Fish number 7 (F7), is a number defined by Japanese googologist Fish in October 2013. It is the largest of the seven Fish numbers. It is based on an extension of Rayo's number.

In Fish number 4, an oracle machine was used to make Rado's sigma function larger. In Fish number 7, an oracle formula is added to Rayo's original micro-language.

A function RR which maps function $$f$$ to function $$RR(f)$$ is defined as follows:

By adding an oracle formula of function $$f$$, $$"f(a)=b"$$, meaning that the ath and bth members of the sequence satisfy the relation $$f(a)=b$$, to the definition of micro-language in Rayo's function, we have a modified version of Rayo's micro-language. We can then define a function $$RR(f)$$, almost identically to Rayo's function, except that we use this modified micro-language.

With that, the new set of formulas in this micro-language is:

1. "ab" means that the ath member of the sequence is an element of the bth member of the sequence.
2. "a=b" means that the ath member of the sequence is equal to the bth member of the sequence.
3. "(¬e)", for formula e, is the negation of e.
4. "(ef)", for formulas e and f, indicates the logical and operation.
5. "∃a(e)" indicates that we can modify the ath member of the sequence such that the formula e is true.
6. "f(a)=b" means that the ath and bth members of the sequence satisfy the relation $$f(a)=b$$

where the 6th formula was added.

The Rayo hierarchy to ordinal $$\alpha$$, $$R_\alpha (n)$$, is defined as follows:

• $$R_0(n) = n$$
• $$R_{\alpha+1} (n) = RR(R_\alpha) (n)$$ (if $$\alpha$$ is a successor)
• $$R_\alpha (n) = R_{\alpha[n]} (n)$$ (if $$\alpha$$ is a limit and $$\alpha[n]$$ is an element of its fundamental sequence)

Therefore,

• $$R_1(n)$$ is on par with Rayo's function.
• $$R_2(n)$$ is like Rayo's function, but using the micro-language which implements $$R_1(n)$$ as an oracle. It is already much more powerful than typical naive extensions of Rayo's function, such as $$Rayo^{Rayo(n)}(n)$$, or $$f_{\varepsilon_0}(n)$$ in a variant of the fast-growing hierarchy where we define $$f_0$$ to be Rayo's function rather than n+1.
• $$R_3(n)$$ is like Rayo's function, but implementing $$R_2(n)$$ as an oracle. Therefore it is much stronger than $$R_2(n)$$.

Fish function 7 is defined by changing the definition of $$m(0,2)$$ in Fish number 6 to $$m(0,2)=RR$$. Therefore,

\begin{eqnarray*} m(0,2)m(0,1)(x) &\approx& R_1(x) \\ m(0,2)^2m(0,1)(x) &\approx& R_2(x) \\ m(0,2)^3m(0,1)(x) &\approx& R_3(x) \\ m(0,3)m(0,2)m(0,1)(x) &\approx& R_\omega(x) \\ \end{eqnarray*}

and the calculation of growth rate is similar to $$F_6$$, except that FGH is changed to Rayo hierarchy. The definition and the growth rate of $$F_7(x)$$ is:

\begin{eqnarray*} F_7(x) &:=& m(x,2)m(x,1) (x) \\ &\approx& R_{\zeta_0}(x) \end{eqnarray*}

Finally, Fish number 7 is defined and approximated as: \begin{eqnarray*} F_7 &:=& F_7^{63}(10^{100}) \\ &\approx& R_{\zeta_0}^{63}(10^{100}) \end{eqnarray*}

## Sources

1. First source in Japanese: ふぃっしゅっしゅ (2013) 『巨大数論』初版 (19 October 2013)
2. Fish, Googology in Japan - exploring large numbers (2013)
3. User_blog:Kyodaisuu/English description of Fish numbers by kyodaisuu (Fish) on 2 November 2013

Original numbers, functions, notations, and notions

By Aeton: Okojo numbers · N-growing hierarchy
By 新井 (Arai): Arai's psi function
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
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

Methodology

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

Proofs, translation maps for analysis schema, and other mathematical contributions

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 Naruyoko Naruyo: Translation map for Extended Weak Buchholz's function and Extended Buchholz's function
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

Entertainments

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

Turing theories: $$\Sigma(1919)$$ (1919th busy beaver number) · Fish number 4 · $$\Xi(10^6)$$ · $$\Sigma_{\infty}(10^9)$$