Talk: Hardy hierarchy

Omgf, no sources! again! (Article.) $Jiawhein$^{\(a\)\(l\)\(t\)} 13:35, May 10, 2013 (UTC)

I see that you have plan to write such messages on any talk page. Ikosarakt1 (talk ^ contribs) 16:55, May 10, 2013 (UTC)

Nah, he won't do it on every talk page. Only on sourceless articles' talk pages. LittlePeng9 (talk) 17:30, May 10, 2013 (UTC)

NO! $Jiawhein$^{\(a\)\(l\)\(t\)} 05:56, May 11, 2013 (UTC)

Incomplete.

Article incomplete. -- A Large Number Googologist -- 19:44, October 11, 2014 (UTC)

Why? LittlePeng9 (talk) 19:50, October 11, 2014 (UTC)

Inconsistencies with FGH

As we know (and is stated in the article), H_w^alpha(n) = f_alpha(n), where alpha < epsilon 0. Therefore, approximations for H_omega^alpha(n) should be equal to f_alpha(n). However, the articles vary by a lot. The approximations for H_omega(n) and f_1 (n) are equal, as are the approximations for H_omega^2(n) and f_2 (n). However, the approximation for f_3 (n) is that it's greater than 2 ^^ n, and the approximation in the Hardy hierarchy is H_omega^3(n) is approximately n ^^ n, a much greater (and I think an overestimated) value. In general (and as defined in the pages), f_m(n) > 2^^m-1n, and H_omega^m(n) is approximately {n,n,m-1} (which was erroneously stated as {n,n,m} before I fixed it an hour or two ago).

My questions are, A: Which approximation is more accurate? B: Should we change the approximations on one page to match?Maybe called Googology Noob (talk) 19:35, November 24, 2015 (UTC)

The approximations on articles are mostly complete guesswork and I have advocated for their removal numerous times. -- vel! 04:41, November 26, 2015 (UTC)

Obviously if it was something like epsilon 0 or even omega^omega^omega or something I would understand, but at such a low level diverging so much is odd. In any case, a better lower bound would be f_w(n) = 2[n+1]n+1, and I'm fairly sure that f_w(n) is approximately 2[n+1]2n-1, but that's besides the point. My question is, should we correct one of the pages? Maybe called Googology Noob (talk) 19:14, November 27, 2015 (UTC)

Just\(n\)?

Why is\(H_0(n)=n\)? Why not\(H_0(n)=n+\frac1n\)? Because extending FGH to negative numbers, we get\(f_{-k}(n)=n+\frac1{n^k}\). I think it's hilarious, but true:\(f_{k+1}(n)\)is\(f_k^n(n)\), thus, it's working as such for negatives.(Thus, it's not\(H_k:N\mapsto N\), but\(H_k:N\mapsto Q\)), same with \(f_k\).80.98.179.160 15:20, November 24, 2017 (UTC)

1. If you are talking about FGH, then why post on HH talk page?

2. Your equality is wrong: Even for \(k=1,n=2\) we have \(f_{-1}^2(n)=f_{-1}(f_{-1}(2))=f_0(\frac{5}{2})=\frac{5}{2}+\frac{2}{5}\neq 3\). Actually, for this to remotely make sense, we need \(f_{-k}\) to be defined on \(\mathbb Q\), not just \(\mathbb N\)). But then we would have to consider \(f_k^n\) for \(n\) not an integer... LittlePeng9 (talk) 21:01, November 24, 2017 (UTC)

Catching up

This paper (Gallier, 1991) seems to be describing "catching up" ordinal from p.255 to Theorem 12.4 (p.256).

https://doi.org/10.1016/0168-0072(91)90022-E

PDF from this link

I have not read through the paper yet, and just leave a note here. 🐟 Fish fish fish ... 🐠 07:33, 25 April 2022 (UTC)

Uh, I got shocked because it says "it is not difficult to show that \(f_{\alpha}(n) = h_{\omega^{\alpha}}(n)\)" in p.255. Since the author is considering any fundamental sequences and any α in the system, the equality does not necessarily hold... Say, it does not hold for Veblen's hierarchy and α = ε_0, and also does not hold for ω[n] = BB(n) and α = 1. I hope that I am making some mistake.

p-adic 08:02, 25 April 2022 (UTC)

In p.254, it is written "The point of ordinal notations is that they allow the definition of standard fundamental sequences. This is particularly simple for the ordinals less than ε_0, where we can use the Cantor normal form.", so the author may considering only for this "standard" FS under ε_0. But the author then writes "Fundamental sequences can also be assigned to certain classes of limit ordinals larger than ε_0, but this becomes much more complicated. In particular, this can be done for limit ordinals less than Γ_o, using the normal form representation given in Theorem 8.2." so the author actually treats ordinals larger than ε_0. So the author should have stated clearly about the condition that the statement is true. 🐟 Fish fish fish ... 🐠 10:15, 25 April 2022 (UTC)

Right. Since the author refered to α in (mathfrac)S, which can be arbitrary large in the current context, the author should clarify the precise condition, before stating the equality...

p-adic 11:00, 25 April 2022 (UTC)

Additional note: Gallier (1991) writes "Theorem 12.4 (Girard). There is an ordinal α such that G_α, and F_α, have the same rate of growth, in the sense that G_α(n) <F_α(n) < G_α(an + b), for some simple linear function an + b." which was first proved in Girard (1981): https://doi.org/10.1016/0003-4843(81)90016-4 (PDF available). 🐟 Fish fish fish ... 🐠 11:34, 25 April 2022 (UTC)

I suspect the possibility that the statement is not precisely cited, i.e. the author dropped conditions on fundamental sequences, because of the points above. Since it is not easy to search results in old paper due to text formatting and I am not accustomed to terminology in formulation of function hierarchies in terms of dilators, I have not found the corresponding statement. Have you found it? (Alwe might know the location, because Alwe knows dilator and Π^1_2-logic very well.)

p-adic 13:53, 25 April 2022 (UTC)

I am trying to guess which part of the Girard's paper may correspond to the theorem, but I am completely at a loss right now. 🐟 Fish fish fish ... 🐠 13:58, 25 April 2022 (UTC)

When an author cites such a long paper, he or she is supposed to clarify the location, because the reader will be suffered like us ┐(ツ)┌

p-adic 14:10, 25 April 2022 (UTC)