1. T. Arai, A system (O(πn),<) of ordinal diagrams, handwritten note, Apr. 1994.
2. Ordinal diagrams for recursively Mahlo universes;Arai;Arch. Math. Logic,2000
3. Ordinal diagrams for Π3-reflection;Arai;J. Symbolic Logic,2000
4. T. Arai, Ordinal diagrams for ΠN-reflection, draft.
5. Proof theory for theories of ordinals I: recursively Mahlo ordinals;Arai;Ann. Pure Appl. Logic,2003