# Googology Wiki

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11,358
pages
 Mumuji Home Talk Edit count Contribs Monday, 29 November 2021

Mumuji Penguin (talk)

## Milestone

700 - 15:33, 19 July 2021 (UTC)

800 - 14:39, 21 July 2021 (UTC) omw to 1K B)

900 - 01:07, 28 July 2021 (UTC) Wowowwoowow. Only 100 to go!!! Let's GOOOOOOOOO

1007- 00:25, 18 August 2021 (UTC) YESSSSSS

yay

b

Boirk

## Random tests

Some random stuff

## Decidec

$$f_100)\ \uparrow 3 <\math> \(100\uparrow100$$ $$10\uparrow10\uparrow100$$

$$f_{10\uparrow^{10}10}(10\uparrow^{10\uparrow^{10}10}10)$$ where $$f_m(n)$$ is the fast growing hierarchy but $$f_0(n) = n\rightarrow_{10}n$$

This number is also called decided

## Random test

$$3\rightarrow3\rightarrow3$$

10<sup>10<sup>10<sup>10<sup>10<sup>100</sup>

10<sup>10</sup>

## Repeated factorial thing?

$$k(n) = b(n,n)$$

Since this is a recursive function, I will add a definition for b(0,n), to ground it and not make it become like infinity.

$$b(0,n) = n!$$

$$b(m,n) = b(m-1,b(m-1,n))$$

So, $$k(3) = b(3,3) = b(2,b(2,3)) = b(1,b(1,b(1,3))) = b(0,b(0,b(0,b(0,3)))) = b(0,b(0,b(0,6) = b(0,b(0,720)$$

$$= 720!! \approx 2.6\cdot10^{1746}! = oh god$$

Yay finally I didn't use ellipses to make recursion!

Moral of the story? Don't underestimate recursion. As long as the first operator is strong enough, recursion makes it immensely more powerful.

But wait, if we have expofactorials, where are the tetrafactorials and so on?

$$p(n) = n\uparrow^{n}(n-1)\uparrow^{n}(n-2)\cdots2\uparrow^{n}1$$

So, we add a new modified function $$k_1(n)$$ and $$b_1(n,m)$$

$$k_1(n) = b_1(n,m)$$

$$b_1(0,n) = p(n)$$

$$b(m,n) = b(m-1,b(m-1,n))$$

So, $$k_1(3) = b_1(3,3) = b_1(2,b_1(2,3)) = b_1(1,b_1(1,b_1(1,3))) = b_1(0,b_1(0,b_1(0,b_1(0,3)))) = b_1(0,b_1(0,b_1(0,$$

$$Tritri) = o no$$

## Penguin Math

Penguin math is denoted with a subscript of p

So, to start things off, we do this:

$$S_p(n) = n\uparrow^nn$$ if n>2, else if $$2\geq n \gt 0 S_p(n) = 3\uparrow^nn$$ else if n = 0 $$S_p(n) = n+1$$

So pengoogol = $$10\uparrow_p100) using the sense that addition is repeated the successor , multiplication is repeated addition, and exponent action is repeated multiplication Say, for example I wanted to do 3+3 in penguin math, we would do it like \(S(S(S(3))) = S(S(tri tri)) = S(tritri\uparrow^{tritri}tritri) = etc.$$

## Random Quotes

"test1" 2021 Reflecting Ordinal

“The more you light your lighter the lighter and lighter it gets until it is too light to light and you will need to buy a new lighter to set your candles alight to produce light.” Me.

care sclera.

Yes.

bruhh.

## Fav num

2. Two is prime, even, it’s the only one with both of those two properties, it’s the only number where addition and exponentation of it result in the same number, my birthday is on the 24th, or 2^2*(2*2+2/2+2/2), with 8 2s or 2+2+2+2, with 4 2s or 2^2.

2 is also one of the numbers required to make 6, the first perfect number.

f_2 (2) is also equal to 8, or 2+2+2+2, with 2^2 2s, and in that there is 2 2s.

2 is the first even number

2 is the base which computers use, and almost all things have computers. Even now I am using a computer to edit this wiki!

It is also the only number where(for talking about amount of objects) you can’t just put the numeral in Chinese（二）, you have to use 兩.

water is H2o, and there is a 2.

I also remember the numbers being taught with animals, and two is the first one to have a bird (swan), and I love birds.

DNA is a DOUBLEhelix, and there is 2!

2 is also needed to make square numbers.

2 is the first superior highly composite number, despite being a prime.

2 is able to turn basically any hyper operation to 4.

My username, Mumuji, has 6 letters, or 2+2*2, with 2+2+2/2 2s or 2^2 2s, or 2 2s.

Also, the only prime followed by a cube is 7, which is followed by 8, or 2 cubed.

2 is the smallest super perfect number.

Because of this, I call numbers that have a multiple of 2 2s half perfect, and ones that are only made of 2 perfect (this is not the real perfect numbers). So 122 is half perfect, and 22 is perfect.

$$\left. \begin{matrix} &&\varphi(10^{100}\underbrace{0,\cdots 0,})(10^{100})\\ & & \varphi(10^{100}\underbrace{0,\cdots 0,})(10^{100}) \\ & & \;\;\underbrace{\quad\;\; \vdots \quad \;\;}\\ & & \varphi(10^{100}\underbrace{0,\cdots 0,})(10^{100}) \\ & & 10^{100}\quad 0's \end{matrix} \right \} \varphi(10^{100},\underbrace{0,0,\cdots,0,0}_{10^{100}})$$ <br />