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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) \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 the s(n) 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, 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*}

And FGH approximation is

\begin{eqnarray*} F_4(x) & \approx & f_{\omega^\text{CK}_{(\omega^{\omega+1}) 63}}(x) \\ F_4 & \approx & f_{\omega^\text{CK}_{(\omega^{\omega+1}) 63}+ 1}(63) \end{eqnarray*}

## Sources Edit

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

Googology in Asia

Fish numbers: Fish number 1 · Fish number 2 · Fish number 3 · Fish number 4 · Fish number 5 · Fish number 6 · Fish number 7
Mapping functions: S map · SS map · S(n) map · M(n) map · M(m,n) map
By Aeton: Okojo numbers · N-growing hierarchy
By BashicuHyudora: Pair sequence number · Bashicu matrix system
Indian counting system: Lakh · Crore · Uppala · Bodhisattva
Chinese, Japanese and Korean counting system: Wan · Yi · Zhao · Jing · Gai · Zi · Rang · Gou · Jian · Zheng · Zai · Ji · Gougasha · Asougi · Nayuta · Fukashigi · Muryoutaisuu
Buddhist text: Tallakshana · Dvajagravati · Mahakathana · Asankhyeya · Dvajagranisamani · Vahanaprajnapti · Inga · Kuruta · Sarvanikshepa · Agrasara · Uttaraparamanurajahpravesa · Avatamsaka Sutra
Other: Taro's multivariable Ackermann function · Sushi Kokuuhen