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m(n) map
Based on\(f^x(x)\)
Growth rate\(f_{\varepsilon_0}(n)\)

m(n) map is a googological function that operates on functions. It was defined by Japanese googologist Fish in 2003[1] and used to define Fish number 5. Its system achieves a growth rate of \(f_{\varepsilon_0}(n)\), similar to Hydra function and tetrational array of BEAF. The name of the map was taken from a word mapping.

Definition and growth rate[]

Each \(m(n)\) is a function with an identical domain and codomain. \(m(1)\) has domain and codomain \(\mathbb{N}\), and each \(m(n + 1)\) is \((D \mapsto D) \mapsto (D \mapsto D)\), where \(D\) is the (co)domain of \(m(n)\). So \(m(1) : \mathbb{N} \mapsto \mathbb{N}\), \(m(2) : (\mathbb{N} \mapsto \mathbb{N}) \mapsto (\mathbb{N} \mapsto \mathbb{N})\), \(m(3) : ((\mathbb{N} \mapsto \mathbb{N}) \mapsto (\mathbb{N} \mapsto \mathbb{N})) \mapsto ((\mathbb{N} \mapsto \mathbb{N}) \mapsto (\mathbb{N} \mapsto \mathbb{N}))\), and so forth.

\begin{eqnarray*} m(1)(x) & = & x^x \\ m(2)f(x) & = & f^x(x) \end{eqnarray*}

where \(m(2)\) matches \(s(1)\) of s(n) map; \(m(2)=s(1)\).

\(m(3)\) map is a function which maps "function which maps functions to function" to "function which maps functions to function", defined and calculated as (parenthesis is written verbosely here to indicate the order of calculation)

\begin{eqnarray*} ((m(3)m(2))f)(x) & = & (m(2)^xf)(x) \\ & = & (s(1)^xf)(x) \\ & = & (s(2)f)(x) \end{eqnarray*} Therefore, \(m(3) m(2) = s(2)\) in \(s(x)\) map, and \(m(3)^2 m(2)\) is calculated as \(s(3)\) as follows. \begin{eqnarray*} ((m(3)^2m(2))f)(x) & = & ((m(3)(m(3)m(2)))f)(x) \\ & = & ((m(3)m(2))^xf)(x) \\ & = & (s(2)^xf)(x) \\ & = & (s(3)f)(x) \end{eqnarray*}

Calculation goes on similarly, with FGH approximation,

\begin{eqnarray*} m(3)^3 m(2) m(1)(x) & = & s(4)f(x) \approx f_{\omega^3}(x) \\ m(3)^4 m(2) m(1)(x) & = & s(5)f(x) \approx f_{\omega^4}(x) \\ m(3)^n m(2) m(1)(x) & = & s(n+1)f(x) \approx f_{\omega^n}(x) \\ m(3)^x m(2) m(1)(x) & = & s(x)f(x) \approx f_{\omega^\omega}(x) \\ \end{eqnarray*}

where \(f(x) = m(1)(x) = x^x\). In general

  • \(M_0\) = set of positive integers
  • \(M_{n+1}\) = set of functions which map \(M_n\) to \(M_n\)

\(m(n)\) map \((n \ge 1)\) belongs to \(M_n\) and is defined as follows.

For \(f_n \in M(n)\), \(m(n+1)(f_n) = g_n\) is defined as follows.
For \(f_{n-1} \in M(n-1)\), \(g_n(f_{n-1}) = g_{n-1}\) is defined as follows.
For \(f_{n-2} \in M(n-2)\), \(g_{n-1}(f_{n-2}) = g_{n-2}\) is defined as follows.
……
For \(f_0 \in M(0)\), \(g_1(f_0) = g_0\) is defined as follows.
\(g_0 = (..((f_n^{f_0}f_{n-1})f_{n-2})...f_1)f_0\)

Therefore, by denoting \(f_1\) as \(f\) and \(f_0\) as \(x\), \begin{eqnarray*} m(1)(x) & = & {x}^{x} \\ (m(2)f)(x) & = & ({f}^{x})(x) \\ ((..((m(n+1)f_n)f_{n-1})...f_2)f)(x) & = & ((..({f_n}^{x}{f_{n-1}})...f_2)f)(x) \end{eqnarray*}

And it works out like this.

\begin{eqnarray*} m(4) m(3) m(2) m(1)(x) & = & m(3)^x m(2) m(1)(x) \approx f_{\omega^\omega}(x) \\ m(3) [m(4) m(3)] m(2) m(1)(x) & = & [m(3)^{x} m(2)]^x m(1)(x) \approx f_{\omega^{\omega+1}}(x) \\ m(3)^2 [m(4) m(3)] m(2) m(1)(x) & \approx & f_{\omega^{\omega+2}}(x) \\ m(3)^a [m(4) m(3)] m(2) m(1)(x) & \approx & f_{\omega^{\omega+a}}(x) \\ [ m(4) m(3) ] ^2 m(2) m(1)(x) & = & [ m(3)^x ] [ m(3)^x m(2)] m(1)(x) \approx f_{\omega^{\omega \times 2} } (x) \\ [ m(4) m(3) ] ^3 m(2) m(1)(x) & \approx & f_{\omega^{\omega \times 3} } (x) \\ [ m(4) m(3) ] ^a m(2) m(1)(x) & \approx & f_{\omega^{\omega \times a} } (x) \\ [ m(4)^2 m(3) ] m(2) m(1)(x) & \approx & f_{\omega^{\omega^2} } (x) \\ m(3) \bigl[ m(4)^2 m(3) ] m(2) m(1)(x) & \approx & f_{\omega^{\omega^2+1} } (x) \\ [ m(4) m(3) ] [ m(4)^2 m(3) ] m(2) m(1)(x) & \approx & f_{\omega^{\omega^2+\omega} } (x) \\ [ m(4) m(3) ] ^2 [ m(4)^2 m(3) ] m(2) m(1)(x) & \approx & f_{\omega^{\omega^2+\omega \times 2} } (x) \\ [ m(4) m(3) ] ^3 [ m(4)^2 m(3) ] m(2) m(1)(x) & \approx & f_{\omega^{\omega^2+\omega \times 3} } (x) \\ [ m(4)^2 m(3) ] ^2 m(2) m(1)(x) & \approx & f_{\omega^{\omega^2 \times 2} } (x) \\ [ m(4)^2 m(3) ] ^3 m(2) m(1)(x) & \approx & f_{\omega^{\omega^2 \times 3} } (x) \\ m(4)^3 m(3) m(2) m(1)(x) & \approx & f_{\omega^{\omega^3} } (x) \\ m(4)^4 m(3) m(2) m(1)(x) & \approx & f_{\omega^{\omega^4} } (x) \\ m(5) m(4) m(3) m(2) m(1)(x) & \approx & f_{\omega^{\omega^\omega} } \end{eqnarray*}

It was shown that \begin{eqnarray*} m(3) m(2) m(1)(x) & = & f_{\omega}(x) \\ m(4) m(3) m(2) m(1)(x) & = & f_{\omega^\omega}(x) \\ m(5) m(4) m(3) m(2) m(1)(x) & = & f_{\omega^{\omega^\omega}}(x) \end{eqnarray*} and similarly, \begin{eqnarray*} m(6) m(5) m(4) m(3) m(2) m(1)(x) & = & f_{\omega^{{\wedge}{\wedge}}4}(x) \\ m(7) m(6) m(5) m(4) m(3) m(2) m(1)(x) & = & f_{\omega^{{\wedge}{\wedge}}5}(x) \\ & … & \\ ((..((m(x)m(x-1))m(x-2))...m(2))m(1))(x) & = & f_{\varepsilon_0}(x-2) \end{eqnarray*}

Examples[]

\begin{eqnarray*} m(3)m(2)m(1)(2) &=& m(2)^2m(1)(2) \\ &=& m(2)[m(2)m(1)](2) \\ &=& [m(2)m(1)]^2(2) \\ &=& [m(2)m(1)][m(2)m(1)](2) \\ &=& [m(2)m(1)]m(1)^2(2) \\ &=& [m(2)m(1)]m(1)(4) \\ &=& [m(2)m(1)](256) \\ &=& m(1)^{256}(256) \\ &>& 10\uparrow\uparrow 257 \\ \end{eqnarray*}

\begin{eqnarray*} m(2)m(1)(x)&=&m(1)^x(x) \approx x \uparrow \uparrow x \\ m(2)^2m(1)(x)&=&[m(2)m(1)]^x(x) \approx (x \uparrow^3 x) \\ m(2)^3m(1)(x)&=&[m(2)^2m(1)]^x(x) \approx (x \uparrow^4 x) \\ m(2)^nm(1)(x)&\approx& \{x,x,n+1\} \\ m(3)m(2)m(1)(x) &=& m(2)^xm(1)(x) \approx \{x,x,x\} \approx A(1,0,x) \\ \end{eqnarray*}

Using Taro's multivariable Ackermann function for approximation.

\begin{eqnarray*} m(4)m(3)m(2)m(1)(2) &=& m(3)^2m(2)m(1)(2) \\ &=& [m(3)m(2)]^2m(1)(2) \\ &=& m(2)^2[m(3)m(2)m(1)](2) \\ &\approx& m(2)^2[A(1,0,x)](2) \\ &\approx& m(2)[A(1,1,x)](2) \\ &\approx& [A(1,2,x)](2) \\ &\approx& A(1,2,2) \\ \end{eqnarray*}

\begin{eqnarray*} m(2)m(3)m(2)m(1)(x) &\approx& [A(1,0,x)]^x \approx A(1,1,x) \\ m(2)^2m(3)m(2)m(1)(x) &\approx& [A(1,1,x)]^x \approx A(1,2,x) \\ [m(3)m(2)]^2m(1)(x) &=& m(2)^x[m(3)m(2)]m(1)(x) \approx A(1,x,x)(x) = A(2,0,x) \\ m(3)^2m(2)m(1)(x) &=& [m(3)m(2)]^xm(1)(x) \approx A(x,0,x) = A(1,0,0,x) \\ m(2)m(3)^2m(2)m(1)(x) &\approx& [A(1,0,0,x)]^x \approx A(1,0,1,x) \\ m(3)m(2)[m(3)^2m(2)]m(1)(x) &\approx& m(2)^x[A(1,0,0,x)] \approx A(1,1,0,x) \\ m(2)[m(3)m(2)][m(3)^2m(2)]m(1)(x) &\approx& [A(1,1,0,x)]^x \approx A(1,1,1,x) \\ [m(3)m(2)]^2[m(3)^2m(2)]m(1)(x) &\approx& A(1,2,0,x) \\ [m(3)m(2)]^3[m(3)^2m(2)]m(1)(x) &\approx& A(1,3,0,x) \\ [m(3)^2m(2)]^2m(1)(x) &=& [m(3)m(2)]^x[m(3)^2m(2)]m(1)(x) \approx A(2,0,0,x) \\ [m(3)^2m(2)]^3m(1)(x) &\approx& A(3,0,0,x) \\ m(4)m(3)m(2)m(1)(3) &=& m(3)^3m(2)m(1)(3) = [m(3)^2m(2)]^3m(1)(3) \approx A(3,0,0,3) \\ m(4)m(3)m(2)m(1)(4) &=& m(3)^4m(2)m(1)(3) = [m(3)^3m(2)]^4m(1)(4) \approx A(4,0,0,0,4) \\ \end{eqnarray*} \(m(4)m(3)m(2)m(1)(5) = A(5,0,0,0,0,5) \approx \) Quintigol

For F5(x) \(= ((..((m(x)m(x-1))m(x-2))...m(2))m(1))(x)\)

\(F_5(3) \approx 10 \uparrow^3 257 \)

\(F_5(4) \approx A(4,0,0,0,4) \approx f_{\omega^4}(4) \approx \{4,4,4,4,4,4\} \approx \) Quadreegol

\(F_5(5) \approx f_{\omega^{\omega^5}}(5) \approx \{5,5,(5),2\} = 5^5\ \&\ 5\ \approx \) Petossol

\(F_5(6) \approx f_{\omega^{\omega^{\omega^6}}}(6)  \approx \) E6#^#^######6 \( \approx \) Hexaelgathor

\(F_5(102) \approx f_{\varepsilon_0}(100) \approx 102 \uparrow\uparrow 100 \&\ 102 \approx \) Goppatoth

\(F_5(x) \approx f_{\varepsilon_0}(x-2) \approx x \uparrow\uparrow (x-2) \&\ x \)

Hydra tree[]

Structure of \(m(n)\) is like Kirby-Paris hydra and corresponding ordinal.[2] In the equation \[[ m(4) m(3) ] ^3 [ m(4)^2 m(3) ] m(2) m(1)(x) \approx f_{\omega^{\omega^2+\omega \times 3} } (x)\]

\([m(4) m(3)]^3\) corresponds to \(\omega \times 3\), \(m(4)^2\) corresponds to 2, and \(m(4)^2 m(3)\) corresponds to \(\omega^2\).

Ordinal \(\alpha\) corresponding to the hydra tree matches that of Hardy hierarchy, while the ordinal \(\beta\) calculated in this page was that of FGH. Therefore, \(\alpha = \omega^\beta\). For \(m(3)^3m(2)m(1)(x)\), for example, \(\alpha = \omega^{\omega^3}\) and \(\beta = {\omega^3}\).

Hydra1

In the figure (Kirby and Paris, 1982),[3] the leftmost top node is \(m(3)\), because it is the 3rd node up from the root. The node one segment below it is \(m(2)\), and it has 3 \(m(3)\) above, and therefore it is \(m(3)^3 m(2)\). The node below it is \(m(1)\) node with \(m(3)^3 m(2)\) and \(m(4)m(3)m(2)\), and therefore \([m(3)^3 m(2)][m(4)m(3)m(2)]m(1)\). The root node is \([m(2)[m(3)^2 m(2)]m(1)]\;[[m(3)^3 m(2)][m(4)m(3)m(2)]m(1)]\;(x)\).

In this figure, after cutting one of the top nodes at the \(m(3)\) level, 2 new hydra trees were produced from a node. It corresponds to evaluating \(m(3)=m(2)^x\) with \(x=3\). By comparing the \(m(n)\) structure to the hydra tree before and after cutting the heads,

\[[m(3)^2m(2)m(1)]^2[[m(3)^3m(2)][m(4)m(3)m(2)]m(1)](3) \\ = [[m(3)m(2)]^3m(1)][m(3)^2m(2)m(1)][[m(3)^3m(2)][m(4)m(3)m(2)]m(1)](3)\]

it exactly matches the definition of \(m(n)\) map, because

\[m(3)^2m(2)m(1)(3) = m(2)^3[m(3)m(2)]m(1)(3) \\ = [m(3)m(2)]^3m(1)(3)\]

\(m(n)\) map can thus be correlated with the hydra tree, which is correlated with ordinal of Cantor normal form (see the cited paper of Kirby and Paris, 1982). Therefore, it has the growth rate of \(f_{\varepsilon_0}(x)\).

Sources[]

  1. Fish, Googology in Japan - exploring large numbers (2013)
  2. Visualization of calculation prodcedure of Fish number 5 using Hydra tree (slide PDF), presented by Pochi at googology meeting in Tokyo, November 3, 2007
  3. Kirby, L.; Paris, J. (1982), "Accessible independence results for Peano arithmetic", Bulletin of the London Mathematical Society 14: 285-293.

See also[]

Original numbers, functions, notations, and notions

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


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 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


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


See also: Template:Googology in Asia
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