Fish number 4

Fish number 4 (F4), is a number defined by Japanese googologist Fish in 2002. It is approximately \(f_{\left[\omega^\text{CK}_{(\omega^{\omega+1}) 63}\right] + 1}(63)\) in FGH and the smallest of uncomputational numbers in Fish numbers.

s'(1) map is a functional which maps function \(f\) to function \(g\), as follows.


 * Function \(g\)is a busy beaver function for an having an oracle which calculates function \(f\). That is, with Turing machine + function \(f\), the maximum possible numbers of ones that can be written with an n-state, two-color oracle Turing machine is \(g(n)\).

By comparing with order-n busy beaver function \(\Sigma_n(x)\), which has growth rate of \(f_{\omega^\text{CK}_n}(x)\), let \(f(x) = x+1\) and

\begin{eqnarray*} s'(1)f & = & \Sigma_1(x) \approx f_{\omega^\text{CK}_1}(x) \\ s'(1)^2f & = & \Sigma_2(x) \approx f_{\omega^\text{CK}_2}(x) \\ s'(1)^3f & = & \Sigma_3(x) \approx f_{\omega^\text{CK}_3}(x) \\ s'(1)^nf & = & \Sigma_n(x) \approx f_{\omega^\text{CK}_n}(x) \\ s'(1)^xf & = & \Sigma_x(x) \approx f_{\omega^\text{CK}_\omega}(x) \end{eqnarray*}

For \(n>1\), \(s'(n)\) map is defined similar to s(n) map,

\begin{eqnarray*} s'(n)f & = & s'(n-1)^{x}f(x) (\text{if } n>1) \\ \end{eqnarray*}

Therefore calculation is similar to \(s(n)\) map; \(s'(n)\) map diagonizes \(s'(n-1)\) map. \begin{eqnarray*} s'(2)f & = & s'(1)^xf(x) \approx f_{\omega^\text{CK}_\omega}(x) \\ s'(1)s'(2)f & \approx & f_{\omega^\text{CK}_{\omega + 1}}(x) \\ s'(2)^2f & \approx & f_{\omega^\text{CK}_{\omega \times 2}}(x) \\ s'(3)f & \approx & f_{\omega^\text{CK}_{\omega^2}}(x) \\ s'(4)f & \approx & f_{\omega^\text{CK}_{\omega^3}}(x) \\ s'(n)f & \approx & f_{\omega^\text{CK}_{\omega^{n-1}}}(x) \\ s'(x)f & \approx & f_{\omega^\text{CK}_{\omega^\omega}}(x) \\ \end{eqnarray*}

After this, definition is similar to Fish number 3;

\begin{eqnarray*} ssʹ(1)f & = & sʹ(x)f(x) \\ ssʹ(n)f & = & [ssʹ(n − 1)^{x}]f(x) (\text{if } n>1) \\ F_4(x) & = & ssʹ(2)^{63}f; f(x) = x + 1 \\ F_4 & = & F_4^{63}(3) \end{eqnarray*}

And the calculation is also similar to F3. \begin{eqnarray*} F_4(x) & \approx & f_{\omega^\text{CK}_{(\omega^{\omega+1}) 63}}(x) \\ F_4 & \approx & f_{\left[\omega^\text{CK}_{(\omega^{\omega+1}) 63}\right] + 1}(63) \end{eqnarray*}