User blog comment:DontDrinkH20/Some Explanations of Certain Large Cardinals/@comment-11227630-20181017041904/@comment-25601061-20181019212530

> On the other hand, if such countable ordinals are just defined by using a property mentioning the countable ordinals corresponding to large cardinals through the ordinal notation system associated to \(\psi\), then you need to use \(\psi\) and collapse the large cardinals again in order to define the resulting OCF.

As nonintuitive as it might seem, there's actually a way to define an OCF that collapses countable ordinals by defining the countable ordinals using the OCF itself. Here's an example of such an OCF:

\(C_{0}(\alpha,\beta) = \beta\cup\{0,A\}\)

\(C_{n+1}(\alpha,\beta) = C_{n}(\alpha,\beta)\cup\{\gamma+\delta,\omega^{\gamma},\psi(\varepsilon)|\gamma,\delta,\varepsilon\in C_{n}(\alpha,\beta)\wedge\varepsilon < \alpha\}\)

\(C(\alpha,\beta) = \bigcup_{n<\omega}C_{n}(\alpha,\beta)\)

\(\psi(\alpha) = \min\{\pi|\sup(C(\alpha,\pi)\cap A) = \pi\}\)

\(A = \min\{\pi|\forall{\kappa<\varepsilon_{\pi+1}}(\pi > \psi(\kappa))\}\)

This gives us A = the BHO. I don't have time to explain why right now, but I'll edit this message when I can. (Hint: what's the difference in the evaluation of \(\psi(\kappa)\) when \(\kappa < \pi\) and \(\kappa \geq \pi\)?)