Fish number 5 (F5) is a number defined by Japanese googologist Fish in 2003.[1] It is one of Fish numbers.
Fish function 5 uses m(n) map.
Definition and growth rate of Fish function 5, \(F_5(x)\), is \begin{eqnarray*} F_5(x) & := & ((..((m(x)m(x-1))m(x-2))...m(2))m(1))(x) \\ F_5(x) & \approx & f_{\varepsilon_0}(x) \end{eqnarray*}
Then Fish number 5 is defined as: \[F_5 := F_5^{63}(3)\]
Therefore, Fish number 5 is greater than Fish number 3 and is approximately \(f_{\varepsilon_0+1}(63)\).
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 5 requires a deep understanding of the notions of functions. On the other hand, there are computer programs by Japanese googologists 猫山にゃん太 and Okkuu for calculating Fish number 1, and hence people can understand the behaviour, even if they do not have sufficient knowledge of functions.
- 猫山にゃん太, ふぃっしゅ数バージョン5の機械的展開方法, Japanese Googology Wiki user blog.
- 猫山にゃん太, Fish Number 5 - ふぃっしゅ数バージョン 5, GitHub page. (javascript code)
- Okkuu, ふぃっしゅ数バージョン5を計算するC++プログラム, paiza. (C++ code)
In particular, Fish number version 5 is computable.
Approximations in other notations[]
Fish number 5 is comparable to grantethrathoth \(\approx f_{\varepsilon_0+1}(100)\).
| Notation | Approximation |
|---|---|
| BEAF | \(\{63,63,2 (X \uparrow\uparrow X) 2\}\)[2] |
| Cascading-E Notation | E63#^^#63#63 |
| Bird's array notation | \(\{63,63,2 [1 \backslash 2] 2\}\) |
| Hyperfactorial array notation | \(63![2,1,1,1,2]\) |
| Fast-growing hierarchy | \(f_{\varepsilon_0+1}(63)\) |
| Hardy hierarchy | \(H_{\varepsilon_0 \omega}(63)\) |
| Slow-growing hierarchy (using Buchholz's function) | \(g_{\psi_0(\Omega_2 + \psi_1(\Omega_2))}(63)\) |
Sources[]
- ↑ Fish, Googology in Japan - exploring large numbers (2013)
- ↑ Using particular notation \(\{a,b (X \uparrow\uparrow X) 2\}\) for \(X \uparrow\uparrow b \&\ a\)
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