Talk:Slow-growing hierarchy

I've tried to create comparisons with BEAF and SGH, assuming that \(g_{\varepsilon_0}(n) = n \uparrow\uparrow n\), \(g_{\zeta_0}(n) \approx n \uparrow\uparrow\uparrow n\), \(g_{\eta_0}(n) \approx n \uparrow\uparrow\uparrow\uparrow n\), etc. First two comparisons are correct, but I have doubts about \(g_{\eta_0}(n) \approx n \uparrow\uparrow\uparrow\uparrow n\), although it seems reasonable. Just look at these comparisons past \(\zeta_0\):

\(g_{\zeta_0}(n) \approx n \uparrow\uparrow\uparrow n\)

\(g_{\zeta_0^{\zeta_0}}(n) \approx (n \uparrow\uparrow\uparrow n) \uparrow\uparrow 2\)

\(g_{\varepsilon_{\zeta_0+1}}(n) \approx (n \uparrow\uparrow\uparrow n) \uparrow\uparrow n\)

\(g_{\varepsilon_{\zeta_0+2}}(n) \approx ((n \uparrow\uparrow\uparrow n) \uparrow\uparrow n) \uparrow\uparrow n \approx (n \uparrow\uparrow\uparrow n) \uparrow\uparrow (2n)\) (using LAPL).

\(g_{\varepsilon_{\zeta_0+3}}(n) \approx (n \uparrow\uparrow\uparrow n) \uparrow\uparrow (3n)\)

\(g_{\varepsilon_{\zeta_0+\omega}}(n) \approx (n \uparrow\uparrow\uparrow n) \uparrow\uparrow (n^2)\)

\(g_{\varepsilon_{\zeta_0+\omega^\omega}}(n) \approx (n \uparrow\uparrow\uparrow n) \uparrow\uparrow (n^n)\)

\(g_{\varepsilon_{\zeta_0+\varepsilon_0}}(n) \approx (n \uparrow\uparrow\uparrow n) \uparrow\uparrow (n \uparrow\uparrow n)\)

\(g_{\varepsilon_{\zeta_0 2}}(n) \approx (n \uparrow\uparrow\uparrow n) \uparrow\uparrow (n \uparrow\uparrow\uparrow n) \approx n \uparrow\uparrow\uparrow (n+1)\)

\(g_{\varepsilon_{\zeta_0 3}}(n) \approx n \uparrow\uparrow\uparrow (n+2)\)

\(g_{\varepsilon_{\zeta_0 \omega}}(n) \approx n \uparrow\uparrow\uparrow (2n)\)

\(g_{\varepsilon_{\zeta_0 \omega 2}}(n) \approx n \uparrow\uparrow\uparrow (3n)\)

\(g_{\varepsilon_{\zeta_0 \omega^2}}(n) \approx n \uparrow\uparrow\uparrow (n^2)\)

\(g_{\varepsilon_{\zeta_0 \omega^\omega}}(n) \approx n \uparrow\uparrow\uparrow (n^n)\)

\(g_{\varepsilon_{\zeta_0 \varepsilon_0}}(n) \approx n \uparrow\uparrow\uparrow (n \uparrow\uparrow n)\)

\(g_{\varepsilon_{\zeta_0^2}}(n) \approx n \uparrow\uparrow\uparrow (n \uparrow\uparrow\uparrow n)\)

I'm not sure, but isn't it that \(g_{\varepsilon_{\zeta_0^\omega}}(n) \approx n \uparrow\uparrow\uparrow\uparrow n\)?

Ikosarakt1 (talk ^ contribs) 15:59, April 4, 2013 (UTC)

Other
Can add more? $Jiawhein$\(a\)\(l\)\(t\) 11:00, April 21, 2013 (UTC)

Fundamental sequences
Can you guys state exactly what fundamental sequences you are using? It's important that we are all on the same page. Deedlit11 (talk) 11:34, April 21, 2013 (UTC)

I used the same fundamental sequences as for fast-growing hierarchy. By the way, I made a mistake in my previous comparisons, now I improved it. Ikosarakt1 (talk ^ contribs) 11:41, April 21, 2013 (UTC)

Okay, but what fundamental sequences are those? It's rather important that we know the details. Deedlit11 (talk) 12:11, April 21, 2013 (UTC)

Here is the list of some of them that I believe to be reasonable:

\(\omega = lim(1,2,3,\cdots)\)

\(\varepsilon_0 = lim(\omega,\omega^\omega,\omega^{\omega^\omega},\cdots)\)

\(\varepsilon_1 = lim(\varepsilon_0,\varepsilon_0^{\varepsilon_0},\varepsilon_0^{\varepsilon_0^{\varepsilon_0}},\cdots\)

\(\zeta_0 = lim(\varepsilon_0,\varepsilon_{\varepsilon_0},\varepsilon_{\varepsilon_{\varepsilon_0}},\cdots)\)

\(\zeta_1 = lim(\varepsilon_{\zeta_0+1},\varepsilon_{\varepsilon_{\zeta_0+1}},\varepsilon_{\varepsilon_{\varepsilon_{\zeta_0+1}}},\cdots)\)

\(\eta_0 = lim(\zeta_0,\zeta_{\zeta_0},\zeta_{\zeta_{\zeta_0}},\cdots)\)

\(\varphi(\omega,0) = lim(\varepsilon_0,\zeta_0,\eta_0,\cdots)\)

\(\varphi(1,0,0) = lim(\varepsilon_0,\varphi(\varepsilon_0,0),\varphi(\varphi(\varepsilon_0,0),0))\)

\(\varphi(1,0,0,0) = lim(\varphi(1,0,0),\varphi(\varphi(1,0,0),0,0),\varphi(\varphi(\varphi(1,0,0),0,0),0,0))\)

\(\vartheta(\Omega^\omega) = lim(\vartheta(\Omega) = \Gamma_0,\vartheta(\Omega^2),\vartheta(\Omega^3),\cdots)\)

I believe that it is easy to fill the intermediate terms, for example \(\zeta_6 = lim(\varepsilon_{\zeta_5},\varepsilon_{\varepsilon_{\zeta_5}},\varepsilon_{\varepsilon_{\varepsilon_{\zeta_5}}},\cdots)\), following this pattern.

Ikosarakt1 (talk ^ contribs) 13:19, April 21, 2013 (UTC)

Catching ordinal
Is the official opinion wrong about catching ordinal with SGH and FGH? It really turns out for me that it is LVO. Ikosarakt1 (talk ^ contribs) 11:39, April 21, 2013 (UTC)

I'm pretty sure they aren't wrong. There are papers on the subject available on JSTOR, if you have access. Some papers are here and here. Deedlit11 (talk) 12:30, April 21, 2013 (UTC)

We can notice the effect from replacing all omega into n's in FGH: it gives very good approximation in googological sense. Chris Bird determined that at LVO ordinal numbers in his array notation catches up the finite numbers: his separator \([1 [1 \neg 1 \neg 2] 2]\) has level LVO, and LVO is also equal to \(\{\omega,\omega [1 [1 \neg 1 \neg 2] 2]\}\), when visualised in array notation. It means that \(g_{\vartheta(\Omega^\Omega)}(n) \approx \{n,n [1 [1 \neg 1 \neg 2] 2] 2\}\), and \(f_{\vartheta(\Omega^\Omega)}(n) \approx \{n,n [1 [1 \neg 1 \neg 2] 2] 2\}\), because Bird's array hierarchy grows roughly as fast as fast-growing one, and at LVO they are catched up. It means that catching ordinal for SGH and FGH is \(\vartheta(\Omega^\Omega)\).

Also I can say that LVO is just first ordinal when SGH catches up FGH. After it, two hierarchies start to be very different again.

Ikosarakt1 (talk ^ contribs) 13:28, April 21, 2013 (UTC)

I think your result may indeed be right. We all know how SGH is sensitive to fundamental sequences. Bird used sequences based on collapsing function and both papers mentioned by Deedlit use sequences based on abstract tree-representations, which, while much stronger, may lead to diametrally different sequences. LittlePeng9 (talk) 14:31, April 21, 2013 (UTC)

I don't know about that. It's possible, but I don't think Ikosarakt's analysis is deep enough to say for sure that it is correct. It depends on a case by case analysis of succeeding ordinals using intuition to guide the way. I think what is needed is a rigorous proof. I understand this is hard to come by, but until we have one it is better to stick by the established results. Note that the Wainer paper lists a bunch of papers by a bunch of different authors that prove that F_{epsilon_0} is at the same level as H_{BHO}, and I don't think they all use tree ordinals - take this paper, for instance. Deedlit11 (talk) 15:02, April 21, 2013 (UTC)

I also personally doubt Ikosarakt's result, I just said it might be true. One big lack in his "proof" is that we don't know if "replace \(\omega\)'s with n's" extends that far into hierarchy. It may break near Feferman-Schutte ordinal, for example. By the way, I think you meant G_{BHO}, as Hardy hierarchy meets Wainer hierarchy at epsilon_0. LittlePeng9 (talk) 15:15, April 21, 2013 (UTC)
 * No matter what, we can always mangle the fundamental sequence so everything turns out okay.
 * I believe that under the standard definition of the Veblen hierarchy, \(\Gamma_0\) is actually the first fixed point of \(\alpha \mapsto \omega\ \{\alpha + 1\}\ \omega\), so it's just a hair away from expansion. Of course we can just change the Veblen function so it really does come out to \(\omega\ \{\{1\}\}\ \omega\). FB100Z &bull; talk &bull; contribs 15:22, April 21, 2013 (UTC)

But why it is not related to Chris Bird's proved fact that \(f_{\omega,\omega [1 [1 \neg 1 \neg 2] 2] 2}(n) = f_{\text{LVO}}(n) \approx \{n,n [1 [1 \neg 1 \neg 2] 2] 2\}\)? Also, why you said that replacing all \(\omega\)'s to n's in SGH is not true after some ordinal? SGH must behave equally with any ordinals, since it is defined equally for all ordinals. I shall, of course, develop comparisons with SGH and Bird's array notation more carefully. Ikosarakt1 (talk ^ contribs) 19:08, April 21, 2013 (UTC)

FB100Z pointed out that \(\Gamma_0\) isn't equal to \(\omega\ \{\{1\}\}\ \omega\). It is in fact close, but it isn't exactly the same. Going even further difference may become much more significant. And when we reach level when we can't use Veblen hierarchy anymore, there is no single definition of fundamental sequence then, so \(\omega\)'s to n's may not work for all of them. LittlePeng9 (talk) 19:41, April 21, 2013 (UTC)

Fundamental sequences
Why the most googologists think that minor changes for fundamental sequences can significantly affect the growth rates? For example, we know that fund. sequence for \(\epsilon_1\) can be defined in two significantly different ways, and both will lead to \(f_{\epsilon_1}(n) \approx n \uparrow\uparrow (2n)\). Ikosarakt1 (talk ^ contribs) 14:21, July 2, 2013 (UTC)

I believe you heard result that \(f_{\varepsilon_0}\approx g_{BH}\) under standard sequences. There are definitions of fundamental sequences based on tree representations such that \(f_{\varepsilon_0}\approx g_{\varepsilon_0}\). LittlePeng9 (talk) 14:46, July 2, 2013 (UTC)