Fish number 3 (F3) is a number defined by Japanese googologist Fish in 2002.[1][2] It is one of the seven Fish numbers.
Define s(n) map as: \begin{eqnarray*} s(1)f & := & g; g(x)=f^x(x) \\ s(n)f & := & g; g(x)=[s(n-1)^x]f(x) (\text{if }n>1) \end{eqnarray*}
where s(n) is a functional, and the growth rate in FGH is
\begin{eqnarray*} s(x)f(x) \approx f_{\omega^\omega}(x) \end{eqnarray*}
ss(n) map is defined as: \begin{eqnarray*} ss(1)f & := & g; g(x)=s(x)f(x) \\ ss(n)f & := & g; g(x)=[ss(n-1)^x]f(x) (\text{if }n>1) \\ \end{eqnarray*}
and the growth rate is \begin{eqnarray*} ss(1)f(x) = s(x)f(x) & \approx & f_{\omega^\omega}(x) \\ ss(n)f(x) & \approx & f_{\omega^{\omega+n-1}}(x) \end{eqnarray*}
Definition and growth rate of Fish function 3, \(F_3(x)\), is \begin{eqnarray*} F_3(x) & := & ss(2)^{63}f; f(x)=x+1 \\ F_3(x) & \approx & f_{\omega^{\omega+1}\times63}(x) \end{eqnarray*}
Finally, \begin{eqnarray*} F_3 := F_3^{63}(3) \approx f_{\omega^{\omega+1}\times63 + 1}(63) \end{eqnarray*}
Computation[]
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 3 requires a deep understanding of the notions of functions. On the other hand, p進大好きbot formulated an alternative computable notation equivalent to this system,[3] and hence people can understand the behaviour, even if they do not have sufficient knowledge of functions. In particular, Fish number 3 is computable.
Estimation[]
A Googology Wiki user Tetramur pointed out that Fish number 3 can be exactly expressed via slightly modified Wainer hierarchy-based FGH as \(f'^{63}_{\omega^{\omega+1}63}(3)\). This version of FGH has only a single modification of fundamental sequences, mainly \(\omega^\omega[n] = \omega^{n-1}\) instead of \(\omega^\omega[n] = \omega^n\) in standard Wainer hierachy.[4]
Approximations in other notations[]
Fish number 3 is smaller than and comparable to godthroogahlah \(\approx f_{(\omega^{\omega+1}) 99}(100)\).
| Notation | Approximation |
|---|---|
| BEAF | \(\{63,63 (1) 1,63\}\) |
| Bird's array notation | \(\{63,63 [2] 1,63\}\) |
| Cascading-E Notation | \(E63\#^\#*\#\#63\) |
| Hyperfactorial array notation | \(63![1,[63],1,2]\) |
| Fast-growing hierarchy | \(f_{(\omega^{\omega+1}) 63 + 1}(63)\) |
| Hardy hierarchy | \(H_{\omega^{(\omega^{\omega+1}) 63 + 1}}(63)\) |
| Slow-growing hierarchy (using Buchholz hierarchy) | \(g_{\psi_0(\Omega^{\Omega^{\Omega + 1} \omega})}(63)\) |
Computer programs[]
There are computer programs for calculating Fish number 3 and its old version:
- aycabta, Ruby program for calculating Fish number 3
- Okkuu, C++ program for calculating Fish number 3
- Okkuu, C++ program for calculating an old version of Fish number 3
Sources[]
- ↑ Fish, Googology in Japan - exploring large numbers (2013)
- ↑ Fish "巨大数の世界 (The world of googology)" 数学セミナー (Mathematics seminar) July, 2019. pp. 28--31.
- ↑ ja:ふぃっしゅ数バージョン3#計算 (Japanese)
- ↑ Tetramur's post on talk page 17 July 2022
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