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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.[1][2]

Définition

[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 \(F_1(x)\).

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.[3] 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.[4]

Calcul

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,[5] and hence people can understand the behaviour, even if they do not have sufficient knowledge of functions. In particular, Fish number 1 is computable.

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 in FGH.

Références

  1. Fish 『巨大数論』 , 1st edition 2013, 2nd edition 2017
  2. ふぃっしゅ数バージョン1が考案された2ch数学板のログ
  3. 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.
  4. 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
  5. Ruby program for calculating Fish number 1
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