Investigations of subsystems of second order arithmetic and set theory in strength between Π 11 - CA and ∆ 1 2 - CA + BI : Part I

This paper is the first of a series of two. It contains proof–theoretic investigations on subtheories of second order arithmetic and set theory. Among the principles on which these theories are based one finds autonomously iterated positive and monotone inductive definitions, Π1 transfinite recursion, ∆ 1 2 transfinite recursion, transfinitetely iterated Π 1 1 dependent choices, extended Bar rules for provably definable well-orderings as well as their set-theoretic counterparts which are based on extensions of Kripke-Platek set theory. This first part introduces all the principles and theories. It provides lower bounds for their strength measured by the amount of provable transfinite induction. In other words, it determines lower bounds for their proof-theoretic ordinals which are expressed by means of ordinal representation systems. The second part of the paper will be concerned with ordinal analysis. It will show that the lower bounds established in the present paper are indeed sharp, thereby providing the proof-theoretic ordinals. All the results were obtained more then 20 years ago (in German) in the author’s PhD thesis [42] but have never been published before, though the thesis received a review (MR 91m#03062). I think it is high time it got published.