User blog comment:Ecl1psed276/BM2 Analysis - A Summary/@comment-30754445-20180709051740/@comment-35470197-20180712152338

@Rpakr

Since the term "OCF" is a little ambiguous, I use the terms "ordinal function" and "ordinal notation" in this reply. First of all, UNOCF is an ordinal notation, but not an ordinal function.

Since the construction from ordinal functions to ordinal notations is not one-to-one, it is obvious to some extent that symbols appearing UNOCF do not have to correspond to inaccessible ones by an ordinal function which yields UNOCF (if exists).

However, given an ordinal notation \(OT\) with such symbol \(I\)'s, the existence of an ordinal function which sends them to certain explicit inaccessible cardinals and which yields the ordinal notation is often necessary to ensure that \(OT\) is (or will be) actually well-defined or well-founded.

Therefore the refutation that the symbols in UNOCF do not have to correspond to inaccessible ones is non-sense, unless you state that UNOCF can be (in the future) derived from an actual ordinal function defined without using inaccessible ones.

Remember that known (mathematically correct) ordinal functions defined with using inaccessible ones are well-defined by the transfinite induction, which strictly using them. Therefore constructing an ordinal function without inaccessible which has the growth rate of inaccessible level.