<p>An <em>ordinal hierarchy</em> 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.</p>

<p>Two well-known definitions in the field are the <em>slow-growing hierarchy</em> $(g_{\alpha})$ and the <em>fast-growing hierarchy</em> $(h_{\alpha})$. 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 $\epsilon_0 &lt; FS_0 &lt; BH_0 &lt; LN_0$ where $FS_0$ and $BH_0$ respectively denote the Fefermann-Schütte and the Bachmann-Howard ordinals.</p>

<p>Specifically, it seems that S.S. Wainer first claimed that $LN_0 = FS_0$, then <a href="http://journals.cambridge.org/action/displayAbstract?fromPage=online&amp;aid=9079915">retracted</a> its claim after J.Y. Girard <a href="http://www.sciencedirect.com/science/article/pii/0003484381900164">proved</a> that $LN_0$ corresponded to the ordinal of the theory $ID_{&lt;\omega}$. I have two questions in this respect:</p>

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

<p>(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 $LN_0$...</p>

CC BY-SA 3.0
2014-04-12T23:26:21.597Z
Shouldn't this be on [Math Overflow](http://www.mathoverflow.net)?
David Richerby
CC BY-SA 3.0
2014-04-13T19:14:12.117Z
@DavidRicherby, I think this one is fine here.
Kaveh
CC BY-SA 3.0
2014-04-25T09:47:01.250Z
For question (2) look at http://en.wikipedia.org/wiki/Ordinal_analysis; several notation systems are indicated for $\mathit{ID}_{<\omega}$.  I would add another one by Stan Wainer in BSL: http://www.math.ucla.edu/~asl/bsl/0503/0503-004.ps
Sylvain
<computability><order-theory>
Exact catchup point between SGH and FGH of ordinals?
NisaiVloot
# Exact catchup point between SGH and FGH of ordinals?

(Asked by: username_0 on 2014-04-11)

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](http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=9079915) its claim after J.Y. Girard [proved](http://www.sciencedirect.com/science/article/pii/0003484381900164) 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 $LN\_0$...

**Tags:** computability, order-theory
> **Comments:**
> **username_1**: Shouldn't this be on [Math Overflow](http://www.mathoverflow.net)?  
> **username_2**: @DavidRicherby, I think this one is fine here.  
> **username_3**: For question (2) look at http://en.wikipedia.org/wiki/Ordinal_analysis; several notation systems are indicated for $\mathit{ID}_{<\omega}$.  I would add another one by Stan Wainer in BSL: http://www.math.ucla.edu/~asl/bsl/0503/0503-004.ps  

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## 0 answers

