Ein Wohlordnungsbeweis für das Ordinalzahlensystem T(J). (A proof of the wellordering of the ordinal number system T(J))
type: Item
instance of: scholarly article
title: Ein Wohlordnungsbeweis für das Ordinalzahlensystem T(J). (A proof of the wellordering of the ordinal number system T(J))
zbMATH Open document ID: 0644.03029
DOI: 10.1007/BF01625829
published in: Archive for Mathematical Logic
publication date: 1988-01-01T00:00:00Z
review text: In an earlier paper ``Majorisierungsrelationen und Fundamentalfolgen eines Ordinalzahlensystems von G. Jäger'' [ibid. 26, 29-55 (1986/7; Zbl 0617.03038)], the author presented a recursive system T(J) of ordinal notations using the ordering functions \(I_{\alpha}\) of the strongly inaccessible cardinals and so-called collapsing functions \(\psi_{\alpha}\), previously introduced by Buchholz. This system is as strong as the system of \textit{G. Jäger} given in ``\(\rho\)-inaccessible ordinals, collapsing functions and a recursive notation system'' [ibid. 24, 49-62 (1984; Zbl 0545.03031)]. The ordinal J is the supremum of the sequence \(J_ n\), defined recursively by \(J_ 0:=0\), \(J_{n+1}:=I_{Jn}0\). The system T(J) is developed set-theoretically assuming the existence of the inaccessible cardinals which the functions \(I_{\alpha}\) enumerate. In the paper under review the author proves that the system T(J) is well- ordered. The proof is carried through in a system of second order arithmetic with \(\Pi^ 1_ 2\)-comprehension axiom schema by means of so-called distinguished sets. These sets are defined by extending the same concept \textit{W. Buchholz} had introduced in ``Normalfunktionen und konstruktive Systeme'' [Lect. Notes Math. 500, 4-25 (1975; Zbl 0342.02021)] to show the well-ordering of a weaker system. In consequence of this proof it is not necessary to presuppose the existence of strongly inaccessible cardinals when the system T(J) is defined. Moreover the order type of T(J) is smaller than the proof- theoretical ordinal of second order arithmetic with \(\Pi^ 1_ 2\)- comprehension.
Mathematics Subject Classification ID: 03F15
zbMATH DE Number: 4049636
zbMATH Keywords: ordinal notations
zbMATH Keywords: inaccessible cardinals
zbMATH Keywords: second order arithmetic
zbMATH Keywords: comprehension
zbMATH Keywords: distinguished sets
zbMATH Keywords: well-ordering
zbMATH Keywords: order type
zbMATH Keywords: proof- theoretical ordinal
author: Kurt Schütte
reviewed by: Helmut Pfeiffer
MaRDI profile type: Publication
cites work: ϱ-inaccessible ordinals, collapsing functions and a recursive notation system
cites work: Majorisier Ungsrelationen und Fundamentalfolgen eines Ordinalzahlensystems von G. Jäger
full work available at URL: https://doi.org/10.1007/bf01625829
OpenAlex ID: W2318189663
description: scientific article; zbMATH DE number 4049636