Rekursion über Dilatoren und die Bachmann-Hierarchie. (Recursion over dilators and the Bachmann hierarchy)
Let ON be the category consisting of the ordinals as objects and strictly monotone functions as morphisms. J.-Y. Girard introduced dilators to be functors from ON to ON, which commute with direct limits and with pull- backs. Let g be a function from On to On. In the paper under review a hierarchy \({\mathbb{J}}^ g_ D\) of functions is defined by primitive recursion on the collection of dilators such that to each dilator D a function \({\mathbb{J}}^ g_ D: On\to On\) is associated. This hierarchy is related to the so- called Bachmann hierarchy based on g, \((\Phi^ g_{\alpha})_{\alpha <\epsilon_{\Omega +1}}\), \(\epsilon_{\Omega +1}\) being the least fixed point of the function \(\lambda \xi \omega^{\xi}\) which is greater than \(\Omega\). The hierarchies cannot be compared immediately because the concepts used to construct them do not correspond directly. Therefore another hierarchy \({\mathbb{J}}^*\) has to be put in between. If g satisfies certain conditions, the functions of the hierarchy \({\mathbb{J}}^*\) can be compared with those of the Bachmann hierarchy and the functions of \({\mathbb{J}}\) can be bounded by those of \({\mathbb{J}}^*\). Thus the Bachmann- Howard ordinal can be characterized in the theory of dilators.
scientific article; zbMATH DE number 4104946
scientific article