's how large infinity is.