Rekursion über Dilatoren und die Bachmann-Hierarchie. (Recursion over dilators and the Bachmann hierarchy)
type: Item
instance of: scholarly article
title: Rekursion über Dilatoren und die Bachmann-Hierarchie. (Recursion over dilators and the Bachmann hierarchy)
zbMATH Open document ID: 0675.03033
DOI: 10.1007/BF01624083
published in: Archive for Mathematical Logic
publication date: 1989-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 03F15
Mathematics Subject Classification ID: 03D55
zbMATH DE Number: 4104946
zbMATH Keywords: dilators
zbMATH Keywords: primitive recursion
zbMATH Keywords: Bachmann hierarchy
zbMATH Keywords: Bachmann-Howard ordinal
author: Peter Paeppinghaus
reviewed by: Helmut Pfeiffer
MaRDI profile type: Publication
cites work: Some uses of dilators in combinatorial problems. III: Independence results by means of decreasing F-sequences (F weakly finite dilator)
cites work: Q3772214
cites work: Describing ordinals using functionals of transfinite type
cites work: Q5800868
cites work: A simplification of the Bachmann method for generating large countable ordinals
cites work: An extension of Schütte's Klammersymbole
cites work: Π12-logic, Part 1: Dilators
cites work: Set recursion and \(\Pi ^ 1_ 2\)-logic
cites work: Functors and ordinal notations. II: A functorial construction of the Bachmann hierarchy
cites work: Les premiers recursivement inaccessible et Mahlo et la theorie des dilatateurs
cites work: Recursion on Homogeneous Trees
cites work: Q3695279
cites work: Ptykes in Gödels T und Definierbarkeit von Ordinalzahlen. (Ptykes in Gödel's T and definability of ordinal numbers)
cites work: Q3684119
cites work: Functors and ordinal notations. IV: The Howard ordinal and the functor ∧.
description: scientific article; zbMATH DE number 4104946