This paper gives a recursive generalization of a strong notation system of ordinals, which was devellopped by Jäger [3]. The generalized systemT(V′) is based on a hierarchy of Veblen-functions for inaccessible ordinals. The definition ofT(V′) assumes the existence of a weak Mahlo-ordinal. The wellordering ofT(V′) is provable in a formal system of second order arithmetic with the axiom schema ofΠ 2 1 -comprehension in a similar way, as it is proved in [6] for the weaker notation systemT(V′).

In this paper we introduce a recursive notation system O(Π 3 ) of ordinals. An element of the notation system is called an ordinal diagram. The system is designed for proof theoretic study of theories of Π 3 -reflection. We show that for each $\alpha in O(Π 3 ) a set theory KP Π 3 for Π 3 -reflection proves that the initial segment of O(Π 3 ) determined by α is a well ordering. Proof theoretic study for such theories (...) will be reported in [4]. (shrink)