From hierarchies to well-foundedness

We highlight that the connection of well-foundedness and recursive definitions is more than just convenience. While the consequences of making well-foundedness a sufficient condition for the existence of hierarchies (of various complexity) have been extensively studied, we point out that (if parameters are allowed) well-foundedness is a necessary condition for the existence of hierarchies e.g. that even in an intuitionistic setting \({(\Pi_1^0-\mathsf{CA}_0)_\alpha \vdash \mathsf{wf}(\alpha)\, {\rm where}\, (\Pi_1^0-\mathsf{CA}_0)_\alpha}\) stands for the iteration of \({\Pi^0_1}\) comprehension (with parameters) along some ordinal \({\alpha}\) and \({\mathsf{wf}(\alpha)}\) stands for the well-foundedness of \({\alpha}\) .