A NOTATION SYSTEM FOR ORDINAL USING Ψ‐FUNCTIONS ON INACCESSIBLE MAHLO NUMBERS
Helmut Pfeiffer, H. Pfeiffer · Mathematical logic quarterly · 1992
Abstract G. Jäger gave in Arch. Math. Logik Grundlagenforsch. 24 (1984), 49‐62, a recursive notation system on a basis of a hierarchy Iαß of α‐inaccessible regular ordinals using collapsing functions following W. Buchholz in Ann. Pure Appl. Logic 32 (1986), 195‐207. Jäger's system stops, when ordinals α with Iα0 = α enter. This border is now overcome by introducing additional a hierarchy Jαß of weakly inaccessible Mahlo numbers, which is defined similarly to the Jäger hierarchy. An ordinal μ is called Mahlo, if every normal‐function f : μ → μ has regular fixpoints. Collapsing is defined for both Mahlo and simply regular ordinals such that for every Mahlo ordinal μ out of the J‐hierarchy Ψμα is a regular σ such that Iσ0 = σ. For these regular σ again collapsing functions Ψσ are defined. To get a proper systematical order into the collapsing procedure, a pair of ordinals is associated to σ and α, and the definition of Ψσα is given by recursion on a suitable well‐ordering of these pairs. Thus a fairly large system of ordinal notations can be established. It seems rather straightforward, how to extend this setting further.