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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, Aycabta has created a Ruby program 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. 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 $$E63\#\#\#63\#\#2$$
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
4. Ruby program for calculating Fish number 1

Fish numbers: Fish number 1 · Fish number 2 · Fish number 3 · Fish number 4 · Fish number 5 · Fish number 6 · Fish number 7
Mapping functions: S map · SS map · S(n) map · M(n) map · M(m,n) map
By Aeton: Okojo numbers · N-growing hierarchy
By BashicuHyudora: Primitive sequence number · Pair sequence number · Bashicu matrix system
By Kanrokoti: KumaKuma ψ function
By 巨大数大好きbot: Flan numbers
By Jason: Irrational arrow notation · δOCF · δφ · ε function
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 Nayuta Ito: N primitive
By p進大好きbot: Large Number Garden Number
By Yukito: Hyper primitive sequence system · Y sequence · YY sequence · Y function
Indian counting system: Lakh · Crore · Tallakshana · Uppala · Dvajagravati · Paduma · Mahakathana · Asankhyeya · Dvajagranisamani · Vahanaprajnapti · Inga · Kuruta · Sarvanikshepa · Agrasara · Uttaraparamanurajahpravesa · Avatamsaka Sutra · Nirabhilapya nirabhilapya parivarta
Chinese, Japanese and Korean counting system: Wan · Yi · Zhao · Jing · Gai · Zi · Rang · Gou · Jian · Zheng · Zai · Ji · Gougasha · Asougi · Nayuta · Fukashigi · Muryoutaisuu
Other: Taro's multivariable Ackermann function · TR function · Arai's $$\psi$$ · Sushi Kokuu Hen

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