The Ackermann ordinal is equal to \(\varphi(1,0,0,0)\) using phi function, \(\vartheta(\Omega^3)\) using Weiermann's theta function, \( heta(\Omega^2)\) using Bird's theta function and \(\psi_0(\Omega^{\Omega^2})\) using Buchholz's psi function (see ordinal notation). It is first fixed point of map \(\alphaightarrow\varphi(\alpha,0,0)\), and also smallest ordinal beyond reach of 3-argument Veblen function.