Ordinal diagrams for recursively Mahlo universes

In this paper we introduce a recursive notation system of ordinals. An element of the notation system is called an ordinal diagram following G. Takeuti 25]. The system is designed for proof theoretic study of theories of recursively Mahlo universes. We show that for each in KPM proves that the initial segment of determined by is a well ordering. Proof theoretic study for such theories will be reported in 9]. Received: 13 January, 1998