User blog comment:Edwin Shade/An Injection From The Countable Ordinals To The Reals/@comment-1605058-20171219180546

Note that your definition requires you to fix a choice of fundamental sequences for all ordinal numbers - without it, your function is not well-defined.

Furthermore, it's not immediately clear to me that this function is an injection - basically, \(Ordname(\alpha)\) encodes some variant of the slow-growing hierarchy and it's not obvious that for different ordinals the two functions must be distinct. Indeed, for (in)appropriately fundamental sequences, the slow/fast growing hierarchy might have equal functions (see here).