 After the work of Bachmann, the story of ordinal representations becomes very complicated. Significant papers (by Isles, Bridge, Pfeiffer, Schütte, Gerber to mention a few) involve quite horrendous computations to keep track of the fundamental sequences. Also Bachmann’s approach was combined with uses of higher type functionals by Aczel and Weyhrauch. Feferman proposed an entirely different method for generating a Bachmann-type hierarchy of normal functions which does not involve fundamental sequences. Buchholz further simplified the systems and proved their computability. For details we recommend the preface to Buchholz et al. 1981.