Ackermann ordinal

The Ackermann ordinal is equal to \(\varphi(1,0,0,0)\) using phi function, \(\vartheta(\Omega^2)\) using theta function and \(\vartheta(\Omega^{\Omega^2})\) using psi function.

The growth rates of finite forms of that ordinal in different hierarchies are shown below:


 * \(f_{\varphi(1,0,0,0)}(n) \approx \{X,X,1,1,2\} \&\ n\) (fast-growing hierarchy)
 * \(H_{\varphi(1,0,0,0)}(n) \approx \{X,X,1,1,2\} \&\ n\) (Hardy hierarchy)
 * \(g_{\varphi(1,0,0,0)}(n) \approx \{n,n,1,1,2\}\) (slow-growing hierarchy)