 That is what Turing did in his Princeton dissertation (1939) concerning, what he called, ordinal logics. There he considers two ways of achieving the effective representation of ordinals. The first way is via the set \(\rW\) of numbers e for recursive well-orderings \(\leq_e\), the second is provided by the class of Church-Kleene notations for ordinals (Church and Kleene 1936) that used expressions in the λ-calculus to describe ordinals. The latter approach was then modified in Kleene 1938 to an equivalent recursion-theoretic definition that uses numerical codes to denote countable ordinals and is known as Kleene’s \({\cO}\).