Are you interested in Big Numbers? Then read this!

About Googology

What is Googology?

As the name has Googol in it, Googology is a kind of math that is just for fun to make huge numbers.

Examples of famous huge numbers include Googolplex, Graham's Number, and Rayo's Number.

How can I join the community?

To join discord: https://discord.gg/3qY7M2G6tv

To join or visit wiki: https://googology.fandom.com/wiki/Googology_Wiki

About Numbers

Googolplex

First we start with "small" numbers.

Googolplex is defined as 10^Googol, where Googol=10^100.

Why is it small? It's bigger than numbers we use in normal life. The reason is it's actually small in Googology. Googolplex is classified as class 3, and many numbers are more than class 3 numbers.

What is Class n?

Class n is defined like this:

function f: f(0)=6, f(n)=10^f(n-1)
function class: class(n)=min{m|n≤f(m)}

Where n and m are natural numbers. And that "min" returns the minimum number that is not in the set. This time, the set is {m|n≤f(m)} and it means the set of every m such f(m) is bigger than n.

Googolplex is in class 3 because 10^10^100 is smaller than 10^10^10^6 (it's f(3)) and 3 is the minimum number.

Also, you can make big numbers using that f. f makes number big as "Tetrational Functions".

Tetration

Tetration is the next thing comes after exponentiation.

First think about multiplication. multiplication iterates addition, and exponentiation iterates it.

Then Tetration iterates exponentiation, and it'll be stronger than exponential functions.

Also tetration can be written like n^^m. But, don't you think this can be expanded to n^^^m or even n^^^...o ^s...^m? It is.

Knuth's up-arrow notation

Knuth's up-arrow notation is a cool thing that makes big numbers.

It's defined like this:

a↑^1 b=a^b
a↑^c 1=a
a↑^c b=a↑^(c-1)a↑^c(b-1)

As you can see, this arrow thing works like exponentiation, tetration,...

This notation is very strong because ↑^3 iterates tetration, and it's called Pentation, and its iteration is Hexation.

So, a↑^c b iterates ↑^(c-1).

Then you might think, can we nest c?

BEAF

Array notation

It's defined like this:

{a}=a
{a,b}=a^b
{a,1,...}=a
{...,a,1}={...,a}
{a,b,1,1,...,1,1,c}={a,a,a,a,...,a,{a,b-1,1,1,...,1,1,c},c-1}
{a,b,c,...}={a,{a,b-1,c,...},c-1}

{a,b,c} is same as a↑^c b, but after {a,b,1,d}, it get's stronger and STRONGER.

Think about {3,3,1,2}. By the definition, it gets expanded like this:

{3,3,1,2}
={3,3,{3,2,1,2}}
={3,3,{3,3,{3,1,1,2}}}
={3,3,{3,3,3}}
={3,3,{3,{3,2,3},2}}
={3,3,{3,{3,{3,1,3},2},2}}
={3,3,{3,{3,3,2},2}}
={3,3,{3,{3,3,2},2}}
={3,3,{3,{3,{3,2,2}},2}}
...

You might notice that {3,n,1,2} nests c where 3↑^c 3, and this is the strength of BEAF.

I'll write other blog about BEAF.

Fast-growing Hierarchy

Fast-growing Hierarchy (a.k.a FGH) is a cool function.

It's defined like this:

f_0(n)=n+1
f_α+1(n)=f^n_α(n)
f_α(n)=f_α[n](n) If α is a transfinite ordinal

If the k is given (f^k_α(n)):
f^0_α(n)=n
f^k+1_α(n)=f_α(f^k_α(n))

Finite Ordinal

Let's calculate f_α(n) with finite ordinal α first.

f_0(n)=n+1 so it just add 1 to the n.

f_1(n)=f^n_0(n)=f_0(f_0(f_0(...(n)...)))=2n

f_2(n)=f^n_1(n)=f_1(f_1(f_1(...(n)...)))=2^n n

So f_α+1(n) will grow up to 2↑^α n.

It's not stronger as BEAF, but why is it cool? Because it can use Transfinite Ordinals.

Transfinite Ordinal

Transfinite Ordinal is a hard thing to understand.

First, we think about least transfinite ordinal, ω.

ω has many fundamental sequences and this is one of them:

ω[n]=n

Why does this make FGH stronger?

Because using this on FGH will grow up to 2↑^n n.

Using ω expanded be like this:

f_ω(n)
=f_n(n)
=f^n_n-1(n)
=f_n-1(f_n-1(...(n)...))

This looks cool right?

After this, we use ω+1 next:

f_ω+1(n)
=f^n_ω(n)
=f_ω(f_ω(...(n)...))

It grows extremely faster than f_ω(n), but why is it strong?

The reason can be explained like this.

First you think about f_ω(3). It's f_3(3) by the definition.

f_ω+1(3) expand to f_ω(f_ω(f_ω(3))) then f_ω(f_ω(f_3(3))) then f_ω(f_f_3(3)(f_3(3))) then f_f_f_3(3)(f_3(3))(f_f_3(3)(f_3(3))). Wow, it nest at the ω.

Actually, f_ω+1(n) is compared as {n,n,1,2} with BEAF.

I'll write the next in Part 2.

Thanks

Thank you for reading! If you have any questions about this blog, please comment about it.