Eine Erweiterung T(V') des Ordinalzahlensystems \(C_{\Omega}(\Lambda _ 0)\) von G. Jäger. (An extension T(V') of the ordinal system \(C_{\Omega}(\Lambda _ 0)\) by G. Jäger)
In this paper the author develops a system of ordinal notations by extending the system \textit{G. Jäger} introduced in ``\(\rho\)-inaccessible ordinals, collapsing functions and a recursive notation system'' [ibid. 24, 49-62 (1984; Zbl 0545.03031)]. Jäger uses for his system (collapsing functions \(\Psi\alpha\) and) functions \(I\beta\) which map the least weak Mahlo cardinal \(\kappa\) into itself. Instead of these \(I\beta\), the author considers functions IA, where A is a matrix \(\left( \begin{matrix} \alpha_ 0...\alpha_ n\\ a_ 0...a_ 0\end{matrix} \right)\) of ordinals which corresponds to the ordinal \(\kappa^{a_ 0}(1+\alpha_ 0)+...+\kappa^{a_ n}(1+\alpha_ n).\) Thus he returns to a construction which he had employed in his first paper on ordinal notations. The ordinal V' chosen to denote the system is defined to be the least ordinal \(\eta\) for which \(I\left(\begin{matrix} 0\\ \eta \end{matrix} \right)=\eta\) holds. It is the least regular ordinal which is not accessible in the system. It is different from the least ordinal V which cannot be desribed in T(V') and which is considerably smaller than V'. V is the supremum of the sequence \((V_ n)\) defined by \(V_ 0:=0\), \(V_{n+1}:=I(V\) \(0_ n)0\). V is not regular. In a postscript the author communicates that the well-ordering of the system T(V') can be proven in a system of second order arithmetic with the axiom schema of \(\Pi\) \(1_ 2\)-comprehension. Thus the order type of T(V') is shown to be smaller than the proof-theoretic ordinal of \(\Pi\) \(1_ 2-CA\).
hierarchy of Veblen-functions for inaccessible ordinals
scientific article; zbMATH DE number 4051609