Bashicu matrix system | |
---|---|

Type | matrix |

Based on | \(n^2\) |

Growth rate | \(f_{>\psi_0(\Omega_\omega)}(n)\) |

**Bashicu matrix system** (BMS) is a notation designed to produce large numbers.^{[1]}^{[2]} It was invented by Bashicu in 2014.^{[3]} There are several versions of definitions as explained in this article.

A Bashicu matrix is any rectangular matrix, such as

\[\begin{pmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23} \end{pmatrix}\]

with any dimensions, whose only elements are nonnegative integers. Such a matrix can also be expressed as the sequence of transposes of its columns (known as the **sequence expression** of a Bashicu matrix);

\[(a_{11},a_{21})(a_{12},a_{22})(a_{13},a_{23})\]

The function \(BM\) takes a natural number \(n\) along with a Bashicu matrix \(S\) as inputs and evaluates a natural number output, denoted as \(BM[n]\) or \(S[n]\), by repeatedly modifying the matrix until it's empty, following transformation rules explained in this article.

Using the FGH to approximate Bashicu matrix yields 1-row matrices (equivalent to the primitive sequence system) to be bounded by \(f_{\varepsilon_0}\) and 2-row matrices (aka the pair sequence system) by \(f_{\psi_0(\Omega_\omega)}\) with respect to Buchholz's function. The growth rate of matrices with 3 (trio sequence system) or more rows in general, and whether \(BM\) is always a total function, is an unsolved problem in mathematics.

## Definition[]

### Original definition[]

Bashicu originally defined the system in Bashicu's undefined modification of the BASIC programming language.^{[4]} The program was not intended to actually be run because of the undefinedness of the modification of the language and also because the limit of the memory and calculation time does not allow calculating the actual final value of the large number in reality, and hence Fish wrote a program called "Bashicu matrix calculator" to demonstrate the intended calculation process (this program was verified by Bashicu). Therefore, the official definition of Bashicu matrices can be found in the source code of Fish's program.^{[5]}^{[6]} It can calculate BM1, BM2, BM2.1, BM2.2, BM2.3, BM3, BM3.1, BM3.2, and BM4 (these are explained in the "Revision" section of this article).^{[7]}

The Bashicu matrix calculator also has a web interface. It has 4 options of "*incrementing n*": the original definition for Bashicu matrices calls for "*n=n * n*", and the other options are variants. If the "*Simulate Hardy function*" variant is chosen, then the calculation exactly matches that of the Hardy function using the Wainer hierarchy for ordinals below \(\varepsilon_0\).

### Transformation rules[]

The terms and rules below refer to the sequence expressions of matrices.

#### Terminology[]

- The "length" of a sequence is the number of pairs of brackets (i.e. the number of columns).
- An element of a sequence is a list nonnegative integers surrounded by just one pair of brackets.
- \(S\) denotes any sequence.
- \(Z\) denotes an element of the form \((0,0,...,0)\), which has one or more zeros.
- \(f(n) = n^2\). (variants of the notation may use other base functions)
- \(A\frown B\) is the concatenation of two sequences. For example, \((0,0)(1,1)\frown (2,2)(3,3)=(0,0)(1,1)(2,2)(3,3)\)

#### Calculation rules[]

Note: The calculation rule explained in the part is BM1 and it has a bug in which the calculation is not terminated. It was already updated into BM4. See "**Definition with mathematical equations**" below for the newest version.

**Rule 1**. \([n]=n\)

**Rule 2**. \(S Z[n]=S[f(n)]\)

**Rule 3**. If neither Rule 1 or 2 is applicable, apply the rules below in order.

**Rule 3-1**. Denote the i-th element from the left in \(S\) by \(S_i\) and the length of \(S\) by \(n\). That is, \(S=S_1S_2\cdots S_n\). Note that \(S_n\) is not equal to \(Z\) since Rule 2 is not applicable.

- If there is no element that is smaller than \(S_n\), replace \(S_n\) by \(Z\); \(S = S_1 S_2 \ldots S_{n-1} Z\) (and then apply Rule 2).
- If there is at least one element that is smaller than \(S_n\), let the rightmost one be \(S_i\). We define the
*good part*of the sequence to be \(G = S_1 \ldots S_{i-1}\), the*bad part*of the sequence to be \(B = S_i \ldots S_{n-1}\), \(L = S_i\), and \(N = S_n\). Note that now \(S=G\frown B\frown N\).

**Rule 3-2**. From \(L = (L_0, L_1, \ldots ,L_m)\) and \(N = (N_0,N_1, \ldots ,N_m)\), \(\Delta=(\Delta_0,\Delta_1,\cdots\Delta_m)\) (difference) is calculated as:

- \(\Delta_i=0\) if there is a zero among \(N_0\) to \(N_{i+1}\) (letting \(N_{m+1}=0\))
- \(\Delta_i=N_i-L_i\) otherwise

As \(L < N\) by choice of \(L\), \(\Delta_i \ge 0\).

**Rule 3-3**: \(B(i)\) is defined as follows:

- \(B(0) = B\)
- \(B(i+1)\) is \(B(i) + \Delta\); here \(+\) means coordinate-wise addition of \(\Delta\) to every element of \(B(i)\). For example, when \(B(0) = (1,1,1)(2,2,2)(3,3,3)\) and \(\Delta = (3,1,0)\), we have \(B(1) = (4,2,1)(5,3,2)(6,4,3)\) and \(B(2) = (7,3,1)(8,4,2)(9,5,3)\).

**Rule 3-4**: \(S[n] = \{ G \frown B(0) \frown B(1) \frown \ldots\ \frown B(f(n))\}[f(n)] \)

### Definition with mathematical equations[]

Koteitan showed^{[8]}^{[9]} we could also define Bashicu Matrix Number as following \(K\), using these rules. \(\mathrm{expand}()\) in the rule is based on the version BM4.

\begin{eqnarray*} \mathrm{Number:}~K&=&\mathrm{Bm}^{10}(9)\\ \mathrm{Function:}~\mathrm{Bm}(n)&=&\mathrm{expand}((\underbrace{0,0,\cdots,0}_{n+1})(\underbrace{1,1,\cdots,1}_{n+1})[n])\\ \mathrm{Rule:}~\mathrm{expand}([n])&=&n\\ \mathrm{expand}({\boldsymbol S}[n])&=&\left\{\begin{array}{ll} \mathrm{expand}({\boldsymbol S}_0\cdots{\boldsymbol S}_{X-2}[f(n)])&(\mathrm{if}~\forall y~S_{(X-1)y}=0)\\ \mathrm{expand}({\boldsymbol G}{\boldsymbol B}^{(0)}{\boldsymbol B}^{(1)}{\boldsymbol B}^{(2)} \cdots {\boldsymbol B}^{(f(n))}[f(n)])&(\mathrm{otherwise})\\ \end{array}\right.\\ \mathrm{Activation~function:}~f(n)&=&n^2\\ \mathrm{Matrix:}~{\boldsymbol S}&=&{\boldsymbol S}_0{\boldsymbol S}_1\cdots{\boldsymbol S}_{X-1}\\ \mathrm{Vector:}~{\boldsymbol S}_x&=&(S_{x0},S_{x1},\cdots,S_{x(Y-1)})\\ \mathrm{Good~part:}~{\boldsymbol G}&=&{\boldsymbol S}_0{\boldsymbol S}_1\cdots{\boldsymbol S}_{r-1}\\ \mathrm{Bad~part:}~{\boldsymbol B}^{(a)}&=&{\boldsymbol B}_0^{(a)}{\boldsymbol B}_1^{(a)}\cdots{\boldsymbol B}_{X-2-r}^{(a)}\\ {\boldsymbol B}_x^{(a)}&=&(B_{x0}^{(a)},B_{x1}^{(a)},\cdots,B_{x(Y-1)}^{(a)})\\ B_{xy}^{(a)}&=&S_{(r+x)y}+a\Delta_{y}A_{xy}\\ \mathrm{Ascension~offset:}~\Delta_{y}&=&\left\{\begin{array}{ll} S_{(X-1)y}-S_{ry}&(\mathrm{if}~y\lt t)\\ 0 &(\mathrm{if}~y\geq t) \end{array}\right.\\ \mathrm{Ascension~matrix:}~A_{xy}&=&\left\{\begin{array}{ll} 1 &(\mathrm{if}~ \exists a( r=(P_{y})^a(r+x)))\\ 0 &(\mathrm{otherwise}) \end{array}\right.\\ \mathrm{Lowermost~nonzero:}~t&=&\max\{y|S_{(X-1)y}\gt 0\}\\ \mathrm{Bad~root:}~r &=& P_t(X-1)\\ \mathrm{parent~of}~S_{xy}:~P_{y}(x)&=&\left\{\begin{array}{ll} \max\{p|p\lt x \land S_{py} \lt S_{xy} \land \exists a( p=(P_{y-1})^a(x))\} & (\mathrm{if}~y\gt 0)\\ \max\{p|p\lt x \land S_{py} \lt S_{xy} \} & (\mathrm{if}~y=0)\\ \end{array}\right.\\ \end{eqnarray*}

## Calculation Example[]

Let's look at the first expansion step of \((0,0,0)(1,1,1)(2,2,2)(3,3,3)(4,2,0)[2]\) using the instructions above:

\begin{eqnarray*}{\boldsymbol S} &=& {\boldsymbol S}_0{\boldsymbol S}_1{\boldsymbol S}_2{\boldsymbol S}_3{\boldsymbol S}_4\\ &=&(S_{00},S_{01},S_{02})(S_{10},S_{11},S_{12})(S_{20},S_{21},S_{22})(S_{30},S_{31},S_{32})(S_{40},S_{41},S_{42})\\ &=&(0,0,0)(1,1,1)(2,2,2)(3,3,3)(4,2,0) \end{eqnarray*}

**Parent****of the first row**: The rightmost element which is smaller than the last entry in the 1st row is called the**parent**of the 1st row. In this example, the last entry is \(S_{40} = 4\) and the parent is \(S_{30}=3\). The column which includes the parent of \(S_{xy}\) is represented as \(P_y(x)\) (note that it refers to the column index, not the entry itself).**Ancestors**: The**ancestors**of an element are the element itself and its ancestor’s parents, recursively. In this case, the ancestors of \(S_{40} = 4\) are \(S_{40}=4,~S_{30}=3,~S_{20}=2,~S_{10}=1~and~S_{00}=0\).**Parents in the second and lower rows**: For any element \(S_{xy}\) not in the first row, its**parent**is defined as the rightmost element which is both:- in the same column as one of the upper element \(S_{x,y-1}\)'s ancestors, \((P_{y-1})^a(x)\)
- smaller than and to the left of \(S_{xy}\)

**bad root**: The column \(P_t(X-1)\) which has the parent of the rightmost column \(X-1\) of the lowermost nonzero row \(t\) is called the**bad root**. The bad root marks the boundary between the non-copied part of the matrix (known as the**good part**\({\boldsymbol G}\)) and the part of the matrix to be copied (the**bad part**\({\boldsymbol B}^{(0)}\)). In this case, \(S_{41} = 2\) is the lowermost nonzero entry in the rightmost column, and its parent is \(S_{11}=1\), so the bad root of this matrix is the second column (\(r=1\)).**Good Part**and**Bad part**: \({\boldsymbol S}_r = (1,1,1)\) means that \({\boldsymbol G} = (0,0,0)\) and \({\boldsymbol B}^{(0)} = (1,1,1)(2,2,2)(3,3,3)\).**Ascension Offset**: Using the value of the bad root \({\boldsymbol S}_r = (1,1,1)\) and the value of cut children \({\boldsymbol S}_{X-1} = (4,2,0)\), the**ascension offset**is calculated as \((\Delta_0, \Delta_1, \Delta_2) = (3,0,0)\).**Ascension Matrix**: The**Ascension Matrix**has the value 1 if the corresponding entry in \(S\) has the bad root as one of its ancestors, and 0 if not. In this case, we get \(A_{xy}=(1,1,1)(1,1,1)(1,1,1)\).**Copying the Bad Part**: Finally, the modified copies of the bad part \({\boldsymbol B}^{(a)}\) become \({\boldsymbol B}^{(0)} = (1,1,1)(2,2,2)(3,3,3)\), \({\boldsymbol B}^{(1)} = (4,1,1)(5,2,2)(6,3,3)\), \({\boldsymbol B}^{(2)} = (7,1,1)(8,2,2)(9,3,3)\), \({\boldsymbol B}^{(3)} = (10,1,1)(11,2,2)(12,3,3)\) and \({\boldsymbol B}^{(4)} = (13,1,1)(14,2,2)(15,3,3)\), respectively.**Expansion rule**: According to the**Expansion rule**, we get \({\boldsymbol S}[2] = {\boldsymbol G}{\boldsymbol B}^{(0)}{\boldsymbol B}^{(1)}{\boldsymbol B}^{(2)}{\boldsymbol B}^{(3)}{\boldsymbol B}^{(4)}[4]\) and so (0,0,0)(1,1,1)(2,2,2)(3,3,3)(4,2,0)[2] = (0,0,0)(1,1,1)(2,2,2)(3,3,3)(4,1,1)(5,2,2)(6,3,3)(7,1,1)(8,2,2)(9,3,3)(10,1,1)(11,2,2)(12,3,3)(13,1,1)(14,2,2)(15,3,3)[4]. As a proper matrix, the final expansion after one step looks like this:

\[\begin{pmatrix} 0 & 1 & 2 & 3 & 4\\ 0 & 1 & 2 & 3 & 2\\ 0 & 1 & 2 & 3 & 0\\ \end{pmatrix}[2] = \begin{pmatrix} 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15\\ 0 & 1 & 2 & 3 & 1 & 2 & 3 & 1 & 2 & 3 & 1 & 2 & 3 & 1 & 2 & 3 \\ 0 & 1 & 2 & 3 & 1 & 2 & 3 & 1 & 2 & 3 & 1 & 2 & 3 & 1 & 2 & 3 \\ \end{pmatrix}[4]\]

This result matches the calculation result by the Bashicu matrix calculator.

## Termination and Revisions[]

The first question after the creation of BM1 was whether the calculation would always terminate. This went unanswered until User:KurohaKafka posted a supposed proof of termination on the Japanese BBS 2ch.net in 2016.^{[10]} However, User:Hyp cos disproved it in the talk page of this very article by showing a non-teminating sequence, suggesting that the 2016 proof was incorrect.^{[11]}^{[12]} It is still unknown whether the Bashicu Matrix System always terminates as of September 2020.

Consequently, Bashicu updated the system by making Bashicu Matrix version 2 (BM2), implemented as another algorithm in BASIC.^{[13]} BashicuHyudora provided a slide to explain its new expansion rule.^{[14]} Bashicu again updated the definition to BM3 on June 12th, 2018.^{[15]} Unfortunately, User:Alemagno12 showed that there was a non-terminating example in BM3 later that month.^{[16]}

In November 11th, 2018, a Japanese Googology Wiki user P進大好きbot provided a termination proof of pair sequence systems, a special case of Bashicu matrices where the number of the rows is exactly 2.^{[17]}

Koteitan proposed another way to fix BM2 on his twitter and implemented it as a C program; this version is known as BM2.3. He also showed that the particular matrix (0,0,0,0)(1,1,1,1)(2,2,1,1)(3,3,1,1)(4,2,0,0)(5,1,1,1)(6,2,1,1)(7,3,1,1) expands differently between BM2 and BM2.3.^{[18]} On August 28th, Bubby3 confirmed that BM2 indeed doesn't terminate with that same matrix.^{[19]}

Bashicu finally fixed the official definition, yielding BM4. This is the latest version as of September 1st, 2018. Koteitan has analysed the code^{[20]} and stated that the behavior of BM4 is identical to that of BM2.3.^{[21]}

Though the last official revision by Bashicu is BM4, a handful of unofficial variants, such as BM2.2, BM2.5, BM2.6, BM3.1, BM3.1.1, BM3.2, and PsiCubed2's version, have been proposed.^{[22]} To recap, versions of Bashicu matrix followed by whole numbers (namely, 1-4) were defined by Bashicu themselves and all other versions were made by others.

The latest proposals are called **Idealized BMS**^{[23]} and consist of several new and/or changed rules, depending on the version:
(Mar.19(1), Mar.19(2), BR+delta comp, UBRABC, UBRABC+parent check, UBRA+parent check, Rpakr Def.1, Def.2, Def.3, Def.4 and Def.5).^{[24]}

## As Notations of Ordinals[]

Bashicu matrix system is not an ordinal notation system, because a recursive well-ordering on the system is not defined.^{[note 1]} However, it is strongly believed that it can "express" ordinals in the way explained in the article of fundamental sequences. More precisely, it is believed that there is a map \(o\) from standard Bashicu matrices^{[25]} \(M\) with respect to a given version (which has not been known to have infinite loops) to countable ordinals such that \(o((0)) = 0\) and for any standard Bashicu matrix \(S \neq (0)\), \(o(S)\) is the least ordinal greater than \(o(S^{(n)})\) for any \(n \in \mathbb{N}\), where \(S^{(n)}\) denote the standard matrix given as the single "expansion" of \(S[n]\).

The existence of such a map \(o\) implies the termination of the corresponding version, and hence has not been verified yet in a peer-reviewed source, although there have been many who claim to have proven the termination^{[note 2]} For a standard Bashicu matrix \(S\), we call \(o(S)\) *the ordinal corresponding to \(S\)*. Bashicu coined the ordinals corresponding to several standard Bashicu matrices with respect to BM4.^{[28]}

standard Bashicu matrix \(S\) | name of the corresponding ordinal \(o(S)\) |
---|---|

\((0,0,0)(1,1,1)(2,2,0)\) | first back gear ordinal |

\((0,0,0)(1,1,1)(2,2,1)(3,3,0)\) | second back gear ordinal |

\((0,0,0)(1,1,1)(2,2,2)\) | omega back ordinal |

## Analysis of Growth Rate[]

The growth rate of 1-row matrices (aka the primitive sequence system) is \(f_{\varepsilon_0}(n)\)^{[29]} and the growth rate of 2-row matrices (the pair sequence system) is \(f_{\psi_0(\Omega_\omega)}(n)\) with respect to Buchholz's function in BM4.

A complete analysis of the growth rate of matrices with 3 (trio sequence system) or more rows is difficult because the system is so strong. People are analyzing the system to some extent.^{[30]}^{[31]}^{[32]}^{[33]}^{[34]} Unfortunately, many of the analyses are not reproducible, because they include results which are described in terms of unspecified or ill-defined OCFs such as Bashicu's OCF and UNOCF.

Most claims about the strength and well-foundedness of BMS have so far not been formally defined, or they haven't been proven. For example, all analyses past the matrix

\[\begin{pmatrix} 0 & 1 \\ 0 & 1 \\ 0 & 1 \end{pmatrix}\] contain no proof of correctness. The furthest analysis which specifies an algorithm to convert matrices to ordinals, claims that the matrix

\[\begin{pmatrix}
0 & 1 & 2 & 3 & 2 \\
0 & 1 & 1 & 1 & 0 \\
0 & 1 & 1 & 0 & 0
\end{pmatrix}\]
corresponds to the countable limit of extended Buchholz's function in BM4, although no proof exists yet.^{[35]}

Many explicitly written analyses past the above matrix are (usually extremely long) lists of correspondences between matrices and ordinals that lack an algorithm for converting matrices to ordinals. In order to verify the correctness of the correspondences, it is insufficient to observe finitely many small examples. Since the tables are created just by observing small values and expecting that the same pattern holds for more complicated matrices without a proof, they do not give valid evidences of the correspondences. In particular, they do not imply the termination of the Bashicu matrix system. Thus, these analyses should only be treated as lists of conjectures that convey a certain amount of intuition, yet lack an explicit proof strategy. Furthermore, some "analysis" goes far beyond the limit of existing OCFs, or use an unspecified (and perhaps ill-defined) OCF. Thus, they are not even well-defined mathematical conjectures, but rather only heuristic arguments.

Here are some conjectures for the corresponding ordinal of certain matrices:

Matrix | Ordinal + Explanation |
---|---|

\[\begin{pmatrix} 0 & 1 & 2 & 3 \\ 0 & 1 & 1 & 1 \\ 0 & 1 & 1 & 1 \end{pmatrix}\] | It is believed to correspond to \(\psi(I_\omega)\) with respect to certain OCFs. Unfortunately, this conjecture is often stated with no specification of the OCF being used which are sometimes unspecified or ill-defined. As said before, there is no proof or algorithm to support this result. |

\[\begin{pmatrix} 0 & 1 & 2 & 3 & 3 \\ 0 & 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 1 & 1 \end{pmatrix}\] | It is believed to correspond to \(\psi(M_\omega)\) with respect to certain OCFs. The same situation as described above occurs here and below as well. |

\[\begin{pmatrix} 0 & 1 & 2 & 3 & 4 \\ 0 & 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 1 & 1 \end{pmatrix}\] | It is believed to correspond to \(\psi(K_\omega)\) with respect to certain OCFs. |

Futhermore, Bashicu has described the ordinal corresponding to

\[\begin{pmatrix} 0 & 1 \\ 0 & 1 \\ 0 & 1 \\ 0 & 1 \end{pmatrix}\] with respect to a version called BM4, by one of his original OCFs (which is not formally defined). Of course, the termination of BM4 up to that matrix (or even much smaller 3-row matrix) is still an open problem.

The ordinal in FGH approximating the growth of an n-row matrix in the form

\[\begin{pmatrix} 0 & 1 \\ 0 & 1 \\ \vdots & \vdots \\ 0 & 1 \end{pmatrix}\] for \(n\ge4\) is not yet known or even has a widely accepted conjectured value, and its well-definedness is even unknown.

All functions definable by BMS are computable (if they are total), and therefore much weaker than the busy beaver function, \(\Sigma(n)\), and the other uncomputable functions.

## Footnotes[]

- ↑ Sometimes the termination of BMS is identified with the claim that BMS is an ordinal notation system, but it is based on confusion of the well-foundedness of BMS with the well-definedness of the lexicographic order and confusion of a recursive relation with a relation related to an algorithm in some sense.
- ↑ For example, user Ytosk submitted her preprint and claims to have proven the termination of BMS
^{[26]}. The proof has been however criticized by user P進大好きbot, who claimed that it plagiarized his own work and that Ytosk demonstrates poor understanding of the subject based on earlier conversations with her^{[27]}. Note that there is no known mistake in Ytosk's work, although user P進大好きbot suspects it to be incorrect.

## Sources[]

- ↑ Bashicu Matrix Calculator
- ↑ Fish,
*巨大数論発展の軌跡 (History of development of googology)*, 現代思想 (Comtemporary philosophy) December 2019, pp. 19--28. - ↑ Bashicu's GWiki account
- ↑ The summarization of the large numbers in BASIC language (Japanese article)
- ↑ Source code of Bashicu matrix calculator
- ↑ Definition of Bashicu Matrix by reading the source code
- ↑ User:Kyodaisuu/BasmatVersion
- ↑ The recent condition of Bashicu Matrix System (Japanese article)
- ↑ User_blog:Koteitan/Purely_mathematical_definition_of_BMS
- ↑ Proof that calculation of "standard form" Bashicu matrix terminates and copy
- ↑ Something wrong happens
- ↑ Bashicu matrix proof of termination（in comment thread）
- ↑ BASIC program of Bashicu Matrix system
- ↑ explanation of the Bashicu matrix system (Japanese)
- ↑ Bashicu Matrix version 3
- ↑ BM3 has an infinite loop
- ↑ p進大好きbot, ペア数列の停止性, Japanese Googology Wiki user blog.
- ↑ https://twitter.com/koteitan/status/1018931762053828608
- ↑ BM2 doesn't terminate.
- ↑ Bashicu Matrix version 4 complete analysis
- ↑ BM4=BM2.3
- ↑ The recent condition of Bashicu Matrix System (Japanese article) introduces the blog post A list of all BMS versions and their differences and My own version of BMS
- ↑ https://docs.google.com/document/d/16aJKvY3HTsgBwP-FY9b1JkisvpKMiyCBPQtaAfXwEVs/edit
- ↑ Bashicu Matrix of the Cambrian Explosion
- ↑ Fish. User blog:Kyodaisuu/Standard Bashicu matrix 2023-09-08.
- ↑ Well-Orderedness of the Bashicu Matrix System
- ↑ Special:Diff/390323
- ↑ Bashicu, バシク行列の解析, Japanese Googology Wiki user blog, retrieved at 02/07/2020.
- ↑ A blog post introducing primitive sequence system
- ↑ Analysis of trio sequence system by Bashicu with his own ordinal notations
- ↑ Bashicu's calculation in more detail
- ↑ Analysis by KurohaKafka
- ↑ Analysis by Bubby3
- ↑ Analysis by Googleaarex
- ↑ User_blog:Ytosk/Algorithm_that_changes_BMS_matrices_into_ordinals_up_to_SRO

## See also[]

**By Aeton:** Okojo numbers · N-growing hierarchy**By 新井 (Arai):** Arai's psi function**By aster:** White-aster notation · White-aster**By バシク (BashicuHyudora):** Primitive sequence number · Pair sequence number · **Bashicu matrix system** 1/2/3/4 original idea**By ふぃっしゅ (Fish):** Fish numbers (Fish number 1 · Fish number 2 · Fish number 3 · Fish number 4 · Fish number 5 · Fish number 6 · Fish number 7 · S map · SS map · s(n) map · m(n) map · m(m,n) map) · **Bashicu matrix system** 1/2/3/4 formalisation · TR function (I0 function)**By Gaoji:** Weak Buchholz's function**By じぇいそん (Jason):** Irrational arrow notation · δOCF · δφ · ε function**By 甘露東風 (Kanrokoti):** KumaKuma ψ function**By koteitan:** **Bashicu matrix system** 2.3**By mrna:** 段階配列表記 · 降下段階配列表記 · 多変数段階配列表記 · SSAN · S-σ**By Naruyoko Naruyo:** Y sequence formalisation · ω-Y sequence formalisation**By Nayuta Ito:** N primitive · Flan numbers (Flan number 1 · Flan number 2 · Flan number 3 · Flan number 4 version 3 · Flan number 5 version 3) · Large Number Lying on the Boundary of the Rule of Touhou Large Number 4 · Googology Wiki can have an article with any gibberish if it's assigned to a number**By Okkuu:** Extended Weak Buchholz's function**By p進大好きbot:** Ordinal notation associated to Extended Weak Buchholz's function · Ordinal notation associated to Extended Buchholz's function · Naruyoko is the great · Large Number Garden Number**By たろう (Taro):** Taro's multivariable Ackermann function**By ゆきと (Yukito):** Hyper primitive sequence system · Y sequence original idea · YY sequence · Y function · ω-Y sequence original idea

**By バシク (BashicuHyudora):** **Bashicu matrix system** as a notation template**By じぇいそん (Jason):** Shifting definition**By mrna:** Side nesting**By Nayuta Ito and ゆきと (Yukito):** Difference sequence system

**By ふぃっしゅ (Fish):** Ackermann function**By koteitan:** Ackermann function · Beklemishev's worms · KumaKuma ψ function**By Mitsuki1729:** Ackermann function · Graham's number · Conway's Tetratri · Fish number 1 · Fish number 2 · Laver table**By みずどら:** White-aster notation**By Naruyoko Naruyo:** p進大好きbot's Translation map for pair sequence system and Buchholz's ordinal notation · KumaKuma ψ function · Naruyoko is the great**By 猫山にゃん太 (Nekoyama Nyanta):** Flan number 4 version 3 · Fish number 5 · Laver table**By Okkuu:** Fish number 1 · Fish number 2 · Fish number 3 · Fish number 5 · Fish number 6**By rpakr:** p進大好きbot's ordinal notation associated to Extended Weak Buchholz's function · Standardness decision algorithm for Taranovsky's ordinal notation

**By ふぃっしゅ (Fish):** Computing last 100000 digits of mega · Approximation method for FGH using Arrow notation · Translation map for primitive sequence system and Cantor normal form**By Kihara:** Proof of an estimation of TREE sequence · Proof of the incomparability of Busy Beaver function and FGH associated to Kleene's \(\mathcal{O}\)**By koteitan:** Translation map for primitive sequence system and Cantor normal form**By Naruyoko Naruyo:** Translation map for Extended Weak Buchholz's function and Extended Buchholz's function**By Nayuta Ito:** Comparison of Steinhaus-Moser Notation and Ampersand Notation**By Okkuu:** Verification of みずどら's computation program of White-aster notation**By p進大好きbot:** Proof of the termination of Hyper primitive sequence system · Proof of the termination of Pair sequence number · Proof of the termination of segements of TR function in the base theory under the assumption of the \(\Sigma_1\)-soundness and the pointwise well-definedness of \(\textrm{TR}(T,n)\) for the case where \(T\) is the formalisation of the base theory

**By 小林銅蟲 (Kobayashi Doom):** Sushi Kokuu Hen**By koteitan:** Dancing video of a Gijinka of Fukashigi · Dancing video of a Gijinka of 久界 · Storyteller's theotre video reading Large Number Garden Number aloud

See also: Template:Googology in Asia