Fast-growing hierarchy

The fast-growing hierarchy is a hierarchy of functions which is defined as follows:


 * \(f_0(n) = n+1\)
 * \(f_{\alpha+1}(n) = f^{n}_{\alpha}(n)\)
 * \(f_\alpha(n) = f_{\alpha[n]}(n)\)

[n] denotes the n-th term of fundamental sequence assigned to ordinal \(\alpha)\)

Here are some examples:


 * \(f_1 = 2n\)
 * \(f_2 = 2^{n} \times n\)
 * \(f_3 > 2\uparrow\uparrow n\)
 * \(f_4 > 2\uparrow\uparrow\uparrow n\)
 * \(f_m > 2\uparrow^{m-1} n\)
 * \(f_\omega(n) > 2\uparrow^{n-1} n = Ack(n), using unary version of Ackermann function\)
 * \(f_{\omega+1}(n) > {n,n,1,2}, using Bowers' array notation.\)
 * \(f_{\omega+2}(n) > {n,n,2,2}\)
 * \(f_{\omega+m}(n) > {n,n,m,2}\)
 * \(f_{2 \times \omega}(n) > {n,n,n,2}\)
 * \(f_{3 \times \omega}(n) > {n,n,n,3}\)
 * \(f_{m \times \omega}(n) > {n,n,n,m}\)
 * \(f_{\omega^2)(n) > {n,n,n,n}\)
 * \(f_{\omega^3}(n) > {n,n,n,n,n}\)
 * \(f_{\omega^m}(n) > {n,m+2 (1) 2}\)
 * \(f_{\omega^{\omega})}(n) > {n,n+2 (1) 2} > {n,n (1) 2}\)
 * \(f_{\omega^{\omega}+1}(n) > {n,n,2 (1) 2}\)
 * \(f_{\omega^{\omega}+2}(n) > {n,n,3 (1) 2}\)
 * \(f_{\omega^{\omega}+m}(n) > {n,n,m+1 (1) 2}\)
 * \(f_{\omega^{\omega}+\omega)(n) > {n,n,n+1 (1) 2} > {n,n,n (1) 2}\)
 * \(f_{\omega^{\omega}+{\omega+1}(n) > {n,n,1,2 (1) 2}\)
 * \(f_{\omega^{\omega}+{2 \times \ omega}(n) > {n,n,n,2 (1) 2}\)
 * \(f_{\omega^{\omega}+{\omega^2}(n) > {n,n,n,n (1) 2}\)
 * \ f_{2 \times {\omega^{\omega}}}(n) > {n,n (1) 3}\)
 * \ f_(3 \times {\omega^{\omega}}}(n) > {n,n (1) 4}\)
 * \ f_(m \times {\omega^{\omega}}}(n) > {n,n (1) m+1}\)
 * \ f_{\omega^{\omega+1}}(n) > {n,n (1) n+1} > {n,n (1) n}\)
 * \ f_{\omega^{\omega+2}}(n) > {n,n (1) n,n}\)
 * \ f_{\omega^{\omega+3}}(n) > {n,n,n (1) n,n,n}\)
 * \ f_{\omega^{\omega+m}}(n) > {n,m (1)(1) 2}\)
 * \ f_{\omega^{2 \times \omega}}(n) > {n,n (1)(1) 2} = {n,2 (2) 2}\)
 * \ f_{\omega^{3 \times \omega}}(n) > {n,n (1)(1)(1) 2} = {n,3 (2) 2\)
 * \ f_{\omega^{m \times \omega}}(n) > {n,n (1)(1)(1)...m (1)'s...(1)(1)(1) 2} = {n,m (2) 2\)
 * \ f_{\omega^{\omega^2}}(n) > {n,n (2) 2}\)
 * \ f_{\omega^{\omega^3}}(n) > {n,n (3) 2}\)
 * \ f_{\omega^{\omega^m}}(n) > {n,n (m) 2}\)
 * \ f_{\omega^{\omega^omega}}(n) > {n,n (n) 2} = {n,n (0,1) 2}\)
 * \ f_{4^{omega}}(n) > {n,n ((1) 1) 2} = n \uparrow\uparrow 3 & n\)
 * \ f_{5^{omega}}(n) > {n,n ((0,1) 1) 2} = n \uparrow\uparrow 4 & n\)
 * \ f_{6^{omega}}(n) > {n,n (((1) 1) 1) 2} = n \uparrow\uparrow 5 & n\)
 * \ f_{m^{omega}}(n) > n \uparrow\uparrow m-1 & n\)
 * \ f_{epsilon_0}(n) > n \uparrow\uparrow n-1 & n\)