Fish numbers are numbers defined by the Japanese googologist Fish.[1][2] There are seven Fish numbers:
- Fish number 1 is an extension of the Ackermann function and is comparable to \(f_{\omega^2+1}(63)\);
- Fish number 2 is an extension of Fish number 1 and is comparable to \(f_{\omega^3}(63)\);
- Fish number 3 uses s(n) map and is comparable to \(f_{\omega^{\omega+1}\times 63+1}(63)\);
- Fish number 4 combines Fish number 3 with an oracle system;
- Fish number 5 uses a system that reaches \(\varepsilon_0\) and is comparable to \(f_{\varepsilon_0+1}(63)\);
- Fish number 6 uses an extension of that system which reaches \(\zeta_0\) and is comparable to \(f_{\zeta_0+1}(63)\);
- Fish number 7 combines Fish number 6 with an oracle system in Rayo's function.
References[]
Fish introduced Fish number 1, Fish number 2, Fish number 3 and Fish number 5 on the publication "Mathematics Seminar[3]", and he introduced the definitions of S map and S(n) map and the relationship between the Fast-growing hierarchy and s(n) map.
Ohguro introduced Arrow notation, Chained arrow notation and the Fish numbers in the magazine "Contemporary Philosophy[4]", and considered that the fact that Fish numbers are not a byproduct of some other goal-achieving process, but are devised purely for the generation of large numbers, is important in the gestalt change in the treatment of large numbers.
Kondo [5] discussed whether large numbers that did not exist before creation can be said to have existed with Fish number 1 and Fish number 7.
Sources[]
- ↑ Fish, Googology in Japan - exploring large numbers (2013)
- ↑ Fish, "Googology 2nd Edition", NextPublishing Authors Press, 2017
- ↑ Fish "The world of the large number", Nippon Hyoron Sha, Vol.58, No.7, pp.28-31, 2019
- ↑ T. Ohguro, "What are 'numbers' for information society?", Contemporary Philosophy, vol.47, No.15, Seidosha, 2019
- ↑ K. Kondo, "Don't there exist the large numbers no one knows how to count?", Contemporary Philosophy, vol.47, No.15, Seidosha, 2019
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