User blog:Simply Beautiful Art/Weakness of my previous OCF using weakly compact cardinals

My previous OCF faces a major issue, namely the failure to mix types. For example, it does not produce non-regular ordinals which are closed under regular ordinals, or regular non-Mahlo ordinals which are closed under Mahlo ordinals. However, the OCF which predefines all of these seems to be fine, so it's simply a matter of altering the OCF which outputs stuff like countable ordinals to a more traditional style.

Also as far as I can tell, the previous OCF is as good as a good OCF using a Mahlo ordinal.

\(\displaystyle\mathrm C(\alpha,\kappa)_0=\kappa\cup\{0,K\}\)

\(\displaystyle\mathrm C(\alpha,\kappa)_{n+1}=\{\gamma+\delta~|~\gamma,\delta\in\mathrm C(\alpha,\kappa)_n\}\)

\(\displaystyle\hphantom{\mathrm C(\alpha,\kappa)_{n+1}={}}{}\cup\{\psi_\pi^\beta(\eta)~|~\pi,\beta,\eta\in\mathrm C(\alpha,\kappa)_n\land\eta\in\alpha\}\)

\(\displaystyle\mathrm C(\alpha,\kappa)=\bigcup_{n\in\mathbb N}\mathrm C(\alpha,\kappa)_n\)

\(\displaystyle\psi_\pi^\beta(\alpha)=\min\{\kappa\in\Xi(\pi)~|~\kappa=\Theta_{\pi+1}(\beta)\cap\mathrm C(\alpha,\kappa)\}\)

which works more like Buchholz's OCFs, for example.