Mumuji Mumuji 13 hours ago

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.

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P進大好きbot P進大好きbot 1 day ago

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…

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BlankEntity BlankEntity 1 day ago

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.


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.

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BlankEntity BlankEntity 2 days ago

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) = 24

T(3,3) = 3927

T(2,2,2) = 416

T(2,2,3) = 416256

T(2,3,2) = 8512

T(3,2,2) = 981

T(2,3,3) = 8512134,217,728

T(3,2,3) = 9816561

T(3,3,2) = 2719,683

T(3,3,3) = 2719,6837,625,597,484,987

T(2,2,2,2) = 1665536

T(2,2,2,3) = 1665536(2256)

T(10,100) = 101001,000...(1098)(1099)(10100)

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VoidSansXD VoidSansXD 2 days ago

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

despite the shitty title this is a slow growing array notation

means literally any array


  1. A(x) = x+1
  2. A(#, 1, 1, 1, (any amount of 1's)) = A(#)
  3. A(x,y) = x+y
  4. A(x,y,z) = x+y+z
  5. A(x,y,z,w,p,a,b,c,d,e..... etc) = z+y+z+w+p+a+b+c+d+e.... etc
  6. B(1)x = A(x) (this isn't that important)
  7. 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
  8. 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.
  9. 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.
  10. B(y)x = Recursion of B(y-1)x x times until you get down to 2. Fun fact: B(y)x is ju…

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Mumuji Mumuji 3 days ago


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

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A Hippopotatomus A Hippopotatomus 4 days ago

Breaking-ish news


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.

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A Hippopotatomus A Hippopotatomus 4 days ago

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})^…

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A Hippopotatomus A Hippopotatomus 5 days ago

Did any of you know...

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

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Hexirp Hexirp 6 days ago

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…

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Tetramur Tetramur 6 days ago


...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:

  1. 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
  2. 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
  3. Finally, for {L,A}n,n, between n,n and 2 size A array of / sign…

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ARsygo ARsygo 6 days ago

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:

  1. Voting for me as the new admin via this page
  2. 3-out rule (for different type of warnings)
  3. Copyright rule (especially when it comes to licensing)
  4. 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:

  1. This wiki lacks multiple active admins.
  2. I can rollback unconstructive edits/vandalisms.
  3. Edit blog post so that the mainspace categories can be removed from the post (as the mainspace category should contain only the articles)
  4. Checking approximations of any numbers to make sure it's close to it. (In terms of notation…
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P進大好きbot P進大好きbot 6 days ago

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!

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Tetramur Tetramur 7 days ago

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…

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Mumuji Mumuji 8 days ago

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

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Root2222 Root2222 8 days ago

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 …

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Aab5c Aab5c 8 days ago


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))

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A Hippopotatomus A Hippopotatomus 8 days ago

Ultimate mixed chained arrow notation

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

  • a↑1b = ab
  • a↑c1 = a
  • a↑cb = a↑c-1(a↑c(b-1))

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

  • a↓1b = ab
  • a↓c1 = a
  • a↓cb = (a↓c(b-1))↓c-1a

Hyp Cos combined them into Mixed arrow notation (userpage)

  • a↑b = a↓b = ab
  • 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 = ab
  • 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…

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Alexander Nolta Alexander Nolta 9 days ago

Exploder Notation (Probably bad, my first notation)

This notation mainly revolves around Arrow Notation.

As an example for Single E,

And so on.

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Msiajoe74 Msiajoe74 9 days ago

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.

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A Hippopotatomus A Hippopotatomus 9 days ago

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.

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Mumuji Mumuji 10 days ago

Array Notation numbers

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

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

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

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

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

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

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

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A Hippopotatomus A Hippopotatomus 11 days ago

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.

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Hypertetrakulus44 Hypertetrakulus44 12 days ago

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!

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Mumuji Mumuji 13 days ago

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)\…

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Mumuji Mumuji 15 days ago

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-…

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Mumuji Mumuji 15 days ago

MathJax reference table

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

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…

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Mumuji Mumuji 15 days ago


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 \(f_0(n) = n\rightarrow_{10}n\)

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Kanrokoti Kanrokoti 15 days ago

Kuma Worm Sequence

This blog post is translated from

  • 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…

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ReflectingOrdinal ReflectingOrdinal 16 days ago

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 $M(n)$.

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ReflectingOrdinal ReflectingOrdinal 16 days ago

can I edit blog page?


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Ytosk Ytosk 17 days ago

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.


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…

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Hypertetrakulus44 Hypertetrakulus44 17 days ago

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…

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Root2222 Root2222 19 days ago

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?

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Elidad Chen Elidad Chen 20 days ago

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)))))



[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



[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

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Msiajoe74 Msiajoe74 20 days ago

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,

R0(n) = Rayo's function (based on FOST or 1-st order set theory).

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

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

The version below reaches only a growth rate of about

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Root2222 Root2222 20 days ago

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…

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A Hippopotatomus A Hippopotatomus 20 days ago

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})
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Mumuji Mumuji 21 days ago

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


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


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 …

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Mumuji Mumuji 21 days ago

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


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.

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Kanrokoti Kanrokoti 22 days ago

Nest Base psi Function

This blog post is a translation from

  • 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 …

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Root2222 Root2222 22 days ago

I have a question

Questions of Mahlo OCF

  1. what is the difference of ?
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Hypertetrakulus44 Hypertetrakulus44 23 days ago

Super Huge Number


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Tetramur Tetramur 24 days ago

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…

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VoidSansXD VoidSansXD 25 days ago

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, =

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BlankEntity BlankEntity 25 days ago


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 >:(

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P進大好きbot P進大好きbot 27 days ago

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,

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Tetramur Tetramur 28 days ago

Irregular parts of rules in BEAF (temporary)


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Tetramur Tetramur 28 days ago

Problems with notations

(cleaned up until better times get up)

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Tetramur Tetramur 15 April

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?

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