Fish number 7

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

In Fish number 4, was used to make Rado's sigma function larger. In Fish number 7, 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 micro-language than Rayo's original micro-language. Function \(RR(f)\) is the Rayo's function of this modified micro-language.

Therefore, the new set of micro-language is


 * 1) "a∈b" 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) "(e∧f)", 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.

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 a Rayo's function of micro-language which implements \(R_1(n)\) as an oracle. It is faster than any function that is defined with help of original Rayo's function and weaker functions (computational functions, Rado's sigma function, or Xi function). For example, \(R_2(n)\) is much stronger than \(Rayo^{Rayo(n)}(n)\), or \(f_{\varepsilon_0}(n)\) of fast-growing hierarchy where \(f_0\) is Rayo's function.
 * \(R_3(n)\) is a Rayo's function which implements \(R_2(n)\) 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. 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*}