Ordinal notations based on a hierarchy of inaccessible cardinals

Abstract

We became interested in far reaching ordinal notation systems when we noticed that the notation systems that existed in 1979 were insufficient for the proof theoretic treatment of (A&CA) + BI and related systems. We wanted to apply the method of local predicativity to those systems. Since this method is based on the fact that (A&CA) + BI has a model at the first recursively inaccessible ordinal in the constructible hierarchy, we needed a notation system which somehow internalizes this ordinal. Our former experience with ordinal notation systems, however, taught us that it usually is essentially easier to develop a notation system on the basis of cardinals instead of their recursive counterparts. So we have been looking for a notation system which internalizes the first weakly inaccessible cardinal. The obvious idea was to enlarge Buchholz’ notation systems which he developed on the basis of the Feferman-Aczel functions 6$. The theory of those functions has already been studied by Bridge [2]. The functions @E are fairly easy to describe. One defines by induction on a the Skolem hull Cl(&, /3) of the ordinal /3 as the closure of /3 under the function + and @g for f < QI and S, as the function which enumerates the set Cr( a) = {E