User blog comment:P進大好きbot/What is the greatest ordinal notation now?/@comment-31966679-20180623154722/@comment-35470197-20180623234747

As LittlePeng9 said, an ordinal notation is not a function hierarchy.

An ordinal notation is a (mainly primitive recursive) map from a countable set to the class of ordinal numbers, while a function hierarchy is kind of a map from a class of ordinal numbers (with fundamental sequences is a wide sense) to the set of strictly increasing endomorphisms of \(\omega\) such that a larger ordinal number corresponds to a function with greater growth.

By the way, I saw your hierarchy now. You forgot to define \(a_0(n)\). It might be \(a(n)\). If so, for short, your definition is given in the following way: \begin{eqnarray*} a_0(n) & = & a(n) \\ a_{b+1}(n) & = & a_b^{n!}(n) \\ a_b(n) & = & a_{b[n]}(n), \end{eqnarray*} where \(b\) in the last equality is a limit ordinal with a fixed fundamental sequence \((b[n])_{n \in \omega}\) consisting of ordinal numbers for which the hierarchy is already defined.

Therefore it has strictly greater growth than FGH. (But I note that FGH is very useful for analysis, because many large functions are defined using a similar diagonalisation strategy. If you do not use it for analysis, it might be better to define \(a_{b+1}(n) = a_b^{n^n}(n)\), \(a_{b+1}(n) = a_b^{n \uparrow \uparrow \uparrow n}(n)\), \(a_{b+1}(n) = a_b^{a_b(n)}(n)\), or something like that because they are greater than \(n!\).)

Thank you. I like the hierarchy!