Ackermann ordinal

The Ackermann ordinal is equal to \(\varphi(1,0,0,0)\) using Veblen's \(\varphi\) functionD. Madore, A Zoo of Ordinals (#1.17) (2017, accessed 2020-11-19), \(\vartheta(\Omega^3)\) using Weiermann's theta function, \(\theta(\Omega^2)\) using Feferman's theta function and \(\psi_0(\Omega^{\Omega^2})=\psi_0(\psi_1(\psi_1(\psi_1(0)+\psi_1(0))))\) using Buchholz's psi function (see ordinal notation).
It is the first fixed point of the map \(\alpha\mapsto\varphi(\alpha,0,0)\), and also smallest nonzero ordinal closed under the 3-argument Veblen function.
==Appearances in googology==
Using this system of fundamental sequences:
*Fast-growing hierarchy: \(f_{\varphi(1,0,0,0)}(n)\) represents the expandal array in the climbing method of BEAF, save for Saibian's blasphemorgulus, which is approximately \(f_{\varphi(1,0,0,0)}(99)\).
==Sources==
== See also ==
zh:阿克曼序
ja:アッカーマン順序数
Category:Transfinite ordinals