An ordinal hierarchy is a way to assign a function $f_{\alpha} : \mathbb{N} \rightarrow \mathbb{N}$ to each (recursive) ordinal $\alpha$. The corresponding functions are expected to be monotone and correspond to even faster growth rates as the ordinal increases.

Two well-known definitions in the field are the slow-growing hierarchy $(g_{\alpha})$ and the fast-growing hierarchy $(h_{\alpha})$. Although the latter grows much faster than the former, some authors have claimed the existence of a 'catch-up' point, i.e. that both hierarchies attain the same growth rate at some large ordinal $LN_0$ - the notation is mine, and we have the expected relations $\epsilon_0 < FS_0 < BH_0 < LN_0$ where $FS_0$ and $BH_0$ respectively denote the Fefermann-Schütte and the Bachmann-Howard ordinals.

Specifically, it seems that S.S. Wainer first claimed that $LN_0 = FS_0$, then retracted its claim after J.Y. Girard proved that $LN_0$ corresponded to the ordinal of the theory $ID_{<\omega}$. I have two questions in this respect:

(1) is this ordinal $LN_0$ canonical, or does it depend on the choice of a fundamental sequences for the hierarchies?

(2) does there exist an ordinal notation system for the ordinals up to $LN_0$? It seems that the most generic ONS known is tree-based and is due to Takeuti (?) but it's unclear to me whether it reaches $LN_0$...


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