 Bachmann’s bold move of using large ordinals to generate names for small ordinals was a very important idea. To obtain ordinal analyses of ever stronger theories one has to find new ways of defining ordinal representation systems that can encapsulate their strength. The latter goes hand in hand with the development of new cut elimination techniques that are capable of removing cuts in (infinitary) proof systems with strong reflection rules. Ordinal representations, however, appear to pose a considerable barrier to understanding books and articles in this research area. Nonetheless we think that they are the best way to express the proof-theoretic strength of a theory as they provide a scale by means of which one can get a grasp of how much stronger a theory \(S_1\) is than another theory \(S_2\) (rather than the bland statement that \(S_1\) is stronger than \(S_2\)).