Fish number 4

Fish number 4 (F₄) is a number defined by Japanese googologist Fish in 2002. 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*}

See also

Googology in Asia

Indian counting system: Lakh · Crore · Tallakshana · Uppala · Dvajagravati · Paduma · Mahakathana · Asankhyeya · Dvajagranisamani · Vahanaprajnapti · Inga · Kuruta · Sarvanikshepa · Agrasara · Uttaraparamanurajahpravesa · Avatamsaka Sutra · Nirabhilapya nirabhilapya parivarta · Jaghanya Parīta Asaṃkhyāta
Chinese, Japanese and Korean counting system: Wan · Yi · Zhao · Jing · Gai · Zi · Rang · Gou · Jian · Zheng · Zai · Ji · Gougasha · Asougi · Nayuta · Fukashigi · Muryoutaisuu
See also: Template:Googology in Japan