SS map is a function which maps "a set of a natural number and a function and S map" to "a set of a natural number and a function and S map". It was defined by Japanese googologist Fish in 2002[1] and used in the definition of Fish number 1 and Fish number 2.
SS map in \(F_1\)[]
\begin{eqnarray*} SS:[m,f(x),S]→[n,g(x),S2] \end{eqnarray*} Here, S2, n, and g(x) are defined as follows. \begin{eqnarray*} S2 & = & S^{f(m)} \\ S2 & : & [m,f(x)]→[n,g(x)] \end{eqnarray*}
SS map in \(F_2\)[]
\begin{eqnarray*} SS:[m,f(x),S]→[n,g(x),S2] \end{eqnarray*} Here, S2, n, and g(x) are defined as follows. \begin{eqnarray*} S2 & = & S^{f(m)} \\ S2:[m,f(x)] & → & [n,p(x)] \\ S2^x:[m,f(x)] & → & [q,g(x)] \\ \end{eqnarray*}
Difference of SS map in \(F_1\) and \(F_2\)[]
In \(F_1\), SS map repeats S map \(f(m)\) times, while in \(F_2\), SS map diagonizes the repetition time of S map. It looks similar but actually this is a big difference, because the SS map in \(F_1\) has the growing level of \(A(0,1) → A(1,0,1,1) → A(1,0,1,2) → A(1,0,1,3)\), while the SS map in \(F_2\) has the growing level of \(A(0,1)→A(1,0,0,n)→A(2,0,0,n)→A(3,0,0,n)\).
As the essential growing mechanism in \(F_2\) is to "diagonizing the functional", in Fish number 3, the definition was simplified. The idea of SS map, i.e., making higher order functional to produce fast-growing function, was not effective in \(F_1\) and \(F_2\) and therefore discarded in \(F_3\); and it was finally realized effectively in Fish number 5.
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