Ordinal Systems, Part 2: One Inaccessible

We develop an alternative approach to well-ordering proofs beyond the Bachmann-Howard ordinal using transsnite sequences of ordinal notations and use it in order to carry out well-ordering proofs for-ordinal systems. We extend the approach of ordinal systems as an alternative way of presenting ordinal notation systems started in Set98b] and develop ordinal systems, which have in the limit exactly the strength of Kripke-Platek set theory with one recursivly inaccessible. The upper bound is determined by giving well-ordering proofs, which use the technique of transsnite sequences. We derive from the new approach the traditional approach to well-ordering proofs using distinguished sets. The lower bound is determined by extending the concept of ordinal function generators in Set98b] to inaccessibles.