User blog comment:Rgetar/Definition of standard form using iterated fundamental sequence/@comment-35470197-20190107080709

> We can define function ots ("ots" means "ordinal to string"), calculating standard form of an ordinal in this notation:

No. You need to assume the surjectivity of sto below a countable fixed ordinal, and the injectivity of the composite of sto and sf. Otherwuse, ots is ill-defined.

> From my experience it turned out, that calculating of element of fundamental sequence (nfs) is easier than comparison of ordinals, corresponding to strings and converting strings into standard forms (sf); and comparison of ordinals, corresponding to strings, is a bit easier than converting strings into standard forms (sf).

If you do not have recursive comparison, there is no computable way to check whether the fundamental sequence actually works as desired, because we usually need the comparison in order to define sto.