User blog comment:DontDrinkH20/Some Explanations of Certain Large Cardinals/@comment-35870936-20181017021650

I was wondering: Would it be possible to define OCFs without having a definition for the cardinal itself? For example, you don't have to use W="the first uncountable cardinal" in Buccholz's OCF, because you could also use w_1^CK, or W_2, or M, or the first huge cardinal, or whatever. You would just have to tweak the definition a bit, but the values of the countable ordinals returned by the psi function would be the same. My intuition says that this should be possible to do, but I'm not entirely sure how.