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## Comparison Table of Ordinal Collapsing Functions

The following table provides a comparison of ordinal collapsing function. It is intended to be a quick reference.

Comparison Ordinal Collapsing Functions

The table is based on the following comparative formulas presented in the comments section of that blog:

$$\vartheta(\Omega.\alpha + \beta) = \theta(\alpha,\beta)$$

$$\vartheta(1 + \Omega.\alpha + \beta) = \psi(\Omega.\alpha(1 + \beta))$$

Comparison Table

Ordinal $$\theta()$$ $$\vartheta()$$ $$\psi()$$
$$\epsilon_0$$ $$\theta(0,1)$$ $$\vartheta(1)$$ $$\psi(0)$$
$$\zeta_0$$ $$\theta(1,0)$$ $$\vartheta(\Omega)$$ $$\psi(\Omega)$$
$$\Gamma_0$$ $$\theta(\Omega,0)$$ $$\vartheta(\Omega^2)$$ $$\psi(\Omega^{\Omega})$$
Ackermann Ordinal $$\theta(\Omega^2,0)$$ $$\vartheta(\Omega^3)$$ $$\psi(\Omega^{\Omega^2})$$
Small Veblen Ordinal (svo) $$\theta(\Omega^{\omega},0)$$ $$\vartheta(\Omega^{\omega})$$ $$\psi(\Omega^{\Omega^{\omega}})$$
(Least) Ordinal Growth Rate of TREE(n) function $$\theta(\Omega^{\omega}.\omega,0)$$ $$\vartheta(\Omega^{\omega}.\omega)$$ $$\psi(\Omega^{\Omega^{\omega}.\omega})$$
Large Veblen Ordinal (LVO) $$\theta(\Omega^{\Omega},0)$$ $$\vartheta(\Omega^{\Omega})$$ $$\psi(\Omega^{\Omega^{\Omega}})$$
Bachmann-Howard ordinal $$\theta(\Omega\uparrow\uparrow\omega,0)$$ $$\vartheta(\Omega\uparrow\uparrow\omega)$$ $$\psi(\Omega\uparrow\uparrow\omega)$$

WORK IN PROGRESS

Appreciate any comments, additions or corrections to this table.

B1mb0w.