Complexities of Proof-Theoretical Reductions

The present thesis is a contribution to a project that is carried out by Michael Rathjen and Andreas Weiermann to give a general method to study the proof-complexity of Pi_2 sentences. This general method uses the generalised ordinal-analysis that was given by Buchholz, Rueede and Strahm as well as the generalised characterisation of provable-recursive functions of PA with axioms for transfinite induction that was given by Weiermann. The present thesis links these two methods by giving an explicit elementary bound, for the proof-complexity increment that occurs after the transition from the theory that was used by Rueede and Strahm, to PA with axioms for transfinite induction, which was analysed by Weiermann.