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Fish number 7 (F7), is a number defined by Japanese googologist Fish in 2013.[1] It is the largest of the seven Fish numbers. It is based on an extension of Rayo's number.

In Fish number 4, 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 x+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. Fish, Googology in Japan - exploring large numbers (2013)