Troisième nombre de Fish

Le troisième nombre de Fish (F₃) est un nombre défini par le googologue japonais Fish en 2002. C'est l'un des sept nombres de Fish‏‎.

Définition

Define application s(n) as:

• s(1)f = g; g(x)=f^x(x)
• s(n)f = g; g(x)=[s(n-1)^x]f(x) (si n>1)

where s(n) is a functional, and the growth rate in the hiérarchie de croissance rapide is

\begin{eqnarray*} s(x)f(x) \approx f_{\omega^\omega}(x) \end{eqnarray*}

ss(n) map is defined as: \begin{eqnarray*} ss(1)f & := & g; g(x)=s(x)f(x) \\ ss(n)f & := & g; g(x)=[ss(n-1)^x]f(x) (\text{if }n>1) \\ \end{eqnarray*}

and the growth rate is \begin{eqnarray*} ss(1)f(x) = s(x)f(x) & \approx & f_{\omega^\omega}(x) \\ ss(n)f(x) & \approx & f_{\omega^{\omega+n-1}}(x) \end{eqnarray*}

Definition and growth rate of Fish function 3, \(F_3(x)\), is \begin{eqnarray*} F_3(x) & := & ss(2)^{63}f; f(x)=x+1 \\ F_3(x) & \approx & f_{\omega^{\omega+1}\times63}(x) \end{eqnarray*}

Finally, \begin{eqnarray*} F_3 := F_3^{63}(3) \approx f_{\omega^{\omega+1}\times63 + 1}(63) \end{eqnarray*}

Computation

Similar to the systems for other fish numbers, this system uses translations of functions. Therefore, unlike usual systems simply rewriting terms, the understanding of the precise definition of Fish number 3 requires a deep understanding of the notions of functions. On the other hand, p進大好きbot formulated an alternative computable notation equivalent to this system, and hence people can understand the behaviour, even if they do not have sufficient knowledge of functions. In particular, Fish number 3 is computable.