This article reports that some robustness of the notions of predicativity and of
autonomous progression is broken down if as the given infinite total entity we
choose some mathematical entities other than the traditional
ω. Namely, the equivalence between normal transfinite
recursion scheme and new dependent transfinite recursion
scheme, which does hold in the context of subsystems of second order number
theory, does not hold in the context of subsystems of second order set theory
where the universe V of sets is treated as the given totality
(nor in the contexts of those of n+3-th order number
or set theories, where the class of all n+2-th order
objects is treated as the given totality).