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Fish number 4 (F4) is a number defined by Japanese googologist Fish in 2002.[1] It is the smallest of the Fish numbers that is defined using an uncomputable function.

s'(1) map is a function which maps functions to functions, as follows.

Function $$s'(1)f$$ is a busy beaver function for an oracle machine having an oracle which calculates function $$f$$. That is, the maximum possible numbers of ones that can be written with an n-state, two-color oracle Turing machine is $$s'(1)f(n)$$.

By comparing with the order-n busy beaver function $$\Sigma_n(x)$$, let $$f$$ be a computable function. Then it's easy to see that (exponents mean iteration of the map here):

\begin{eqnarray*} s'(1)f & = & \Sigma_1(x)\\ s'(1)^2f & = & \Sigma_2(x)\\ s'(1)^3f & = & \Sigma_3(x)\\ s'(1)^nf & = & \Sigma_n(x)\\ s'(1)^xf & = & \Sigma_x(x)\end{eqnarray*}

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

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

After this, the 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{for } n>1) \\ F_4(x) & = & ssʹ(2)^{63}f; f(x) = x + 1 \\ F_4 & = & F_4^{63}(3) \end{eqnarray*}

Sources

1. Fish, Googology in Japan - exploring large numbers (2013)