On Wikipedia and the Googology Wikia, it’s complicated and long and boring and zzZZZZZZ… so I’ll try my best to simplify it.

Define a function, f_{0}(n) = n + 1. It doesn’t grow so fast, does it?

For f_{a}(n), that is f_{a-1}^{n}(n), where that denotes doing the previous function n times.

f_{1}(n) would equal out to 2n, because iterating f_{0}(n) n times would just be adding n to n, or 2n.

f_{2}(n) is also iterating f_{1}(n) n times, which is doubling each time, so it would equal out to 2^{n}n.

I’ll do a couple more.

f_{3}(n) is… its lower bound is 2^{n}n((2^{(2^n)n})↑↑(n−1)).

f_{4}(n) has a lower bound of f_{3}(n)↑↑↑n.

After the finite ordinals, comes ω, the ordinal infinity. To find a fast-growing f function faster than the g function for Graham’s number, or the TREE function, we need to involve ω.

Let’s think of a table of f functions:

f_{1}(1) = 2

f_{1}(2) = 4

f_{1}(3) = 6

f_{1}(4) = 8

…

f_{2}(1) = 2

f_{2}(2) = 8

f_{2}(3) = 24

f_{2}(4) = 64

…

f_{3}(1) = 2

f_{3}(2) = 2048

f_{3}(3) = According to Numberphile, a number with 121 million digits

f_{3}(4) = uyefwgiefwugewf

…

f_{4}(1) = 2

f_{4}(2) = WTF

f_{4}(3) = Just no.

f_{4}(4) = Let’s not?

…

…

…

…

…

…

The table of fast-growing hierarchy!

The following information is provided originally from Numberphile.

Think of the diagonal from the upper-left corner, descending south-east. It’s faster than any function that came before it. f_{ω}(n) is that diagonal, so:

f_{ω}(n) = f_{n}(n)

It has a lower bound of 2↑^{n-1}n, which is also the single-argument Ackermann function defined by Harvey Friedman.

The next step? ω + 1. f_{ω+1}(n) just does the same thing, so f_{ω}^{n}(n). Why is f_{ω}(n) and f_{ω+1}(n) different? Because ω is an ordinal infinity, where order matters, unlike cardinals.

f_{ω+1}(n) is HUGE. It’s iterating f_{ω}(n) n times, which is so huge. Its lower bound is {n, n, 1, 2} in Bird’s Array Notation.

Interesting fact: Googol is between f_{2}(323) and f_{2}(324).Another one: an approximation for Graham’s number is f_{ω+1}(64). For TREE? Uhh… if you want to know, it’s in the next post about fast-growing hierarchy.

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