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The Ackermann ordinal is equal to $$\varphi(1,0,0,0)$$ using Veblen's $$\varphi$$ function[1], $$\vartheta(\Omega^3)$$ using Weiermann's theta function, $$\theta(\Omega^2)$$ using Bird's theta function and $$\psi_0(\Omega^{\Omega^2})$$ 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.

## Sources

1. D. Madore, A Zoo of Ordinals (#1.17) (2017, accessed 2020-11-19)