 The pattern of definition exhibited in  Definition 5.11  continues for stronger systems, albeit only as a basic template since for theories beyond the level of \((\Delta^1_2{\Hy}\bCA)+{{\BI}}\) substantially new ideas are required. Analogies between large set-theoretic ordinals (cardinals) and recursively large ordinals on the one hand and ordinal representation systems on the other hand can be a fruitful source of inspiration for devising new representation systems. More often than not, hierarchies and structural properties that have been investigated in set theory and recursion theory on ordinals turn out to have proof-theoretic counterparts.