Ein Wohlordnungsbeweis für das Ordinalzahlensystem T(J). (A proof of the wellordering of the ordinal number system T(J))
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.
scientific article; zbMATH DE number 4049636