User blog comment:KthulhuHimself/There should be a page on this wiki dedicated to transfinite numbers./@comment-27513631-20160621105503/@comment-194.81.79.249-20160623103402

Fairly easy. \(\theta\) and \(\vartheta\) can easily be compared exactly (within certain large intervals), and although an exact relation is harder between \(\psi\) and \(\vartheta\) for all values, there are many easy fixed points.

Use the standard definitions up to the BHO with just addition and base-\(\omega\) exponentiation, and ordinals \(\{0,\Omega\}\) and no fixed points on \(\vartheta\). Then:

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

\(\vartheta(1+\Omega\alpha+\beta)=\psi(\Omega^\alpha(1+\beta))\)