| m(m,n) map | |
|---|---|
| Based on | \(f^x(x)\) |
| Growth rate | \(f_{\zeta_0}(n)\) |
m(m,n) map is a function which maps maps to maps. It was defined by Japanese googologist Fish in 2007[1] and used to define Fish number 6. It is a 2 variable extension of m(n) map and has a growth rate of \(f_{\zeta_0}(n)\).
Definition[]
\(M[m,n]\) map \((m=0,1,...; n=1,2,...)\) is defined as follows.
- \(M[0,1]\) is a set of functions of natural numbers to natural numbers.
- \(M[m+1,1]\) is a set that has one element from each \(M[m,n] (n=1,2,...)\); \(M[m+1,1]\) is a direct product of \(M[m,1], M[m,2], M[m,3], ...\).
- Element of \(M[m,1]\) also works as a function in \(M[0,1]\) that is element of element of ... element of \(M[m,1]\).
- \(M[m,n+1] (n=1,2,...)\) is a set of maps from \(M[m,n]\) to \(M[m,n]\).
Element of \(M[m,n]\), \(m(m,n)\) is defined as follows. Here, \(a_i, b_i,f_i\) are elements of \(m(m,i)\) and strict structure of definition is same as m(n) map.
\begin{eqnarray*} m(0,1) (x) & := & x+1 \\ m(m,n+1) f_n f_{n-1} ...f_1 (x) & := & f_n^xf_{n-1}... f_1 (x) \\ & & (m=0; n=1,2,... \text{ or } m=1,2,...; n=2,3,…) \\ m(m+1,1) & := & [m(m,1),m(m,2),m(m,3),…] \\ m(m+1,2)[a_1,a_2,...] & := & [b_1,b_2,…] \text{ where } b_n \text{ is defined as:} \\ b_n f_{n-1}...f_1(x) & := & a_y a_{y-1}...a_n f_{n-1}…f_1(x) (y=max(x,n)) \end{eqnarray*}
Calculation[]
\begin{eqnarray*} m(1,1)(x) & = & [m(0,1),m(0,2),m(0,3),…](x) \\ & = & m(0,1)(x) = x+1 \\ \end{eqnarray*} Let \(m(1,2) m(1,1) = [a_1,a_2,a_3,…]\) and \begin{eqnarray*} a_1(x) & = & m(0,x) m(0,x-1) … m(0,1) (x) \\ & \approx & f_{\varepsilon_0}(x) \\ \therefore m(1,2) m(1,1)(x) & \approx & f_{\varepsilon_0}(x) \\ m(0,2) a_1(x) & \approx & f_{\varepsilon_0 + 1}(x) \\ m(0,3) m(0,2) a_1 (x) & \approx & f_{\varepsilon_0 + \omega}(x) \\ m(0,4) m(0,3) m(0,2) a_1 (x) & \approx & f_{\varepsilon_0 + \omega^{\omega}}(x) \\ m(0,5) m(0,4) m(0,3) m(0,2) a_1 (x) & \approx & f_{\varepsilon_0 + \omega^{\omega^{\omega}}}(x) \\ a_2 a_1(x) & = & m(0,y) m(0,y-1) …m(0,2) a_1(x) (y=max(x,2)) \\ & \approx & f_{\varepsilon_0 \times 2}(x) \\ \end{eqnarray*}
Then,
\begin{eqnarray*} m(0,3) a_2 a_1 (x) & \approx & f_{\varepsilon_0 \times \omega}(x) \\ m(0,4) m(0,3) a_2 a_1 (x) & \approx & f_{\varepsilon_0 \times \omega^{\omega}}(x) \\ m(0,5) m(0,4) m(0,3) a_2 a_1 (x) &\approx & f_{\varepsilon_0 \times \omega^{\omega^{\omega}}}(x) \\ a_3 a_2 a_1(x) & = & m(0,y) m(0,y-1) ... m(0,3) a_2 a_1 (x) (y=max(x,3)) \\ & \approx & f_{\varepsilon_0 ^2}(x) \end{eqnarray*} As for \(a_4\), \begin{eqnarray*} m(0,4) a_3 a_2 a_1 (x) & \approx & f_{\varepsilon_0 ^\omega}(x) \\ m(0,5) m(0,4) a_3 a_2 a_1 (x) & \approx & f_{\varepsilon_0 ^{\omega \times 2}}(x) \\ m(0,6) m(0,5) m(0,4) a_3 a_2 a_1 (x) & \approx & f_{\varepsilon_0 ^{\omega ^ 2}}(x) \\ m(0,7) m(0,6) m(0,5) m(0,4) a_3 a_2 a_1 (x) & \approx & f_{\varepsilon_0 ^{\omega^{\omega}}}(x) \\ a_4 a_3 a_2 a_1(x) & = & m(0,y) m(0,y-1)...m(0,4) a_3 a_2 a_1(x) (y=max(x,4)) \\ & \approx & f_{\varepsilon_0 ^{\varepsilon_0}}(x) \end{eqnarray*}
And similarly, \(a_5\), \(a_6\) can be calculated as \begin{eqnarray*} a_5 a_4 a_3 a_2 a_1(x) & \approx & f_{\varepsilon_0 ^{\varepsilon_0^{\varepsilon_0}}}(x) \\ a_6 a_5 a_4 a_3 a_2 a_1(x) & \approx & f_{\varepsilon_0 ^{\varepsilon_0^{\varepsilon_0^{\varepsilon_0}}}}(x) \\ \end{eqnarray*}
Therefore, \begin{eqnarray*} m(1,2)^2 m(1,1) (x) &=& m(1,2)[a_1,a_2,...](x) \\ & \approx & f_{\varepsilon_1}(x) \end{eqnarray*}
Now let \begin{eqnarray*} m(1,2)^3 m(1,1)(x) = [b_1,b_2,b_3,...](x) \end{eqnarray*} and for calculating \(b_i\), \(\varepsilon_0\) in the calculation of \(a_i\) is changed to \(\varepsilon_1\). Therefore,
\begin{eqnarray*} m(1,2)^3 m(1,1)(x) & \approx & f_{\varepsilon_2}(x) \\ m(1,2)^4 m(1,1)(x) & \approx & f_{\varepsilon_3}(x) \\ m(1,2)^n m(1,1)(x) & \approx & f_{\varepsilon_{n-1}}(x) \\ \end{eqnarray*}
Then calculation goes on similar to \(m(n)\) map. \begin{eqnarray*} m(1,3) m(1,2) m(1,1) (x) & \approx & f_{\varepsilon_\omega} \\ m(1,4) m(1,3) m(1,2) m(1,1) (x) & \approx & f_{\varepsilon_{\omega^\omega}} \\ m(1,5) m(1,4) m(1,3) m(1,2) m(1,1) (x) & \approx & f_{\varepsilon_{\omega^{\omega^\omega}}} \\ m(2,2) m(2,1) (x) &=& m(1,x) m(1,x-1) ... m(1,2) m(1,1) (x) \\ & \approx & f_{\varepsilon_{\varepsilon_0}} \\ \end{eqnarray*}
And then \begin{eqnarray*} m(2,2)^2 m(2,1) (x) & \approx & f_{\varepsilon_{\varepsilon_1}}(x) \\ m(2,2)^3 m(2,1) (x) & \approx & f_{\varepsilon_{\varepsilon_2}}(x) \\ m(2,2)^4 m(2,1) (x) & \approx & f_{\varepsilon_{\varepsilon_3}}(x) \\ m(2,3) m(2,2) m(2,1) (x) & \approx & f_{\varepsilon_{\varepsilon_\omega}}(x) \\ m(3,2) m(3,1) (x) &=& m(2,x) m(2,x-1) ... m(2,2) m(2,1) (x) \\ & \approx & f_{\varepsilon_{\varepsilon_{\varepsilon_0}}}(x) \end{eqnarray*}
As we have calculated to \(m(3,2) m(3,1) (x)\), it is easy to see that \begin{eqnarray*} m(1,2) m(1,1) (x) & \approx & f_{\varepsilon_0} (x) \\ m(2,2) m(2,1) (x) & \approx & f_{\varepsilon_{\varepsilon_0}} (x) \\ m(3,2) m(3,1) (x) & \approx & f_{\varepsilon_{\varepsilon_{\varepsilon_0}}} (x) \\ m(4,2) m(4,1) (x) & \approx & f_{\varepsilon_{\varepsilon_{\varepsilon_{\varepsilon_0}}}} (x) \\ \end{eqnarray*} and therefore
\[m(x,2)m(x,1)(x) \approx f_{\zeta_0}(x)\]
Sources[]
- ↑ Fish, Googology in Japan - exploring large numbers (2013)
See also[]
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
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
By ふぃっしゅ (Fish): Ackermann function
By koteitan: Ackermann function · Beklemishev's worms · KumaKuma ψ function
By Mitsuki1729: Ackermann function · Graham's number · Conway's Tetratri · Fish number 1 · Fish number 2 · Laver table
By みずどら: White-aster notation
By Naruyoko Naruyo: p進大好きbot's Translation map for pair sequence system and Buchholz's ordinal notation · KumaKuma ψ function · Naruyoko is the great
By 猫山にゃん太 (Nekoyama Nyanta): Flan number 4 version 3 · Fish number 5 · Laver table
By Okkuu: Fish number 1 · Fish number 2 · Fish number 3 · Fish number 5 · Fish number 6
By rpakr: p進大好きbot's ordinal notation associated to Extended Weak Buchholz's function · Standardness decision algorithm for Taranovsky's ordinal notation
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
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