User blog comment:Alemagno12/Huge Ordinal Analysis/@comment-11227630-20180203000827/@comment-25601061-20180203003854

I was actually thinking of doing some analysis to find that catching point once I analyze ZFC + there exists a rank into rank ordinal. At that point, I will have most likely found a pattern that helps me find the pattern of ZFC + there exists a for any cardinal a, so I would just need to come up with larger and larger cardinals until I find the catching point.

(btw, you should consider redefining catching function with nested FGH: if you manage to define it properly for any cardinal, or even up to Mahlos and weakly compacts, it will be insanely strong)