User blog comment:PsiCubed2/Question: General breakdown of levels up to Second Order Arithmetic/@comment-1605058-20180203222427/@comment-1605058-20180204092231

1. Let me first answer a question with a question: large cardinals (inaccessibles, Mahlos etc.) are not countable, let alone recursive. How do they connect to all this? The answer is given by the ordinal collapsing functions, and that generalizes to large countable ordinals just as well - in fact, the whole theory of OCFs can be framed without any mention of uncountable ordinals. Every mention of an inaccessible, Mahlo, indescribable cardinal can be replaced by a recursively inaccessible, recursively Mahlo or reflecting ordinal and literally nothing changes apart from how we name things. For things beyond indescribability/reflection, large countable ordinals have turned out to be a more convenient language to speak in, stable ordinals being the most prominent example.

2. The answer is a great variety of various kinds of stable ordinals, which in a way are generalization of reflecting ordinals (for example, the (+)-stable is \(\Pi^1_1\)-reflecting) and go far beyond them. Again, all of those are then subjected to suitable collapsing functions.