I don’t think you have any idea how fast I am

I’m fast as f_Γ0(n) boi

What on Earth is f_Γ0(n)?

Remember the last post, Fast as (insert word here) boi, where I talked about the fast-growing hierarchy? These are just extensions on it.

Let’s take a step down.

Do you remember ω? The ordinal that used the diagonal of the hierarchy table? (I’ll be completely honest, that sounds like a good book title) You can use ω in different ways, like ω + 1, ω2 (ω times two, or ω + ω), ω^ω and much more. Eventually, we will run out of numbers to do those operations with, so we have to move to the next step:

ω^ωωωω… until you run out of ω, in which it becomes epsilon-zero, or ε₀. There also exists a f function for it, f_ε0 that grows so fast. Don’t think about thinking about it. You can do weird things to ε₀, like ε₁, ε_ε0, ε₀^ε₀, ε_ω, ε_εε0, and more.

Surely that’s it?

No.

ζ₀ exists! You can also do “normal” things like powers, epsilon-ζ₀ (I was too lazy to copy and paste), and stuff.

That’s it. Right?

Wrong.

η₀.

Whyyyyyyyyyyyyyyyyy

There’s more.

Feferman-Schütte ordinal, named after Solomon Feferman and Kurt Schütte. That’s the one that was in the beginning of this post.

Yet that wasn’t enough to stop a beast.

TREE.

We have to go one more step further. ϑ.

f_{ϑ(Ω^ω, ω, 0)}(n) is the approximation for the TREE function.

If you ask me, I’ll say, “I don’t know.”