Proof Theory and Intuitionistic Systems and preliminaries A review of Gentzens second consistency proof The intuitionistic system of number theory A formally intuitionistic system as strong as classical analysis Transfinite induction with respect to recursive wellorderings without function parameters A formally intuitonistic theory equivalent to classical transfinite induction with respect to recursive wellfounded trees with function parameters A system containing barinduction with respect to decidable predicates Harrop formulas The Markov principle Relative consistency proof of ZTN with respect to ZTi IN*.