User blog comment:P進大好きbot/ZFC-variant of Rayo's number/@comment-39605890-20191204031223

Is there any particular reason to believe that this function is fast growing? $$\alpha_0$$ might believe that f satisfies a small n, right?

There are probably on a countable number of ordinals that can be an $$\alpha_0$$, and their universes may not have really fast growing functions.