This article reports that some robustness of the notions of predicativity and ofautonomous progression is broken down if as the given infinite total entity wechoose some mathematical entities other than the traditionalω. Namely, the equivalence between normal transfiniterecursion scheme and new dependent transfinite recursionscheme, which does hold in the context of subsystems of second order numbertheory, does not hold in the context of subsystems of second order set theorywhere the universe V of sets is treated as the given totality(nor in the contexts of those of n+3-th order numberor set theories, where the class of all n+2-th orderobjects is treated as the given totality).
