Abstract

In this paper we introduce a recursive notation system O of ordinals. An element of the notation system is called an ordinal diagram. The system is designed for proof theoretic study of theories of $\Pi_3$-reflection. We show that for each $\alpha < \Omega$ in O a set theory KP $\Pi_3$ for $\Pi_3$-reflection proves that the initial segment of O determined by $\alpha$ is a well ordering. Proof theoretic study for such theories will be reported in [4].