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Binary198 Binary198 14 hours ago

My illions- system pt.1

This includes all of my tier 1 -illions (using my own unpublished googological ruler)

  • Ailion (sounds like alien when you pronounce it) - \(10^6\)
  • Billion (coincides with billion in short scale) - \(10^9\)
  • Cillion (c is pronounced like an s) - \(10^{12}\)
  • Dillion - \(10^{15}\)
  • Fillion - \(10^{18}\)
  • Gillion - \(10^{21}\)
  • Jillion - \(10^{24}\)
  • Killion - \(10^{27}\)
  • Nillion - \(10^{30}\)
  • Pillion - \(10^{33}\)
  • Rillion - \(10^{36}\)
  • Sillion - \(10^{39}\)
  • Tillion - \(10^{42}\)
  • Villion - \(10^{45}\)
  • Willion - \(10^{48}\)
  • Zillion - \(10^{51}\)
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Binary198 Binary198 17 hours ago

Pretty pog function PPF(n)

This is a pretty pog function which I coded in python, and took some inspiration from Loader.c. Here is the code:This program calculates PPF(0). To get PPF(1), just replace print(multi_encode([3,5,7],1)) with print(multi_encode([3,5,7],2)), and so on. :

  • PPF(0) = 28672
  • PPF(1) = 41943040
  • PPF(2) = 52776558133248
  • PPF(3) = 22817710804108129226940657696768

  • PPF(4) = 36794784852483217832937571293171829026104371517528449732287767245550213084811834739706791854136911571329560706591588276995872501216523128251636641320996447498969032415258414259849055273763474694433546450399544231792918209163132645395204526260611015996104772383619867867527174082467311278701515847086085073228215819587157852340592800909333447077607070300142454070768415095087235021360330264125…

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Rgetar Rgetar 21 hours ago

Ordinal Explorer update (periods; levels; indentation modes)

Today I updated Ordinal Explorer web page program.

I found errors in its previous (March 2021) version, for example, infinite descending chain, incorrect converting into syntax sugar. In the new version I fixed these issues and many other bugs and made other improvements.

  • 1 Periods
    • 1.1 Levels
    • 1.2 Alphabet
    • 1.3 Strings
    • 1.4 Comparison
    • 1.5 Periodical comparison
    • 1.6 Fundamental sequences
      • 1.6.1 Zero rule
      • 1.6.2 SLO rule
      • 1.6.3 Successor rule
      • 1.6.4 Plain rule
      • 1.6.5 Main rule
      • 1.6.6 Finding endingn
    • 1.7 Standard forms
  • 2 Switching nlevels and initial strings
  • 3 Indentation modes
  • 4 Сontrol of the list from the keyboard
  • 5 Some changes
  • 6 Non-trivial cases of non-minimal standard forms
  • 7 Other issues

In previous version fundamental sequences looked ugly, for example, fundamental sequence of ω2 were


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Hexirp Hexirp 23 hours ago

(2021-11-28) Simple Ordinal Collapsing Functions with Multivariate Functions for counting Inaccessible Cardinals

In order to go beyond it, you have to know it.

I decided to challenge my own limits on defining ordinal collapsing functions.

  • 1 Name
  • 2 Definition
    • 2.1 System
    • 2.2 The Class of Ordinals
    • 2.3 Finite Products
  • 3 Notes

It is "Simple Ordinal Collapsing Functions with Multivariate Functions for counting Inaccessible Cardinals".

It is SOCFwMFfcIC.

I work on \( \mathsf{MK} \).

I define that \( \mathrm{On} \) is the class of ordinal numbers.

I define that \( { A } ^ { \_ < \omega } \) is finite products of \( A \).

I define an indexing function of \( { \mathrm{On} } ^ { \_ < \omega } \) by the below formula.


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Kyodaisuu Kyodaisuu 1 day ago

Cleanup of uncategorized sound files

There are so many uncategorized files. Special:UncategorizedFiles shows only result of 1000 files, and all of them are now sound files. Therefore I am going to move untegorized sound files to Category:Sound, so that we can find uncategorized files easily from the link above.

For this purpose, I created a bot account . As Help:Bots explains, the account was given a bot flag by fandom.

A bot flag is a setting in the MediaWiki software that hides bot edits from the Recent changes list. Bot edits can always be revealed by removing the "human" only filter or clicking "Show bots" in Recent Changes.

The bot account is not autoconfirmed yet, so I have to wait for the account to be autoconfirmed to get write permission. After that, I will do this work…

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Kanrokoti Kanrokoti 2 days ago

Strong K-psi function

Japanese version:

  • 1 Overview
  • 2 Strong K-psi function
    • 2.1 Notation
    • 2.2 Ordering
    • 2.3 Cofinality
    • 2.4 Search function
    • 2.5 Fundamental sequence
    • 2.6 FGH
    • 2.7 Standard form
    • 2.8 Naming

We define Strong K-psi function. This notation extends Ordinal Notation Associated to Extended Buchholz's OCF.

In Weak K-psi function, a value of \(\textrm{rank}\) is determined by \(n\) in \(\psi_{$n}(0)\). Based on Mrna den's idea that fixing a value of \(\textrm{rank}\) for each system makes fully richer structures, Strong K-psi function introduce a new variable to fix a value of \(\textrm{rank}\).

Special thanks: p進大好きbot Mrna den

Let \(T\) a…

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Tongfangnoi1 Tongfangnoi1 2 days ago

My Numbersx

  • Arcon
  • Arconwahanull Trucis
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MultiSoul MultiSoul 3 days ago

Mess Of Illions (final page of illions on my blog post)

Million / duillion / duolillion / unillion - 1,000,000

Billion / gillion / tretillion / dillion - 1,000,000,000

Trillion / tetrillion / tetillion - 10^12

Quadrillion / pentillion / petillion / quattuorillion - 10^15

Quintillion / hexillion / hetillion / quinillion - 10^18

Sextillion / heptillion / sexillion - 10^21

Septillion / oktillion / okyillion / septenillion - 10^24

Octillion / ennillion - 10^27

Nonillion / nontillion / dekillion / decillion / novemillion - 10^30

Decillion / hendekillion / mecillion - 10^33

Undecillion / dodekillion / becillion - 10^36

Duodecillion / umptisqiuddlyzillion / trisdekillion / doedecillion / trecillion - 10^39

Tredecillion / tetradekillion / tetrecillion - 10^42

Quattuordecillion / pentadekillion / pentecillion - 10^4…

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Arcioko Arcioko 3 days ago

Arcioko Hash Notation (AHN)

WIP, i will edit this later

i use sharps instead of hashes because i cant use hashes

  • 1 Definition
    • 1.1 Basics
  • 2 Growth Rates
  • 3 Googolisms
  • 4 Examples

  • ()^,

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Binary198 Binary198 3 days ago


I created a 284-byte python code which computes this number. Try to find the code I made!


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Pintchoter896 Pintchoter896 3 days ago

List Of My Numbers

List Of My Coined Numbers with "gol/ol"

Exol = 1,000,000

Hepol = 10,000,000

Number Group 2: More Of My Coined Numbers!:

Whockabeha = {10, 100, 7 (1) 7} & 10

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NumLynx NumLynx 6 days ago

List of numbers

I've started a comprehensive list of numbers here.


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Hexirp Hexirp 10 days ago

(2021-11-18) Bug

You probably see "User_blog_comment:Windows_NT_6.0/Numbers_about_mathematical_constants/@comment-38470668-20200607132940" in Category:Candidates_for_deletion.

This is probably a bug. In the past, it was possible to use templates in comments, and it did. However, the comment system was switched and this is the only one that fell into the space-time limbo.

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Plain'N'Simple Plain'N'Simple 11 days ago

PDF with the definition of Q

Here is the definition of Q in Letter Notation:

Definition of Q - PDF

I'm only posting this since I promised (and I will not be participating in the discussion here).

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Dokk17 Dokk17 11 days ago

Help needed - legion arrays

So, um, hello?

Been following this wiki for quite a while now, and although I understand several concepts and functions and etc., I still cannot figure out a few things.

One of these is legion arrays, the pinnacle of BEAF.

Now, I know this is ill-defined and therefore doesn't really count the way it is now. However I found its functioning fascinating since it will generate arrays that are really large with little effort.

So.... there goes my question. I understand that:

{3,3/2} = 3&3&3, or in other words, a humongous array with a tritri 3's between brackets.

In this sense, how do you open {3,3/3} up?

I see that in BEAF page it is said that it becomes {3&3&3/2}. Now, this means {3,3,3,3,3......3,3/2} with a tritri 3's before "/2"? Or it opens up i…

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ARsygo ARsygo 13 days ago

Another news

Hello there, I noticed that there is a popup stating that our wiki's URL will be migrated to the standard "" domain, in the coming weeks. And also, I checked it on other sites, they also have the same popup. What do you think about this move?

Update: it is now moved to "" domain. I'll update the news ASAP.

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Skrato333 Skrato333 15 days ago

powers of numbers

256 (2^8)

512 (2^9)

1,024 (2^10)

2,048 (2^11)

4,096 (2^12)

32,768 (2^15)

65,536 (2^16)

131,072 (2^17)

262,144 (2^18)

1,048,576 (2^20)

16,777,216 (2^24)

536,870,912 (2^29)

1,073,741,824 (2^30)

2,147,483,648 (2^31)

4,294,967,296 (2^32)

1,099,511,627,776 (2^40)

1,125,899,906,842,624 (2^50)

9,007,199,254,740,992 (2^53)

72,057,594,037,927,936 (2^56)

1,152,921,504,606,846,976 (2^60)

9,223,372,036,854,775,808 (2^63)

18,446,744,073,709,551,616 (2^64)

295,147,905,179,352,825,856 (2^68)

1,180,591,620,717,411,303,424 (2^70)

1,208,925,819,614,629,174,706,176 (2^80)

77,371,252,455,336,267,181,195,264 (2^86)

79,228,162,514,264,337,593,543,950,336 (2^96)

324,518,553,658,426,726,783,156,020,576,256 (2^108)

85,070,591,730,234,615,865,843,651,857,942,052,864 (2^126)


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Skrato333 Skrato333 15 days ago

i have a problem .-.

Hexataxia-Taxis (10^^^10^^^6)

Folkman’s Number (2^^^2^901)

Grahal (g1) (3^^^^3)


Fish Number Seven (F763(10100)

Big Foot (FOOT10(10100)

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Kyodaisuu Kyodaisuu 16 days ago

MathJax on mobile

Now we can see MathJax equation on mobile.

It implies that MediaWiki:Common.js is read on mobile skin. Right?

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Literally Legendary Literally Legendary 16 days ago

Redpoppy function

The Redpoppy(m,n) function whose domain is the set of pairs (m,n) of positive integers m and n is defined as follows:

Redpoppy(m,n) is the largest positive integer a such that there exists a positive integer y such that Poppy(m,ᵇ n) = a, where Poppy(m,¹ n) = Poppy(m,n) and Poppy(m,ˣ⁺¹ n) = Poppy(m,ˣ Poppy(m,n))

Let's find the value of Redpoppy(2,2):

2*2+1 = 5, 5*2+1 = 11, 11*2+1 = 23, 23*2+1 = 47, 47*2+1 = 95, 95/5 = 19, 19*2+1 = 39, 39/3 = 13, 13*2+1 = 27, 27/3 = 9, 9/3 = 3, 3*2+1 = 7, 7*2+1 = 15, 15/3 = 5. This forms a cycle, and the largest number in this cycle is 95, so Redpoppy(2,2) = 95.

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Literally Legendary Literally Legendary 17 days ago

Poppy function

The Poppy(m,n) function whose domain is the set of pairs (m,n) of positive integers m and n is defined as follows:

If n is 1: Poppy(m,n) = 1

If n is prime: Poppy(m,n) = mn+1

If n is composite, let q be the smallest factor of n which is greater than 1, then Poppy(m,n) = n/q

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Root2222 Root2222 17 days ago

function using prime numbers

I denote \(\mathbb{N}_+\) by the set of positive integers. And \(\mathbb{N}_0:=\{0\}\cup\mathbb{N_+}\).

I defined a sequence \(p_k:\mathbb{N}_0\rightarrow\mathbb{N}_+\) as follows:

If \(k=0\), then \(p_k=1\)

If \(k=1\), then \(p_k=2\)

Suppose \(k>1\).


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Spelpotatis Spelpotatis 20 days ago

My attempt to extend the godstein sequences

  • 1 Introduction
  • 2 My extension
  • 3 Does it terminate?
  • 4 If it does terminate, how powerful would it be?

You guys probably already know how goodsteain sequences work, and if you do, skip to the next seection. If you don't, read on.

A goodstein sequence is a sequence produced by a set of rules:

In order to get started we should first learn about the weak goodstein sequence, which has a limit growth rate of about w^2 (assuming wainer hiearchy fundamental sequences)

Basically, we first start with a "seed" which is the input to the function. Then, every step we take, we increase the base by 1, and represent the new nmber in the new base. We then subtract 1 (using the new base)

Then, we continue until we have reached 0.

The length of a sequence with seed n is w…

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Kanrokoti Kanrokoti 20 days ago

Subspecies primitive difference psi function

Japanese version:

  • 1 Overview
  • 2 Subspecies primitive difference psi function
    • 2.1 Notation
    • 2.2 Ordering
    • 2.3 Depth function
    • 2.4 Ascension function
    • 2.5 Cofinality
    • 2.6 Fundamental sequence
    • 2.7 FGH
    • 2.8 Standard form
    • 2.9 Naming

We define Subspecies primitive difference psi function. This notation extends Ordinal Notation Associated to Extended Buchholz's OCF by introducing Subspecies primitive difference sequence to subscripts of \(\psi\). I expect that \(\psi_0(\psi_{$1}(0)) = (0,0)(1,1)\), \(\psi_0(\psi_{$1}(\psi_{$3}(0))) = (0,0,0)(1,1,1)\), \(\psi_0(\psi_{$1}(\psi_{$3}(\psi_{$6}(0)))) = (…

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CaptainChurchill CaptainChurchill 21 days ago

Binary Zenith Notation

So a lot of people here know that SAN is the strongest array notation that is widely known. A few people from googology discord may also know that me, System Shark, and a few other "trolls" from the server are total nuisances and dumbasses. Well, I want to change these 2 statements. Me, System Shark, Binary198, Veblen Waifu, Aarex Tiaokhiao, and a few other googologists are now developing super OVERPOWERED, COMPACT, AND COMPLEX array notations which are gonna reach at least (0,0,0)(1,1,1)(2,2,0) in BM4. Currently, Aarex Tiaokhiao has been extraordinarily successful, with his notation ASAN obliterating SAN, and even reaching past TRIO SEQUENCING SYSTEM. AAREX, YOU ARE NOW A GOOGOLOGY GOD. And I look up to you and I will follow your footstep…

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Spelpotatis Spelpotatis 21 days ago

pBot,please read this.

This is message personally directed to pBot. I am not here to say you're bad or anything like everyone else, i just wanna offere an opportunity.

You see, i have noticed the discord does not like you. However, i believe basically everyones ways of going about things have been bad. They keep themselves of the perspective that they are superior, while not considering your arguments.

I am a member of the discord. I really want all of this to get sorted out peacefully.

Spelpotatis#1839 is my discord username, so pBot, if you have discord, you could tell me your problems with the discord server, and i can bring it up with them, and hopefully we can eventually get peace between us two, because quite frankly i'm sick of all this discussion, and i wis…

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Sbiis Saibian Sbiis Saibian 21 days ago

New LNS : Official Release Date

So I've finally made a decision about the New LNS. I had tentatively said I would keep up the Classic LNS for as long as google would host it unless I made an official decision about when to perform the upgrade. I have decided to upgrade my site on December 9th, 2021 to commemorate my websites 13th anniversary. This means the Classic version will only be up for approximately one more month before disappearing forever (so if you want to archive it for yourself, this is your last opportunity). The New LNS will be released on an "as is" basis. What this means is whatever the transfer process does to the site will be more or less how it first goes online. The reason for this is that there seems to be issues with saving changes for non-publishe…

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Koteitan Koteitan 22 days ago

True EB sequence system and the sale as crypto googology

\( \newcommand{\bm}[1]{\boldsymbol #1} \newcommand{\len}{ {\rm len}} \newcommand{\cutlen}{ {\rm cutlen}} \newcommand{\if}{~{\rm if}~} \newcommand{\nat}{ {\mathbb N} } \)

I made a large number named "True EB Sequence System Number" on Sept. 29-Nov. 9, 2020 in Japanese Googology Wiki. It's the translation in English.

I'm selling it in opensea, which is the largest NFT market.

Can it be the first sold natural number?

  • 1 Notation
  • 2 Fundamental sequence
  • 3 FGH
  • 4 Large number
  • 5 Explanation by hydra diagram drawing

I defined a notation which is finite natural number sequence \(FT\) as the following.

  1. \((1) \in FT\)
  2. \(\forall k\in \nat~(1 \leq k \rightarrow (1,\underbrace{2,4,6,8,10,\cdots,2k}_{k}) \in FT)\)

I defined the set of True EB sequence \(OT\) as the follow…

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Tsskyx Tsskyx 22 days ago

An extension - recast of Feferman's Theta?

I have created this OCF that is a kind of mashup between Feferman's \( \theta \) and Extended Buchholz's \( \psi \) functions:

\( \textrm{C}_\nu^0(\alpha,\beta) = \Omega_\nu = \left\{\begin{matrix} 1 & \nu = 0 \\ \aleph_\nu & \nu > 0 \end{matrix}\right. \)

\( \textrm{C}_\nu^{n+1}(\alpha,\beta) = \{ \gamma + \delta, \theta_\mu(\xi, \eta) \mid \gamma, \delta, \mu, \xi, \eta \in \textrm{C}_\nu^n(\alpha,\beta) \land (\xi < \alpha \lor \xi = \alpha \land \eta < \beta ) \} \)

\( \textrm{C}_\nu(\alpha,\beta) = \bigcup_{n < \omega} \textrm{C}_\nu^n(\alpha,\beta) \)

\( \theta_\nu(\alpha,\beta) = \min \{\gamma \in \textrm{On} \mid \gamma \notin \textrm{C}_\nu(\alpha,\beta) \} \)

(Keep in mind that an ordinal is identified with the set of all ordinals sma…

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Kanrokoti Kanrokoti 22 days ago

Subspecies primitive difference sequence

Japanese version:

  • 1 Overview
  • 2 Subspecies primitive difference sequence
    • 2.1 Notation
    • 2.2 Ordering
    • 2.3 Ascension function
    • 2.4 Cofinality
    • 2.5 Fundamental sequence
    • 2.6 FGH
    • 2.7 Standard form
    • 2.8 Naming

We define Subspecies primitive difference sequence. Subspecies primitive difference sequence is expected to be a notation which translates Subspecies Primitive psi Function into sequences. As Subspecies Primitive psi Function was made by introducing ascending system to subscripts in Buchholz's function, Subspecies primitive difference sequence is also made by introducing ascending system to diff…

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Tetramur Tetramur 22 days ago

Question about Rayo(2) and powerset

Can we directly apply powerset operation to current best Rayo name of 2 with 56 symbols? If it is possible, we can get even better estimation on Rayo(n) for really small n.

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Binary198 Binary198 22 days ago

Really weird but powerful linear algebra thing

I just thought of this yesterday.

For two matrices or tuples \(A, B\), I define \(\Delta(A, B)\) as the sum of the differences of their values if they have the same size. If the first is larger, \(\Delta(A, B)\) as the sum of the differences of their values plus the sum of all the elements which the first matrix doesn't have, and vice versa.

For example, \(\Delta((2,2,3),(3,4)) = Abs(2-3) + Abs(2-4) + Abs(3) = Abs(-1) + Abs(-2) + Abs(3) = 1+2+3 = 6\)

Another example: \(\Delta((1,2),(3,4,5)) = Abs(1-3) + Abs(2-4) + Abs(5) = Abs(-2) + Abs(-2) + Abs(5) = 2+2+5 = 9\)

\(\Delta\) is commutative unless I am mistaken.

I then define the function \(LINEAR(n)\) like so:

  • For any matrix \(M\), a vector \(\vec{V}\) and a point \(P\) on \(\vec{V}\).
  • Transform \…

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Kanrokoti Kanrokoti 25 days ago

Embedded exponentiation-based psi function

Japanese version:

  • 1 Overview
  • 2 Embedded exponentiation-based psi function
    • 2.1 Notation
    • 2.2 Abbreviation
    • 2.3 Ordering
    • 2.4 Cofinality
    • 2.5 Fundamental sequence
    • 2.6 FGH
    • 2.7 Limit function
    • 2.8 Standard form
    • 2.9 Naming

We define Embedded exponentiation-based psi function. If we will purely introduce exponentiation-base \(\verb|^|\) to Ordinal Notation Associated to Extended Buchholz's OCF as an extension, \(\psi_a(b) \verb|^| \psi_c(d)+e\) will have two way of interpretations: \((\psi_a(b) \verb|^| \psi_c(d))+e\) and \(\psi_a(b) \verb|^| (\psi_c(d)+e)\). O…

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Tsskyx Tsskyx 26 days ago

Recasting Rathjen's Recast

This will be my (not too serious) attempt at understanding Rathjen's Recast of Bachmann's psi function. The aim here is to somehow analyze it and see what properties it has.

Rathjen's recast of Bachmann's function goes as follows:

  • Let \(\Omega\) be an uncountable ordinal such as \(\aleph_1\). Then define \(C^\Omega(\alpha, \beta)\) as the closure of \(\beta \cup \{0, \Omega\}\) under \(+, (\xi \mapsto \omega^\xi), (\xi \mapsto \psi_\Omega(\xi))_{\xi < \alpha}\), and \(\psi_\Omega(\alpha) = \min \{\rho < \Omega : C^\Omega(\alpha, \rho) \cap \Omega = \rho\}\).

We can formally restate this as:

  • \(\textrm{C}_0(\alpha,\beta)=\beta \cup \{0,\Omega\}\)
  • \(\mathrm{C}_{n+1}(\alpha,\beta)=\{\gamma+\delta, \omega^\gamma, \psi(\xi) \mid \gamma,\delta,\xi \i…
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Tsskyx Tsskyx 28 days ago

Feferman function examples?

I am having a difficult time deriving the values of Feferman's Theta function, or rather, it seems like the results I am getting do not agree with the Veblen function, which it's supposed to be equivalent to. Could someone update the article to include examples similar to Madore's Psi and Buchholz's Psi function examples in their respective articles?

Also, could we consider the following alternative formulation of the function? It seems more laborate to me regarding the uncountable ordinals:

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Hexirp Hexirp 29 October

(2021-10-29) Maintenance

I'm going to delete no-license files.

I do not delete files that use templates like Template:Selfcc. To make the work easier, I made it so that categories are automatically assigned to files when using such templates.

There are many files in Special:UncategorizedFiles that are not categorized. I will be deleting files from these that meet some of the criteria.

  1. The license is not specified.
  2. Not used in normal articles.
  3. There is no hope of contacting him/her. Specifically, the uploader has been blocked from posting indefinitely, with his/her own talk page also uneditable. Or, his/her account has been closed.

This should allow us to quickly find files that are not licensed.


I have viewed over 1000 files and deleted 431 of them. However, I am …

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Binary198 Binary198 29 October

Any idea for growth rate?

This is a function I created a while ago. I believe it is powerful, but I'm not sure how strong it is. Does anyone have any clue, at least an approximation?

In BIP-array notation, \(\{a,b,c\} = a \uparrow \uparrow b\) topped with \(c\). For example, \(a^b = \{a,0,b\}\), or googol-n-plex is \(\{10,n,100\}\). Then I define line, plane, ... separators like so (although they are much weaker than those in BEAF):

\(\{a,b,c\textrm{#}_1(n)\textrm{#}_2\}=\{a,\{\{\textrm{#}_1\textrm{#}_2\},b,1\},c\}\). I use hashtags to denote the remainder of the array, and I add subscripts so you don't confuse where to put each. For example, \(\{10,0,1(1)10,0,1(1)100,0,1\}=\{10,\{\{10,0,1(1)100,0,1\},0,1\},1\}=\{10,\{\{10,\{\{100,0,1\},0,1\},1,\},0,1\},1\}=\)giggolp…

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Literally Legendary Literally Legendary 27 October

Corvid function

This is inspired by the look-and-say sequence.

Corvids are a family of birds and I'm going to make a function named after them. This function is based off of a notation which I will call the Raven notation (raven is a type of corvid) which is described as follows:

The notation has the symbols 0, 1, 2, 3, -, and ;. Each command, which is of the form (x) denotes function iteration.

What is a good upper bound of this function? Is it computable?

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Binary198 Binary198 26 October

Idea for fixing the first bit of UNOCF

  • 1 Original
  • 2 What went wrong
  • 3 How I fixed it
  • 4 Examples

\(\psi(0) = 1\)

\(\psi(\alpha+1)[n] = \psi(\alpha)n\)

\(\psi(\alpha)[n] = \psi(\alpha[n])\) if \(\alpha\) is limit.

If \(\alpha\) has cofinality \(\Omega\) and \(\alpha\) is the output of some function \(f\), \(\psi(\alpha)[0] = \psi(f(0))\) and \(\psi(\alpha)[n+1] = \psi(f(\psi(\alpha)[n]))\).

The function \(f\) is undefined, so the output can vary wildly defined on the choice of \(f\). That is the main problem.

A hyperoperator is either the successor function, or the iteration of a previous hyperoperator via the rule: \(H_{n+1}(a,b) = H_n(a,H_n(a,...H_n(a,a))))\) w/ \(b\) copies. \(H_n(a,b)\) with limit \(b\) is \(\textrm{sup}\{H_n(a, c): c < b\}\) and \(H_n(a,b)\) with limit \(a\) is \(\textrm{su…

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Literally Legendary Literally Legendary 25 October

Ply function

Ply(x) for any non-negative integer x is defined as follows:

Ply(0) = 0

Ply(x+1) is the smallest positive integer greater than Ply(x) such that li(Ply(x+1))-x) is the prime counting function of x and li(x) is the logarithmic integral of x.

The name is a combination of "pi" and "li".

The -ply suffix can mean the Ply function of the original number, so Googolply = Ply(10^100) for example

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Googleaarex Googleaarex 25 October

I'm back, again!

Everything have been changed for last few years. This year, I recently came back to Googology, better than I was young! I like to thank Spampingabot for motivating me for coming back.

Significantly, I made 2 strong array notations including methods you haven't seen before. These array notations might beat SAN, and I will try my best to beat one of strongest notations, Omega-Y Sequence.

  • Aarex's Superstrong Array Notation (I am pretty confident that this might beat TSS.)
  • Aarex's Oddly Strong Notation
  • (plus a new OCF embedding aSBN: Superstrong OCF)

Lastly, I separated my googological work into a new separate page. It's hosted in the same site where you can play incremental games.

  • Aarex's Googology

Unfortunately, I am not going to involve in uncompu…

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Literally Legendary Literally Legendary 24 October

xelpmoC function

First I define complexsum(n)

Should Imaginary Boundless Constant be considered an infinite number?

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Literally Legendary Literally Legendary 24 October

Quartz function

Quartz(m,n) for any integer m>1 and any non-negative integer n, is defined as follows:

Quartz(m,0) = m+1

Quartz(m,n+1) = the smallest positive integer p such that p>Quartz(m,n) and Quartz(m,n) is a factor of m^p-1. If there is no such number, then Quartz(m,n+1) = the smallest positive integer p such that p>Quartz(m,n) and Quartz(m,n)+1 is a factor of m^(p)-1.

Example values (correct me if i'm wrong):

Quartz(2,0) = 3

Quartz(2,1) = 4

Quartz(2,2) = 8

Quartz(2,3) = 12

Quartz(2,4) = 24

Quartz(2,5) = 40

what is this function

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Binary198 Binary198 24 October

OwO function

A function based off of Turing Machines.

We denote the output of a given Turing machine \(T\) with input \(x\) as \(o(T, x)\).

An enumeration of Turing machines is a map \(\mathbb{N} \rightarrow T; n \mapsto f_n\), where \(T\) is the set of all Turing machines, assigning each natural number to the \(n\)th Turing machine in that enumeration.

A family of Turing machines is a map \(\mathbb{N} \rightarrow E; n \mapsto F_n^m\), where \(E\) is the set of all enumerations of Turing machines, assigning each natural number to the \(n\)th enumeration of Turing machines in that family.

An indexed family of Turing machines is a map \(\mathbb{N}^2 \rightarrow T; (n, m) \mapsto F_n^m\), assigning each natural number to the \(m\)th Turing machine in the \(n\…

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Kanrokoti Kanrokoti 24 October

Buchholz sequence

Japanese version:

  • 1 Overview
  • 2 Buchholz sequence
    • 2.1 Notation
    • 2.2 Ordering
    • 2.3 Ascension function
    • 2.4 Cofinality
    • 2.5 Fundamental sequence
    • 2.6 FGH
    • 2.7 Standard form
    • 2.8 Naming

We define Buchholz sequence. Buchholz sequence is expected to be a notation which translates Buchholz's function into sequences. To do that, Buchholz sequence involves difference sequences to define \(\textrm{dom}\), since simply setting \(\textrm{dom}\) for each element of sequences only creates \(0\)-starting Beklemishev's worms. In other words, by letting difference sequences as \(0\)-starting Beklemishev'…

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30jl02 Jason Liao 30jl02 Jason Liao 24 October

Hyper X notation

This notation is used to make large numbers like most other notation like hyper E notation(ExE)

X10= 10{10}10 (Tridecal)

Xn= n{n}n


Xn#m=XX.....(m X's)XX.....((m-1)X's)..................................XXX(3 X's)XX(2 X's)X(1 X)n


Xn##m=Xn#n#n#....n(m #'s)

1.Xn=10(n=0), Otherwise Xn= n{n}n

2.XXn=20(n=0), Otherwise Xn= n{n}n

3.Every X(s) that you add in the front when n=0, you add 10

4.Xn#m=X10(n,m=0),Otherwise Xn#m=XX.....(m X's)XX.....((m-1)X's)..................................XXX(3 X's)XX(2 X's)X(1 X)n






Xn{#,4}m=Xn{Xn{Xn{Xn{Xn{n^^m}m}m}m}m}m(add one ^)



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Hexirp Hexirp 22 October

(2021-10-23) New Policy

I'm currently rewriting the policy for this wiki. You can find a draft of it at User:Hexirp/sandbox/Googology_Wiki:Policy.

First, I would like to clarify the division of roles between policies and guidelines. This is because I believe that the current policy is too detailed.

Secondly, I think it would be a good idea to lift the ban on using Googology Wiki blog posts as a source for articles altogether.

  • This was a restriction to prevent people from creating large numbers of their own articles, but it has been easily circumvented by Google Site and others.
  • Personal sites such as Google Site are difficult to keep track of changes, but Googology Wiki's blog posts keep track of all changes.
  • Personal sites may contain inappropriate content, but blog …
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Arcioko Arcioko 22 October

unnamed factorial

im bad at extending this to make it sound "formal" so i will just dump the definition

Notated as: (not so good approximation) E25#539

5? = no idea how to calculate

any sugestions to how i shall name this? i have considered interrogatorial (or ?torial) and detonatorial (detonate + factorial, kinda obvious), i also thought of making it the fourth superfactorial, since it beats all of them

edit: i dont think i wanna name it detonatorial, so either i name interrogatorial (or ?torial) or i make a superfactorial

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Binary198 Binary198 20 October

Sushi number🍣🐠🐡

One of the world's biggest numbers, dedicated to one of the world's best foods! My attempt at making the largest valid googologism. Leave any corrections in the comments, and I will make sure to fix them because I won't let a tiny mistake get in my way :)

I guess I could keep going, but I'm fine with stopping now. Hopefully the fourth sushi number is the largest valid googologism :)

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Clyde1000^1000 Clyde1000^1000 18 October

Is this correct?

Today, I will do a good meaning towards E# and xE#

E A=10^A

A E B=A*10^B (following rules of equations...)

E A#B=E (E A#(B-1) )

E A#B#C=E A#E (A#B#(C-1))

? will now mean unknown part of txt.

E ?#X#Y = E ?# E ?#X#(Y-1)

E X##Y = E X#(X##(Y-1))

E X##1 = E X

E X #^{Y} Z=E X #...# Z Y #'s { because i hate writing ##########################################}

E X#^{Y}Z=E X#^{Y-1}X#^{Y-1}......X#^{Y-1}X#^{Y-1}X Z X's

GODGAHLAH= E 100#^{100}100

that all

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