Does Koepke's notion of ordinal computability admit an analogue of the Kleene $T$-predicate? If so, is the existence of such a $T$-predicate independent of $ZFC$? Also, if one assumes the existence of such a $T$-predicate, can an analogue of the Kleene hierarchy be defined for Koepke's ordinal computability, and can one define a set theory based on that analogue of the Kleene hierarchy rather than, say, the cumulative hierarchy?