A simpli version of local predicativity

The method of local predicativity as developed by Pohlers in [10],[11],[12] and extended to subsystems of set theory by Jager in [4],[5],[6] is a very powerful tool for the ordinal analysis of strong impredicative theories. But up to now it suers considerably from the fact that it is based on a large amount of very special ordinal theoretic prerequisites. This is true even for the most recent (very polished) presentation of local predicativity in (Pohlers [15]). The purpose of the present paper is to expose a simplied and conceptually improved version of local predicativity which | besides some very elemen-tary facts on ordinal addition, multiplication, and exponentiation | requires only amazingly little ordinal theory. (All necessary nonelementary ordinal theoretic prerequisites can be developed from scratch on just two pages, as we will show in section 4.) The most important feature of our new approach however seems to be its conceptual clarity and exibility, and in particular the fact that its basic concepts (i.e. the innitary system RS 1 and the notion of an H-controlled RS 1-derivation) are in no way related to any system of ordinal notations or collapsing functions. Our intention with this paper is to make the fascinating eld of `admissible proof theory ' (created by Jager and Pohlers) more easily accessible for non-prooftheorists, and to provide a technically and conceptually well developed basis for further research in this area. We think a good way to accomplish this goal is to apply our method to one particularly interesting (and strong) theory, namely the system KPi rst analyzed by Jager and Pohlers in [9], and to carry out the ordinal analysis for this theory in full detail. Accordingly the whole paper is devoted to the proof of the following 1 The nal version of this paper was written while the author was visiting Carnegie Mellon University during the academic year 1990/91. I would like to express my sincere thanks to Wilfried Sieg (who invited me) and all members of the Philosophy Department of CMU for their generous hospitality