A recursive notation system of a strong segment of ordinals was developped by Jäger [3]. An unessential modified versionT(J) of this notation system was described in [4]. In the following, the well-ordering ofT(J) is proved in a formal system of second order arithmetic with the axiom schema ofΠ 2 1 -comprehension. It follows, that the proof theoretical ordinal ofΠ 2 1 -analysis is greater than the order type ofT(J).

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′).