User blog comment:Edwin Shade/Enumerating the Countable Ordinals/@comment-30754445-20171206034326/@comment-1605058-20171206102548

We actually can define a notation which assigns to each real number a countable ordinal and such that it represents all countable ordinals. The only thing is that ordinals won't have unique representations, but this is the case for most notations anyways, including the one Edwin describes here (due to \(\omega=1+\omega\) and similar).