User blog comment:Deedlit11/The slow-growing hierarchy and other hierarchies/@comment-78.22.170.27-20130619191359/@comment-5529393-20130619225101

Wow, I'm amazed that the slow-growing hierarchy can catch with the fast at \(\omega^2\) using nonpathological sequences! Is the growth rate the same as that of the fast-growing hierarchy at \(\omega^2\) using standard sequences?

A question I think we are all interested in is how the Hardy and fast-growing hierarchies can vary as the fundamental sequecnes change. Are they as fragile as the slow-growing hierarchy seems to be? Or do you always get, for example, that the PA-recursive functions are those less than \(\varepsilon_0\), the Pi-1-1-CA_0 recursive functions are those less than \(\psi(\Omega_\omega)\), and so on? I'm assuming "natural" fundamental sequences, of course.

Anyway, it's great to hear from an expert at our humble wiki!