User blog:Planterobloon/The Fast-Growing Hierarchy Explained

Do you ever see the FGH everywhere and think "man, I wish there was a community of large number fans to explain it to me"? Learning it by comparing it to other notations is all well and good, but if you're like me, you don't have time for all that. I had to learn BEAF while stroking my BEAF, for Munafo's sake, and judging by the recent posts, I'm not the only one! I always wanted a quick explanation for what all those weird letters mean, and now I'm giving it to you!

Getting started
It starts off with the successor function, f0(n), because it looks like fun. This thing just adds 1 to n. Obviously we want to get a bit bigger, and that's where recursion comes in. The FGH just does the same thing over and over again. If we go to f1(n), it runs f0(n) twice (so n+2). If we do f2(n), it runs f1(n) twice (n+4) So fx(n) = n+2x. This might not seem very powerful at first, but imagine a number like f50(n). You'd have to increase n by 250, or in other words, 1,125,899,906,842,624. That's a 16-digit number!

Now that your mind is blown, let's use the FGH to figure out how big some numbers are. We all know the grangol, right? It's Sbiis Saibian's most famous googolism, but how big is it? How do you compare it to other numbers? That's what the FGH is for! If we go to its wiki page, we see that the grangol is around f3(100). All we have to do is add 23 to 100, and voila, the grangol is equal to 108. The grangol is actually way smaller than a googol, which is f2(323), or 327, but for some reason the wiki says it's bigger. Must be those sockpuppets again!

You may notice we're not using MathJax. That's because, as a wise woman once said, ain't nobody got time for that. We don't need "university standards" here, amirite?

The first ordinal: w
So we already know that this thing is really powerful, but I'm about to show you something so gigantically huge it cannot be described. See, there's this thing called w that can turn itself into whatever n is. I'm not sure why they couldn't just put n down there too, but math people are weird like that. *sigh*

fw(n) grows really fast. If we put fw(10), that's only 1,034, but if we go up to fw(100), that's 1,267,650,600,228,229,401,496,703,205,476. If that isn't fast-growing, I dont know what is! Fun fact: the gugold is equal to f101(100), so you basically just have to double the number we just got and you get a really big number.

There's also things like fw+1(n) or f2w(n), which make w bigger than n. So fw+1(100) is f101(100) and f2w(100) is f200(100). You can even put in powers of w, so fw3(100) is f1000000(100). Holy crap, that's big! Hypercalc says that number has over 300,000 zeroes. Bet you've never seen anything that big, huh?

We can even combine operators, so we could have 2(w3)+1, for example. If you wanna get really crazy, we could even put w in the exponent, like ww or even ww^w! What would fww^w(100) look like, you ask? Well, first we take 100 to the 100th power, so that's 10200. Then, we take the hundredth power of that, so it ends up being f1020000(100). Hypercalc says that's almost 1010^20000! That's clearly overestimating it, since we took the power of 2, not 10, so really it'd be way smaller, but it'd still be really huge.

But what if we could go even further?

The second ordinal: E
These next numbers leave the last ones in the dust. Don't read this one if you have liver problems, are pregnant or may become pregnant, or have been to a region where certain fungal infections are common, because these will blow your mind and you'll never be able to stop thinking about them. Don't say I didn't warn you.

Unlike the last ordinal, this one's a capital because it's way more powerful. E0 is equal to w^w^w^w^w... with n w's. If we took fE0(100), our number would be as big as 2^100^100^100^100... with 100 100's, plus 100! A lot of notations cap out at E0, and it's easy to see why when it's so powerful! You can do all the operations with E0 that you can do with w, like E0+1, 2E0, E0^w and even E0^E0. To make things simpler, when you have something like E0^E0, you just merge the power towers together, so fE0E0(100) would have 200 100's in it, fE0E0^E0(100) would have 300 100's in it.

After E0, we have E1. E1 is E0^E0^E0...E0 with n E0's. It resolves into a power tower of w's with n2 w's in it, and then a 2 on the bottom, plus n. fE1(100) is 2^100^100^100^100... with 10,000 100's plus 100, for example. Just like E0, you can have E1^E1, E1^E1^E1 and so on, until you get E2, which is to E1 as E1 is to E0. Therefore, E2 has n3 w's in the power tower. The pattern keeps going like this. After E2 you have E3, which has n4 w's in the power tower, E4 (which has n5 w's) in the tower, and so on until you get to Ew.

Ew is exactly what it sounds like. The number of w's in the power tower is nn+1. You can carry out the same operations on the w as you normally could, so you can have E(2w), E(w^w^w) and so on. You can also carry out operations on the whole Ew, so Ew+1, Ew^w, and so on. Eventually, you'd get to EE0. Let's try a small number for this one: fEE0(3). This turns into Ew^w^w, which turns into E(27^3) = E19683. You now have a power tower of 319684 3's! If you wanna get even crazier, you can do things like EEw, EEE0, EEEE0, andso on.

There are bigger ordinals, like Z0 and U0, but no one really uses those, so I think it's fine to stop here. Thanks for reading, and I hope you understand the FGH a lot better now!

But seriously though...
I've had this idea in my head for a while of "what if someone wrote about the FGH, but they didn't understand it at all?". I threw in every possible misconception I could think of. I spent a little more time than I should've on this. If some newbie reads this way after April Fools' and thought this was serious, sorry I wasted your time, but hopefully when you know more about the FGH you can come back and laugh at this. I'm too lazy to figure out how powerful the "wrong-growing hierarchy" is right now, but I'd be interested in knowing how it stacks up to the real FGH. I'm pretty sure Ew is around f3(f3(n)), but I'm not sure how strong it is after. How long would it take to get to fω(n)? If someone could do a quick analysis that would be awesome!