Fast-Growing Hierarchy
General rule:
\(f_\alpha(n)=f_{\alpha-1}^n(n)\)
\(f_0(n)\)
Equivilent to \(n+1\)
\(f_0(1)=1+1=2\)
\(f_0(2)=2+1=3\)
\(f_0(3)=3+1=4\)
\(f_1(n)\)
Equivilent to \(2n\)
\(f_1(1)=f_0(1)=1+1=2\)
\(f_1(2)=f_0(f_0(2))=2+1+1=2+2=4\)
\(f_1(3)=f_0(f_0(f_0(3)))=3+1+1+1=3+3=6\)
\(f_2(n)\)
Equivilent to \({2^n}n\)
\(f_2(1)=f_1(1)=f_0(1)=1+1=2\)
\(f_2(2)={2^2}2=8\)
\(f_2(3)={2^3}3=24\)
\(f_3(n)\)
Equivilent to \({2^n}n^2\)
\(f_3(1)=f_2(1)=f_1(1)=f_0(1)=1+1\)
\(f_3(2)=f_2(f_1(f_0(f_0(2))))\)
\(f_3(3)=f_2(f_2(f_1(f_1(f_0(f_0(f_0(3)))))))\)
\(f_\omega(n)\)
\(f_\omega(n)=f_n(n)\)
\(f_\omega(1)=f_1(1)=2\)