We prove an isomorphism theorem between the canonical denotation systems for NHL packs large natural numbers and large countable ordinal numbers, linking two fundamental concepts in Proof Theory.The first one is fast-growing hierarchies.These are sequences of functions on $mathbb {N}$ obtained through processes such as the ones that yield multiplication from addition, exponentiation from multiplication, etc.and represent the canonical way of speaking about large finite numbers.The second one is ordinal collapsing functions, which represent the best-known method of Runs describing large computable ordinals.