Fish number 1

Fish number 1 (F1), the smallest of the seven Fish numbers, is a number defined by Japanese googologist Fish in 2002.Fish, Googology in Japan - exploring large numbers (2013) 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 the "SS map", which iterates S map, and is calculated with \(SS^{63}(3,x+1,S)\).
== History ==
Fish number 1 was posted to an anonymous Japanese textboard 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.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. 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.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 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.
[1] 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*}
[2] 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*}
[3] 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.
* aycabta, Ruby program for calculating Fish number 1
* izmktr, C++ program for calculating Fish number 1
* セカタロウ, PHP program for calculating Fish number 1
* tomo-ukisu, Java program for calculating Fish number 1
* Okkuu, C++ program for calculating Fish number 1
* Okkuu, C++ program for calculating Fish number 1
* Okkuu, C++ program for calculating an old version of Fish number
* Mitsuki1729, scratch program for calculating Fish number 1
* Shintaro Ishimine, Elixir program for calculating Fish number 1
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 	\(63![2,1,2]\)
Ampersand Notation 	\(63[\&_{1}1]\)
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 ==
== See also ==
ja:ふぃっしゅ数バージョン1fr:Premier nombre de Fish
Category:Numbers
Category:Googology in Asia
Category:Non-powers
Category:Quadratic omega level
Category:Fish number
Category:Numbers by Fish
Category:Eponymous numbers