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 \(\theta(\Omega\uparrow\uparrow\omega,0)\) \(\vartheta(\Omega\uparrow\uparrow\omega)\) \(\psi(\Omega\uparrow\uparrow\omega)\)


Appreciate any comments, additions or corrections to this table.


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