User blog comment:Simplicityaboveall/Extremely Large Numbers 4/@comment-28606698-20160801212403

Hi, Joe, For FGH definition usually used next rules $$ f_0(a)=a+1$$ $$ f_{\alpha+1}(a)= f_\alpha^a(a)$$ $$ f_\alpha(a)= f_{\alpha[a]}(a)$$  When I wrote [] I found some problem: using rules above it is enough hard to calculate and write exact meaning of  functions for $$\alpha>2$$) . Your rules I formulated in next form $$ f_0(a)=10\times a$$ $$ f_{\alpha+1}(a)= f_\alpha^{a-1}(10)$$ $$ f_\alpha(a)= f_{\alpha[a]}(a)$$ It give possibility to easy express any $$ f_\alpha(a)$$ (for all $$\alpha<\omega$$) with help uparrow notation (Knuth, 1976) Analysis $$ f_0(a)=10\times a$$ $$ f_0(10)=10\times 10=10^2=10\uparrow 2$$ $$f_0^2(10)= f_0(f_0(10))=10\times (10\times 10)=10^3=10\uparrow 3$$ $$ f_1(a)=f_0^{a-1}(10)=\underbrace{ 10\times 10\times...\times10\times 10}_{a \quad tens}=10\uparrow a=10^a$$ $$f_1(10)= 10^{10}=10\uparrow\uparrow 2$$ $$f_1^2(10)= f_1(f_1(10))=10^{10^{10}}=10\uparrow\uparrow 3$$ $$ f_2(a)=f_1^{a-1}(10)=\underbrace{ 10^{10^{...{^{10^{10}}}}}}_{a \quad tens }=10\uparrow\uparrow a$$  $$ f_2(10)=10\uparrow\uparrow 10$$ $$f_2^2(10)= f_2(f_2(10))=10\uparrow^2 (10\uparrow^2 10)=10\uparrow\uparrow\uparrow 3=$$ $$=\underbrace{ 10^{10^{...{^{10^{10}}}}}}_{\underbrace{ 10^{10^{...{^{10^{10}}}}}}_{10 \quad tens}}$$ $$ f_3(a)=f_2^{a-1}(10)=  \left. \begin{matrix}     &&\underbrace{10^{10^{...{^{10^{10}}}}}}\\    & &\underbrace{10^{10^{...{^{10^{10}}}}}} \\     & & \underbrace{\quad \;\; \vdots \quad\;\;}\\     & &\underbrace{10^{10^{...{^{10^{10}}}}}} \\    & &10 \end{matrix} \right \} \text {a layers}=10\uparrow\uparrow\uparrow a $$   $$ f_3^2(10)=f_3(f_3(10))=\left. \left.\begin{matrix} &&\underbrace{10^{10^{...{^{10^{10}}}}}}\\& &\underbrace{10^{10^{...{^{10^{10}}}}}} \\ & & \underbrace{\quad \;\; \vdots \quad\;\;}\\ & &\underbrace{10^{10^{...{^{10^{10}}}}}} \\& &10\end{matrix} \right \} \begin{matrix} &&\underbrace{10^{10^{...{^{10^{10}}}}}}\\& &\underbrace{10^{10^{...{^{10^{10}}}}}} \\ & & \underbrace{\quad \;\; \vdots \quad\;\;}\\ & &\underbrace{10^{10^{...{^{10^{10}}}}}} \\& &10\end{matrix} \right \} \text {10 layers}=10\uparrow\uparrow\uparrow\uparrow 3 $$   $$f_4(a)=f_3^{a-1}(10)=$$ $$=\underbrace{\left. \left.\left.\begin{matrix} &&\underbrace{10^{10^{...{^{10^{10}}}}}}\\& &\underbrace{10^{10^{...{^{10^{10}}}}}} \\ & & \underbrace{\quad \;\; \vdots \quad\;\;}\\ & &\underbrace{10^{10^{...{^{10^{10}}}}}} \\& &10\end{matrix} \right \} \begin{matrix} &&\underbrace{10^{10^{...{^{10^{10}}}}}}\\& &\underbrace{10^{10^{...{^{10^{10}}}}}} \\ & & \underbrace{\quad \;\; \vdots \quad\;\;}\\ & &\underbrace{10^{10^{...{^{10^{10}}}}}} \\& &10\end{matrix} \right \}\cdots \begin{matrix} &&\underbrace{10^{10^{...{^{10^{10}}}}}}\\& &\underbrace{10^{10^{...{^{10^{10}}}}}} \\ & & \underbrace{\quad \;\; \vdots \quad\;\;}\\ & &\underbrace{10^{10^{...{^{10^{10}}}}}} \\& &10\end{matrix} \right \} \text {10 layers}}_a=10\uparrow\uparrow\uparrow\uparrow a $$ $$f_\omega (a)=f_\omega[a] (a)$$ $$f_\omega (10)=f_{\omega[10]} (10)=f_{10} (10)=$$ $$=\underbrace{10\uparrow\uparrow\cdots\uparrow\uparrow 10}_{10  \uparrow' s}$$ $$f_{\omega}^2 (10)=f_\omega (f_\omega (10))=f_{\omega[f_{\omega[10]} (10)]} (f_{\omega[10]} (10))=$$ $$=\underbrace{ 10\uparrow\uparrow\uparrow\uparrow\cdots\uparrow\uparrow\uparrow\uparrow}_{\underbrace{10\uparrow\uparrow\cdots\uparrow\uparrow 10}_{10  \uparrow' s}}(\underbrace{10\uparrow\uparrow\cdots\uparrow\uparrow 10}_{10  \uparrow' s})$$ $$f_{\omega+1} (a)=f_\omega^{a-1} (10)>$$ $$>  \left. \begin{matrix}     &&\underbrace{10\uparrow\uparrow\cdots\uparrow\uparrow 10}\\    & &\underbrace{10\uparrow\uparrow\cdots\uparrow\uparrow 10} \\     & & \underbrace{\qquad \;\; \vdots \qquad\;\;}\\     & &\underbrace{10\uparrow\uparrow\cdots\uparrow\uparrow 10} \\    & &10 \uparrow's \end{matrix} \right \} \text {a layers}$$ $$f_{\omega+2}(a)=f_{\omega+1}^{a-1}(10)>$$ $$>\underbrace{\left. \left.\left.\begin{matrix} &&\underbrace{10\uparrow\uparrow\cdots\uparrow\uparrow 10}\\& &\underbrace{10\uparrow\uparrow\cdots\uparrow\uparrow 10} \\ & & \underbrace{\qquad \;\; \vdots \qquad\;\;}\\ & &\underbrace{10\uparrow\uparrow\cdots\uparrow\uparrow 10} \\& &10\end{matrix} \right \}\begin{matrix} &&\underbrace{10\uparrow\uparrow\cdots\uparrow\uparrow 10}\\& &\underbrace{10\uparrow\uparrow\cdots\uparrow\uparrow 10} \\ & & \underbrace{\qquad \;\; \vdots \qquad\;\;}\\ & &\underbrace{10\uparrow\uparrow\cdots\uparrow\uparrow 10} \\& &10\end{matrix} \right \}\cdots \begin{matrix} &&\underbrace{10\uparrow\uparrow\cdots\uparrow\uparrow 10}\\& &\underbrace{10\uparrow\uparrow\cdots\uparrow\uparrow 10} \\ & & \underbrace{\qquad \;\; \vdots \qquad\;\;}\\ & &\underbrace{10\uparrow\uparrow\cdots\uparrow\uparrow 10} \\& &10\end{matrix} \right \}  \text {10 layers}}_a$$