## Dimensional Busy Beaver??

This is just a random though I had.

\(DBB(n,m)\) has two inputs:

n = dimension of the tape

m = number of states.

So it computes a Turing machine with m states in a tape with n dimensions.

I also add the functionality of using a subscript to denote Turing machines of higher levels (ie super Turing machines etc.)

So for n = 1, \(DBB(1,m)\) grows at the same speed of \(BB(m)\)

If we decide to increase the dimensions, say, to two, now we are at a plane of boxes, not just a line.

So that means we can have the tape head move up and down. Though am not sure if it is any more powerful.

We can go up to any dimension, adding possible ways of movement in a state.

## List of common misconceptions about Rathjen's psi

This is a list of common misconceptions related to Rathjen's \(\psi\) based on the least weakly Mahlo cardinal. Since old members irresponsibly spread wrong informations on \(\psi\) without knowing the precise definition, there are so many common misconceptions, which I repeated to correct so many times. One starting point is to check the precise definition before stating wrong informations. Seriously, please do not believe that you can understand an OCF without reading the definition. See User blog:p進大好きbot/List of common mistakes in googology#I fully understand it although I do not know the precise definition! for details about the cheating.

I first wrote this list in an old version of the article on OCFs,
and moved it to Rathjen's psi#C…

## Blank's Notation Function

I've got an idea for a notation, of course it **probably can't
be defined properly**, but it doesn't matter for now.

**Definition**:

D(n) is equal to how fast-growing the notation is, I'll explain what I mean

D(1) would equal the slowest growing function ever created

D(2) would equal the second slowest growing function ever created, and so on

D(∞) would then have to be the *fastest function that could
ever be created*.

Like I said before, I highly doubt this could ever be properly defined because there's infinite possibilities for creating notations, and you can always create an even faster growing function even if it grows +1 faster.

## T Notation

\(T(a,b)=(a^1)^{(a^2)^{(a^3)^{.^{.^.}}}}\) b times

\(T(a,b,c)=(a^b)^{(a^{(b^2)})^{(a^{(b^3)})^{.^{.^.}}}}\) c times

\(T(a\{b\}c)=T(a,b,b,\cdots,b,b)\) with c entries

T(2,2) = 2^{4}

T(3,3) = 3^{927}

T(2,2,2) = 4^{16}

T(2,2,3) = 4^{16256}

T(2,3,2) = 8^{512}

T(3,2,2) = 9^{81}

T(2,3,3) = 8^{512134,217,728}

T(3,2,3) = 9^{816561}

T(3,3,2) = 27^{19,683}

T(3,3,3) = 27^{19,6837,625,597,484,987}

T(2,2,2,2) = 16^{65536}

T(2,2,2,3) = 16^{65536(2256)}

T(10,100) =
10^{1001,000...(1098)(1099)(10100)}

## does astolfo is gay or european??????

despite the shitty title this is a slow growing array notation

means literally any arrayRules.)

- A(x) = x+1
- A(#, 1, 1, 1, (any amount of 1's)) = A(#)
- A(x,y) = x+y
- A(x,y,z) = x+y+z
- A(x,y,z,w,p,a,b,c,d,e..... etc) = z+y+z+w+p+a+b+c+d+e.... etc
- B(1)x = A(x) (this isn't that important)
- B(2)x = A(x, x, x..... etc with x amount of x's in the array), i.e B(2)2 = A(2,2) = 2+2 = 4. It's basically just squaring lmao
- B(3)x = B(2)B(2)B(2)...B(2)x with x B(2)x's. i.e B(3)2 = B(2)((B(2)2) = B(2)4 = A(4,4,4,4) = 4^2 = 16.
- B(4)x = B(3)B(3)B(3)...B(3)x with x B(3)x's. i.e B(4)2 = B(3)((B(3)2) = B(3)4 = B(2)((((B(2)(((B(2)((B(2)2) = B(2)(((B(2)((B(2)4) = B(2)((B(2)16) = 16^2 = 256.
- B(y)x = Recursion of B(y-1)x x times until you get down to 2. Fun fact: B(y)x is ju…

## Subcursion

Subcursion comes from the word sub, as in subscript, and cursion, as in recursion. It is a higher level of recursion.

So, lets make a simple function. \(f(x) = x+1\) (note that the subcursed function can be anything.)

if n= 0 then it is the same as no subcursion.

unfortunately I don’t know how to avoid ellipses here.

q is for quaiL

Let p be a large countable ordinal such to every limit ordinal m < p there is assigned a fundamental sequence (a strictly increasing sequence of ordinals whose supremum is m). A quick growing hierarchy of functions, \(q^m : \mathbb N \rightarrow \mathbb N\), for m

## Breaking-ish news

**JONATHAN BOWERS' WEBSITE GOT A NEW UPDATE!!!!**

BUT. There is a problem. This update, massive as it is, is confined to the polytope section of the website.

**BUT**. There is some good news too. The polychoron page
hints at there being future **monthly updates** to fill in
information that this update excluded. **You heard that right,
monthly updates.** The last update was more than a year ago, so
this is an unprecedented thing. And some of them may also include
googological content.

We return to our regularly scheduled program.

## More random numbers

I never stop thinking of these things.

- 1 Rubicol
- 2 Factoriatrix
- 3 Vroom math
- 4 Calculus???

The rubicol is equal to the number of permutations of a googol×googol×googol Rubik's cube. The rubicolplex is equal to the number of permutations of a rubicol×rubicol×rubicol Rubik's cube. The rubiplexol is equal to the number of permutations of a googolplex×googolplex×googolplex Rubik's cube.

According to here, the number of permutations of an n×n×n rubik's cube is \(\lceil36410\cdot11771943321600^{n\text{ mod }2}\cdot620448401733239439360000^{\lfloor\frac{n-2}{2}\rfloor}\cdot3246670537110000^{\lfloor(\frac{n-2}{2})^2\rfloor}\rceil\), so a rubicol equals \(36410\cdot620448401733239439360000^{\frac{10^{100}-2}{2} }\cdot3246670537110000^{(\frac{10^{100}-2}{2})^…

## Did any of you know...

\(\text{...that right-clicking on a MathJax formula brings up a "secret" menu?}\)

## Sikigami Googology 2021

Sikigami Googology 2021 is a competition with making programs that calculates large numbers. There is the rule in Japanese.

Sikigami Googology 2021 has three goals:

- I enjoy it.
- Everyone enjoy it.
- Everyone competes by creating programs to calculate large numbers.

Sikigami Googology has many rules, for everyone to enjoy.

Sikigami Googology is for program calculate large numbers.

Sikigami Googology is divided into ORIGINAL and PROGRAMIZED sections.

- 1 Rule
- 2 Question
- 3 Duration
- 4 ORIGINAL section
- 5 PROGRAMIZED section
- 6 Operator
- 7 Organizer
- 8 Judge
- 8.1 Judge VALID or INVALID
- 8.2 Judge HALTING or NON-HALTING
- 8.3 Give APPROXIMATION

- 9 Table
- 10 Ranking
- 11 Participant
- 11.1 Post

- 12 Article
- 13 Program
- 13.1 Name of the program
- 13.2 Source of the program
- 13.3 Information of the large numbers output…

## Finally...

...I found a way to define BEAF past {L,X}n,n.

{L,X}n,n = {n,n(1)/2}

I chose left placement of new & sign, so {n,n(1)/2} = 1&(1)&n. Assuming all previous layers being already formalized, this layer is expanded similarly.

New rules:

- If there is at least one / before first separator in /-chain, arrays with internal /-chains resolve into corresponding &-chains, otherwise instead of this corresponding block of / with separators is added before first separator and one / is deleted after first separator available
- For A(&)-chains, it is based on the A(/)-array, only with last / before first separator in chain deleted: X&&(1)&3 = {R/(1)/R/(1)/R}, where R is {3,3/(1)/2} array fully expanded
- Finally, for {L,A}n,n, between n,n and 2 size A array of / sign…

## A notice (or announcement)

Hello there. Normally I won't do any of my blog post, but today, I needed to do a blog post for an announcement, as there are updates on the policy including:

- Voting for me as the new admin via this page
- 3-out rule (for different type of warnings)
- Copyright rule (especially when it comes to licensing)
- Personal websites (as the definition might change over time, so, retrieval date is needed)

Here's the reason why I needed to be an admin:

- This wiki lacks multiple active admins.
- I can rollback unconstructive edits/vandalisms.
- Edit blog post so that the mainspace categories can be removed from the post (as the mainspace category should contain only the articles)
- Checking approximations of any numbers to make sure it's close to it. (In terms of notation…

## Announcement: voting for admins

This is an unoffical announcement of voting for admins.

We will hold voting to choose a new admin and releave inactive
admins. **Only non-blocked users that were registered for at least
100 days, were active at least 10 days, and have edited at least
100 times may vote to avoid abusing the voting system. Voting holds
for 10 days.** (See Googology Wiki:Policy#Voting.)

**Update 15/05/2021:**

Now we have the voting!

## FPCI(a) - continue

I continue work of Wythagoras and use his convention:

> Note that I think that we should replace all greater than and lesser thans with greater than or equals and lesser than or equals. This gives the first values a bit nicer.

Link for his investigation: User blog:Wythagoras/Finite Promise Games

The game lasts two rounds, degrees of the functions are at most 2.

m = 3, so we can reject 1 and 2

**First round:**

s = 3

x = (w!)! = 2

choose P = X+Y and Q = X^2+Y^2

we reject this and play P' = [1,1] but then we promise that none of the integers that we ever play are 2

**Second round:**

s = 3

x = (w!)! = 1

we accept this and win because we never played 2.

This can't change if we choose any other P and Q because we must choose from polinomials degree no more than 2 and…

## Duckie number

Seconds passed since 8th May, 2021 00:00:00 UTC => S

m = S/60 when S mod 60 = 0

h = m/60 when m mod 60 = 0

d = h/24 when h mod 24 = 0

duckie = \(10\uparrow2\uparrow4\uparrow8\uparrow16\uparrow32\uparrow64\uparrow128\uparrow256\uparrow512\uparrow1024\) \(\uparrow2048\uparrow4096\uparrow8192\uparrow16384\uparrow32768\)

Duckie Number = \(S\uparrow^{(m+1)\uparrow^{(h+1)\uparrow^{(d+1)\uparrow^{duckie}(d+1)}(h+1)}(m+1)}S\)

## Brace Function

This is a basic function defined for my array function(This is not yet complete). It is a simple function consisting only of addition and recursion. But as FGH shows, it doesn't increase so slowly. The growth rate of this function is exactly the same as \(f_\omega(n)\).

For a positive integers n,m and non-negative integer k, I define a positive integer \(\{n_k\}^m\) in the following recursive way:

1. If k=0, then \(\{n_k\}^m=n\)

2. Suppose k=1

2-1. if m=1, \(\{n_k\}^m=n+1\). And you can remove m, k. \(\{n_k\}^m=\{n\}\)

2-2. if m>1, \(\{n_k\}^m=\{n_n\}^{m-1}\)

2-3. Also, you can remove k. \(\{n_k\}^m=\{n\}^m\)

3. If k>1, \(\{n_k\}^m=\{(\{n_{k-1}\}^m)_1\}^m\)

n is the variable, m is the number of braces, and k is the subscript to make the function …

## pangolin

pangolin is a formula:

pangolin(x) =

if pangolin is used multiple times in one equation, it is
equivalent to pangolin(y) where y is equal to each previously
recited x multiplied by each other

for example:

pangolin(1) = 2392031250

pangolin(2) = 15728001190723584 (2 multiplied by 1 is 2 so in this case pangolin(2) is pangolin(2))

pangolin(3) = 79654273022752657268734013065399553751082787143680000000 (this time 3 is multiplied by 2 and 1 so we use what would have normally been pangolin(6))

## Ultimate mixed chained arrow notation

Alright, so you probably know about up-arrow notation, which notates the "strong" hyper-operators:

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

And you might also know about down-arrow notation, which produces weak operators:

- a↓
^{1}b = a^{b} - a↓
^{c}1 = a - a↓
^{c}b = (a↓^{c}(b-1))↓^{c-1}a

Hyp Cos combined them into Mixed arrow notation (userpage)

- a↑b = a↓b = a
^{b} - a#1 = a
- a#↑b = a#(a#↑(b-1))
- a#↓b = (a#↓(b-1))#a

Where # is any sequence of either up or down arrows.

Rgetar created an an extension that adds more symmetry:

- a|b = a↑b = a↓b = a↕b = a
^{b} - a#1 = a
- a#|b = a#a
- a#↑b = a#(a#↑(b-1))
- a#↓b = (a#↓(b-1))#a
- a#↕b = (a#↕(b-1))#(a#↕(b-1))

The | symbol is just for symmetry and doesn't exactly produce larger numbers.

John Conway created chained arrow notation. It can express everything in normal ar…

## Exploder Notation (Probably bad, my first notation)

This notation mainly revolves around Arrow Notation.

As an example for Single E,

**And so on.**

## ITTM vs Gap ordinals

May I know between these two groups of ordinals, which is larger?

ITTM ordinals: ? or gap ordinals?

Edit: Never mind, I just found out gap ordinals are below stable ordinals, based on info from David A. Madore. According to this wiki, stable ordinals are below ITTM ordinals.

## Weird BEAF expansion

This expansion seems more factorial-like than hyperoperator-like
despite BEAF being hyperoperator-based. This will add yet another
complication to trying to define BEAF for ordinals; I don't think
either the climbing or non-climbing method could handle this
well.

## Array Notation numbers

\(p_3\{3,3,3,3\} = quintatri\)

\(p_1\{2,3,4,5\} = \text increasing \, array \, number\)

\(p_{10}\{10,10,10,100\} = \text Quadragoogolplex\)

\(p_4\{4,4,4,4\} = \text Quintaquad\)

\(p\{2,4,6,8,10\} = \text Even \, array \, number\)

\(p\{1,3,5,7,9\} = \text Odd \, array \, number\)

\(p_2\{4,8,16,32\} =\text Two \, array \, number\)

## Random numbers!

Alright here they are:

- 394839587248453857
- 4938482948394748384
- 348738274782821828888800

Nah, just kidding. Here are some numbers I randomly thought of:

The **chain factorial** is defined as n→! =
n→(n-1)→(n-2)→...→3→2→1.

The **arrayorial** is defined as n{!} =
{n,n-1,n-2,...,3,2,1}.

The **expactorial** is defined as n:! =
n{(n-1){(n-2){...{3{2{1}2}3}...}(n-2)}(n-1)}n (not to be confused
with the expandofactorial, n =
√(E100#100#100#100#100#100#100#100#100). This was an extremely
important discovery that will change the study of large numbers
forever and may possibly bring this wiki back to its former glory.
Thanks for watching.

## My system of numbernames

My analogs of googol

So the n-ary analog of googol is equal to a^b, ^ means exponent, a means any natural number, b is the square of a

Here we go!

Unak = 1^1

Binak = 2^4

Trinak = 3^9

Quartinak = 4^16

Quinak = 5^25

Senak = 6^36

Septenak = 7^49

Octenak = 8^64

Novak = 9^81

Decak = 10^100

Undecak = 11^121

Dodecak = 12^144

Tredecak = 13^169

Quardecak = 14^196

Quidecak =15^225

Hexdecak = 16^256

Hepdecak = 17^289

Octdecak = 18^324

Ennadecak = 19^361

Vigesak = 20^400

Trigesak = 30^900

Sarak = 40^1,600

Penanak = 50^2,500

Exatak = 60^3,600

Eptatak = 70^4,900

Ogdatak = 80^6,400

Enanak = 90^8,100

Hectak = 100^10,000

Chiliak = 1,000^1,000,000

Myriak = 10,000^100,000,000

That's all!

## Array notation

This is my first array notation, also called Penguin array notation. (PAN)

I denote all NON-NEGATIVE, WHOLE numbers as set \(\mathbb N\), and in the following all variables (a,b,c,d...) are all terms within the set \(\mathbb N\).

Core recursion function for core notation:

(I only did this because I want to avoid ellipses)

I denote function C(m,n) in the following recursive way:

\(C(m,n,o,p,q) = n\rightarrow_qo\rightarrow_qp\) if m = 0

\(C(m,n,o,p,q) = C(m-1,C(m-1,n,o,p,q),C(m-1,n,o,p,q),C(m-1,n,o,p,q),\)

\(C(m-1,n,o,p,q))\) if m is a positive whole number larger than 0.

Core notation:

\(a(b)c = C(0,a,b,c,c\)

\(a((b))c = C(a(b)c,a(b)c,a(b)c,a(b)c,a(b)c)\)

In general, \(a(b)^dc = C(a(b)^{(d-1)}c,a(b)^{(d-1)}c,a(b)^{(d-1)}c,a(b)^{(d-1)}c,a(b)^{(d-1)}c)\…

## Recursive Factorials!

Repeated factorial thing?

"n" must be a whole positive number.

\(k(n) = b(n,n)\)

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.

If m = 0: \(b(m,n) = n!\)

If m > 0: \(b(m,n) = b(m-1,b(m-1,n))\) where \(m > 0\)

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-…

## MathJax reference table

Since I can't find anything about how to use MathJax on this wiki, I will write it here

https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference

This is a reference table of different codes in MathJax. Use ctrl+f and search what you need.

Common codes:

Note: all of them use \ and NOT /

Opening: In order to use mathjax, you need this "\(" (Note, do not add the "")

Ending. This ends the area of the mathjax \). It is exactly reverse from the opening, so it is easy to remember.

Some common codes.

\uparrow Used for Arrow notation.

x^{yz} X with superscript y an z.\(x^{yz}\) Note that if there is only one number eg 3^3, \(3^3\) you do not need to use {}.

x_{yz} X with subscript yz. \(x_{yz}\) Same rules regarding {} ab…

## Decidec

- This is just some random big number I thought of. It is probably ill defined, but whatever.

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

## Kuma Worm Sequence

This blog post is translated from https://googology.wikia.org/ja/wiki/%E3%83%A6%E3%83%BC%E3%82%B6%E3%83%BC%E3%83%96%E3%83%AD%E3%82%B0:Kanrokoti/%E3%81%8F%E3%81%BE%E3%82%80%E3%81%97%E6%95%B0%E5%88%97

- 1 Overview
- 2 Kuma Worm
Sequence
- 2.1 Notation
- 2.2 List of non-negative integers
- 2.3 Expansion Rule
- 2.4 FGH
- 2.5 Naming
- 2.6 Expectation

p進大好きbot accidentally generated a sequence system while calculating Worm psi Function. With p進大好きbot's permission, I decided to formalise it. After the formalisation, infinite loop was found by japanese googologist Okkuu. Therefore, I and p進大好きbot remade the notation. In order to remake this, I referred to Reflection principles and provability algebras. Slides of LC2002 tutorial. Münster, August 3-9, 2002, p.28, p.29. defined by L…

## Notations created by me

The definition of M-notation.

Definition 4** Googology function $M(n)$

Given a pair of a term and a positive integer $(A,n)$, if $A$ is succeedly, let $o(A,n)=(A-1,n+1)$; if $A$ is limitly, let $o(A,n)=(A[n],n+1)$.

Let $B=\psi_M(0),A_n=B\circ B\circ ...\circ B$（$n$ $B$s）. Let $(A_n,n)$ becomes $(0,M(n))$ after we repeat the operation $o$ for some times. In this way we have given the definition of $M(n)$.

## Greatly Mahlo limits of Greatly Mahlos make no sense

The existence of Greatly Mahlo limits of Greatly Mahlo ordinals
implies some very strange stuff - ordinals that are stationary on
the class of hyper-Mahlo limits of Greatly Mahlo ordinals are
unbounded in every ordinal that is stationary on the class of
Greatly Mahlo ordinals. In this blog post, I will prove a slightly
weaker (but still unexpected) condition. As far as I know, all
weakly compacts are Greatly Mahlo, which means this strange
property should be true, although it doesn't seem to make
sense.

Definitions

First, I should define some words I use, because other stuff is sometimes used instead of "stationary on" and Greatly Mahlos probably aren't very commonly known.

An ordinal α is stationary on a set or class A of ordinals iff for ev…

## My NOtation

This is my notation. I will show you how to use.

Line(n) = f₁(n)

Triangle(n) = f₃(n)

And you keep going like this

n inside an n-gon is equal to fw(n) in FGH.

So n inside a megafuga-n gon is equal to fε₀(n).

And so on...

Zeta-zero is pentation, eta-0 is hexation, phi(4,0) is heptation, etc, so it means that n inside a gigafuga-n is equal to zeta-zero in FGH, n inside a petafuga-n is equal to eta-zero, and so on, and so forth. Use the sequence of hyperoperators.

Start with Ew^w(n) w is omega

Ew^w(n) = En#^#n

Ew^w^w(n) = En#^#^#n

Keep going like this

Eε₀(n) = n#^^#n

Eζ₀(n) = n#^^^#n

Eη₀(n) = n#^^^^#n

Eφ(4,0) = n#^^^^^#n

Keep going like this. Using the clue in my notation, φ(99,0) in my hierarchy is equal to n#^^^...^^^#n with 100 #^s.

We can link this to Gods…

## Question of googology

Which is the larger googolism?

1. Tarintar vs Y sequence, Bashicu Matrix System

According to this list, Y sequence and Bashicu Matrix System are larger, but in Googol Maps 2.0, Tarintar appears to be larger. So which one is more accurate?

2. \(\text{Rayo}(10^6)\) vs \(\Xi(10^6)\)

3. \(\text{Rayo}(10^9)\) vs \(\Sigma_\infty(10^9)\)

4. BIG FOOT vs Large Number Garden Number

I know that BIG FOOT was ill-defined. But If BIG FOOT worked perfectly, which one is larger?

1. What is the smallest positive integer n that satisfying BB(n)>Loader's number?

2. What is Large Number Mansion Number?

4. What is the name of these OCF?

## Elidad Chen's Array Notation

Elidad Chen's Array Notation 1(ECAN1) (My first array notation)

[a]b'c = b*(a^c)

[a]b'c'd = b*(a^(c*(a^d)))

[a]b'c'd'e = b*(a^(c*(a^(d*(a^e)))))

.....

Example:

[3]4'5 = 4*3^5 = 4*243 = 972

[2]6'1'1 = 6*2^1*2^1 = 6*2^1*2 = 6*2*2 = 24

[5]2'2'4'3 = 2*5^2*5^4*5^3 = 2*5^2*5^4*125 = 2*5^2*5^500 > 5^5^500 = ~10^10^349

[a]

Example:

[3]2'6 = [ [3]2'6 ]2'6 = [1458]2'6 = ~1.92121133*10^19

[5]2'2 = [ [ [5]2'2 ]2'2 ]2'2 = [ [50]2'2 ]2'2 = [5000]2'2 = 50000000

[2]1'2 = [ [ [ [ [2]1'2 ]1'2 ]1'2 ]1'2 ]1'2 = [ [ [ [4]1'2 ]1'2 ]1'2 ]1'2 = [ [ [16]1'2 ]1'2 ]1'2 = [ [256]1'2 ]1'2 = [65536]1'2 = 4294967296

More update coming soon

## Infinite Order Set Theory

Since we have first and second order set theory, naturally we should have third and higher order set theories, though I am not sure how they are defined. Anyway, let's assume they are well defined to produce this fundamental sequence:

**Further Improved version:**

For non-negative integer k and positive integer function R,

R_{0}(n) = Rayo's function (based on FOST or 1-st order
set theory).

R_{k-1}(n) is defined as the function based on k-th
order set theory.

Further growth shall be achieved through R_{O}(n) where
O denotes any combination of countable ordinals.

The version below reaches only a growth rate of about

## N function

N function is the strongest function ever I made. It's inspired by R function.

For a positive integer n and non-negative integers m and h, we define a positive integer \(n\text{N}^{h}m\) in the following recursive way:

1. If \(h = 0\), then \(n\text{N}^{h}m = m\).

2. Suppose \(h = 1\).

2-1. if \(m=0\), then \(n\text{N}^{h}m = n+1\).

2-2. if \(m>0\), then \(n\text{N}^{h}m = n\text{N}^{n}(m-1)\).

3. If h > 1, then \(n\text{N}^{h}m = n\text{N}^{1}(n\text{N}^{h-1}m)\).

4. For a positive integer n, we define a positive integer \(n\text{N}\{0\}\) as \(n\text{N}^{1}n\).

So in the basic level, the growth rate is \(f_{\omega}(n)\)

5. nN{m+1}=nN{m}{m}...{m}(n times)

You can apply rule 4 and 5 inside of brace like

The limit is nN{0{0{..{0{0*}0}...*}0*}0}=nN{0…

## Rayo challenge

I challenge those of you more familiar with FOST to find bounds for these numbers:

- M = min({x|Rayo(x)≥x})
- N = min({x|Rayo(x)>x})
- M' = min({x|∀y≥x Rayo(y)≥y})
- N' = min({x|∀y≥x Rayo(y)>y})

## Chick number

It is equal to 10@(c1@(c1@(c1)c1)c1...c1 layers... )10 = chick number

eg 3 layers = c1@(c1@(c1@(c1)c1)c1)c1 (total 3 c1@('s）with 2 c1)s and 1c1 at end

4 layers =c1@(c1@(c1@(c1@(c1)c1)c1)c1)c1

etc.

N layers = total N c1@(‘s, excluding the innermost c1, N-1 c1)s, 1 c1 at end

where c1 =

10@(10@(10)10)10 = mini chick number = c1

the notation @ is defined as follows

a@b = a^(ab)b = a^...(a*b arrows)...^b

eg.

3@3 = 3^(3*3)3 = 3^(9)3 = 3^^^^^^^^^3

method: solve ab to find amount of arrows and put ab arrows between a and b

a@@b = a@b^(a@b)a@b

in order to solve a@...@b (where the amount of @s are more than 1) deconstruct it into a similar equation of a@b, except replace a and b with a@...(one less than original)...@b.

equation: x^(xy)y =x^…xy arrows…^y

in the …

## Power towers

Let's define the tower function. (denoted as tow(n) )

tow(n）= h(n,n)

h(1,n) = m=>m=>m (chained arrow notation）

h(n,m) = h(n-1,h(n-1,m)) if n> 1

eg

tow(3) = 3^(3^(3^3)3)3 = 3^(3^(27)) = o no

Power tower （p-tow(n)(m))

p-tow(1)(n) = tow(tow(n))

if n >1

p-tow(n)(m) = k(n,n)

k(1,n) = p-tow(1)(tow(n))

k(n,m) = k(k(n-1,m),k(n-1,m))

Θ(n) = p-tow(n)(n)

it grows kinda fast.

## Nest Base psi Function

This blog post is a translation from https://googology.wikia.org/ja/wiki/%E3%83%A6%E3%83%BC%E3%82%B6%E3%83%BC%E3%83%96%E3%83%AD%E3%82%B0:Kanrokoti/%E3%83%8D%E3%82%B9%E3%83%88%E3%83%99%E3%83%BC%E3%82%B9%CF%88%E9%96%A2%E6%95%B0

- 1 Overview
- 2 Nest Base psi
Function
- 2.1 Notation
- 2.2 Abbreviation
- 2.3 Ordering
- 2.4 Cofinality
- 2.5 Fundamental Sequence
- 2.6 FGH
- 2.7 Limit of the notation
- 2.8 Standard Form
- 2.9 Naming

This is an attempt to make Nest Base psi Function. In order to make this, I modified the Ordinal Notation Associated to Extended Buchholz's OCF defined by p進大好きbot.

Here, we define character strings used for the notation.

Let \(T\) and \(PT\) are sets of formal strings consisting of \(0\), \(+\), \(\psi\), \((\), and \()\), which are simultaneously defined in the …

## Super Huge Number

10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000…

## My progress

Finally I have found a way to understand Rathjen's original
"small psi" OCF (I clearly distinguish it, simplified Rathjen's
function and Rathjen's "large psi"). I am interested only in
ordinals now. I'm also fully aware of uselessness of ordinals
without accompanying FSs in computable googology, but I have not
much time for studying them. I heavily used this excellent blogpost
for reference. For now, I independently did obtain some values
between gamma-zero and φ(1,2,0) (in that time, I assumed **without
proof which I knew** that χ_0(0) was indeed the first uncountable
ordinal).

I used only C function as B function is needed only for defining chi function and I used value of χ_0(0) as given.

For example, I want to get ψ_{χ_0(0)}(0).

*Original defini…*

## function (bad)

this is not finished i will keep editing until done ok ok good

rule of function:::::

1.) = x+1

2.) .

13.) =

14.) =

15.) Overall, =

## BlAnK OrDiNaL

**This page is a joke, please don't take it seriously.**

The blank odinal 1 is eqalu to \(\vartheta_{\psi_0(\omega_1^{CK})}(\Omega_{\Omega_\eta}^{\Omega_{\varepsilon_{\varepsilon_0}}^\omega})\)

THe blank odinal 2 is equla to \(\vartheta_{\psi_0(\omega_1^{CK})}(\Omega_{\Omega_\eta}^{\Omega_{\varepsilon_{\varepsilon_0}^{\mu_{\kappa^{\theta_{\lambda_{\varpi^\beta}}^\delta}}}}^\omega})\)

Yes, I already know this is probably not a proper odinal, you dot'n need to tele me >:(

## Analysis of 二関数

This is an English translation of my Japanese blog post about analysis of 二関数.

In this article, I assume the well-foundedness of the notation system \((OT,

## Rathjen's Psi over K

Are there any non-trivial descriptions of how ordinal notation associated with this OCF could work (i.e. FSs, its limit) like this?