Fish number 1 (F1), the smallest of the seven Fish numbers, is a number defined by Japanese googologist Fish in 2002. It uses modified Ackermann function, defined as the following:

• $$S_1(0,y) = y+1$$
• $$S_1(x,0) = S_1(x - 1,1)$$
• $$S_1(x,y) = S_1(x - 1,S_1(x,y - 1))$$

After that, we define the $$S_2$$ function that uses $$S_1$$ as the base function and in general, we have an $$S_n$$ function that uses $$S_{n-1}$$ as the base:

• $$S_1(0,y) = y+1$$
• $$S_z(0,y) = S_{z-1}(y,y)$$ for $$z > 1$$
• $$S_z(x,0) = S_z(x - 1,1)$$
• $$S_z(x,y) = S_z(x - 1,S_z(x,y - 1))$$

The Fish function $$F_1(x) = S_x(x,x)$$ grows about as fast as $$f_{\omega^2}(x)$$. The Fish function is equivalent to Taro's 3-variable Ackermann function, namely $$S_z(x,y) = A(z,x,y)$$.

The Fish number 1 is defined with "SS map", which iterates S map, and calculated with $$SS^{63}(3,x+1,S)$$.

## History

Fish number 1 was posted to an anonymous Japanese textboard 2channel in 2002, in a recreational thread to create a number larger than Graham's number. After Fish posted the number, people discussed how to evaluate the size of the number. Among those people, Doom Kobayashi (小林銅蟲), anonymous poster at the time but currently known as the author of Sushi Kokuu Hen, was especially fascinated with the analysis of the Fish number. In the thread, googological concepts such as chained arrow notation, busy beaver function, and the fast-growing hierarchy were discussed, and other new numbers and functions such as other versions of Fish numbers and Taro's multivariable Ackermann function were invented, and programs of primitive sequence number and pair sequence number were posted. When Doom Kobayashi published Sushi Kokuu Hen, googology was popularized in Japan.

## Original definition

The original definition of Fish number 1 is actually more complicated than above, and the number is also slightly different. The order of the number in FGH is same. Here is the original definition.

 Define S map, a map from "a pair of number and function" to "a pair of number and function", as follows. \begin{eqnarray*} S:[m,f(x)]→[g(m),g(x)] \end{eqnarray*}

Here, $$g(x)$$ is given as follows. \begin{eqnarray*} B(0,n) & = & f(n) \\ B(m+1,0) & = & B(m, 1) \\ B(m+1,n+1) & = & B(m, B(m+1, n)) \\ g(x) & = & B(x,x) \end{eqnarray*}

 Define SS map, a map from "a set of number, function and S map" to "a set of number, function and S map" as follows.

\begin{eqnarray*} SS:[m,f(x),S]→[n,g(x),S2] \end{eqnarray*}

Here, $$S2$$, $$n$$, and $$g(x)$$ are given as follows. \begin{eqnarray*} S2 & = & S^{f(m)} \\ S2 & : & [m,f(x)]→[n,g(x)] \end{eqnarray*}

 Apply SS map 63 times to [3,x+1,S] and we get Fish number $$F_1$$ and Fish function $$F_1(x)$$.

## Computation

Similar to the systems for other fish numbers, this system uses translations of functions. Therefore, unlike usual systems simply rewriting terms, the understanding of the precise definition of Fish number 1 requires a deep understanding of the notions of functions. On the other hand, there are computer programs for calculating Fish number 1, and hence people can understand the behaviour, even if they do not have sufficient knowledge of functions.

In particular, Fish number 1 is computable.

## Approximations in other notations

Fish number 1 is comparable to (slightly larger than) $$A(1,0,1,63)$$ in Taro's 4-variable Ackermann function. Therefore, it is in the order of $$f_{\omega^2+1}(63)$$ in FGH. Thrangol is in the order of $$f_{\omega^2+1}(100)$$, and therefore slightly larger than Fish number 1. In BEAF, it can also be bounded more precisely by {10,63,1,1,2} < F1 < {10,64,1,1,2}, and {4,64,1,1,2} < F1 < {5,64,1,1,2}. Using the approximations in thrangol, Fish number 1 can be approximated as follows.

Notation Approximation
Chained arrow notation $$3 \rightarrow_2 63 \rightarrow_2 2$$
Notation Array Notation $$(63,2\{4,2\}2)$$
BEAF $$\{4,64,1,1,2\}$$
Strong array notation s(3,64,2,1,2)
Extended Hyper-E Notation $$\textrm E63\#\#\#63\#\#2$$
Hyperfactorial array notation $$63![2,1,2]$$
Taro's multivariable Ackermann function $$A(1,0,1,63)$$
s(n) map $$s(1)s(3)[x+1](63)$$
m(n) map $$m(2)[m(3)^2m(2)]m(1)(63)$$
Fast-growing hierarchy $$f_{\omega^2+1}(63)$$
Hardy hierarchy $$H_{\omega^{\omega^2+1}}(63)$$
Slow-growing hierarchy $$g_{\varphi(1,0,0,0)}(63)$$

## Sources

1. Fish, Googology in Japan - exploring large numbers (2013)
2. Fish "巨大数論発展の軌跡 (Trajectory of the development of googology)", in Japanese, 現代思想 (Contemporary Philosophy,) Decembre 2019, pp. 19-28.
3. Shinji Suzuki and Fish. "討議 有限と無限のせめぎあう場所 (Discussion: Battlefield of finite and infinite)", in Japanese, 現代思想 (Contemporary Philosophy,) Decembre 2019, p. 11

Original numbers, functions, notations, and notions

By Aeton: Okojo numbers · N-growing hierarchy
By 新井 (Arai): Arai's psi function
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
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

Methodology

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

Implementation of existing works into programs

Proofs, translation maps for analysis schema, and other mathematical contributions

By ふぃっしゅ (Fish): Computing last 10000 digits of mega · 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 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

Entertainments

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