How about renaming instead?[]
I suggest not deleting this page, because a list of ordinals is something that this wiki has needed but didn't have for a long time, other than as a draft on project space. If you are so insistent on terminology, how about renaming this page to "List of ordinals" or "List of transfinite numbers"? -- ☁ I want more clouds! ⛅ 05:00, October 24, 2016 (UTC)
- Maybe changing to "List of countable ordinals" and also make "List of uncountable cardinals", as we already have Template:Transfinite_ordinals. 🐟 Fish fish fish ... 🐠 15:17, December 7, 2016 (UTC)
- List of ordinals sounds good . i dont think ordinals should be in the list of googologisms' 'Chronolegends (talk) 16:44, December 7, 2016 (UTC)
An uncountable error[]
I found omega one 3D infinite-piece chess in the list, but it's \(\omega_1\). —Preceding unsigned comment added by 80.98.179.160 (talk • contribs) 13:48, December 19, 2017
- While it is believed to be equal to \(\omega_1\), it hasn't been proven yet. So let's leave it here. However, I've added a note about it. -- ☁ I want more clouds! ⛅ 15:36, December 19, 2017 (UTC)
- Actually, it has been proven to be equal to \(\omega_1\). See https://arxiv.org/abs/1302.4377. Deedlit11 (talk) 15:42, December 19, 2017 (UTC)
Lack of References[]
The result on KPM (and maybe several other results) seems to be unpublished, and hence conflicts the policy of this wiki. At least, it should be clearly written that there is no academic reference unless anyone adds a reference. -- p-adic 23:54, August 26, 2018 (UTC)
- This result was proven in "Proof-theoretic Analysis of KPM" by Michael Rathjen. Deedlit11 (talk) 03:11, August 27, 2018 (UTC)
- No. Haven't you ever read it? -- p-adic 04:22, August 27, 2018 (UTC)
- What do you mean? The result is given in Theorem 7.15. If you are talking about the fact that the expression for the ordinal is somewhat different than what is listed on this page, you can look at "The Realm of Ordinal Analysis", which has as a corollary the equality \(|KPM| = \psi^{\varepsilon_{M+1}}(\Omega)\). (Yes, the variables are in different places, but that should hardly matter.) Deedlit11 (talk) 04:42, August 27, 2018 (UTC)
- The standard OCF defined in "Proof-theoretic Analysis of KPM" is not the OCF in "The Realm of Ordinal Analysis". This is not a problem on expression. Or do you have an explicit comparison result on those? Also, the latter note is not an evidence, because it does not refer to the existing proof. Also, when the former one was written, the result was still open. Therefore it is far from a reference of the result. In addition, please answer my quention at the reply of your blog post. Sincerely, you know the proof of your statement, right? -- p-adic 05:21, August 27, 2018 (UTC)
- You're being very pedantic here. In "The Realm of Ordinal Analysis" he states the result that you want, along with an overview of the proof, referring back to the previous paper for the full details. If you're looking for a paper in which he gives the same full proof using a slightly different notation, that would never ever happen, since that would not be remotely publishable, nor of particular interest, since it would be the same proof just with notational differences. The best we could possibly hope for after the first paper would be another paper which gives the result using the OCF that we want, referring back to the first paper, and that's what we got. It was enough for Michael Rathjen to cite the first paper for the proof, and it should be enough for you. I don't know what you mean by "when the former one was written, the result was still open. Therefore it is far from a reference of the result." Generally when original papers are written, the results that they prove are still open. Then the papers prove them, and they are no longer open. Then you cite those papers when you reference those results. So I don't know what you are talking about.
- I denote by \(P\) the non-standard OCF in "The Realm of Ordinal Analysis" in order to distinguish it from the standard \(\psi\) in "Proof-theoretic Analysis of KPM". I though that the equality \(\textrm{PTO}(\textrm{KPM}) = \psi_{\Omega_1}(\varepsilon_{M+1})\) in this page is a theorem on \(\psi\), but not on \(P\), because proof theorists often refer to \(\psi\) when they mention to the equality. Since you also refered to "Proof-theoretic Analysis of KPM", I thought that you are mentioning \(\psi\). On the other hand, your another source is on another equality \(\textrm{PTO}(\textrm{KPM}) = P_{\Omega}(\varepsilon_{M+1})\). Do you have any proof of \(P_{\Omega}(\varepsilon_{M+1}) = \psi_{\Omega}(\varepsilon_{M+1})\)? Or you are talking about \(P\), right? Then it is better to emphasise that the OCF in this page is not the standard one.
- > I don't know what you mean by "when the former one was written, the result was still open. Therefore it is far from a reference of the result."
- Could I ask you the question again? Haven't you ever read the paper? Rathjen did not verify the equality in "Proof-theoretic Analysis of KPM", but just wrote that it was written an unpublished preprint, which is still in preparation. At least, in "A Note on the Ordinal Analysis of KPM" written by Bucchholz in 1991, only the inequality \(\leq\) is regarded as a theorem. Therefore the paper is not a reference of the proof of the equation at all. After all, "The Realm of Ordinal Analysis" has no precise reference to the proof.
- > The statement in my blog post refers to the result of Rathjen
- Oh... Why didn't you write the citation? -- p-adic 11:19, August 27, 2018 (UTC)
Stegert's and Arai's OCFs[]
Does Stegert's 1st OCF have the same countable limit as Arai's OCF? C7X (talk) 15:44, July 31, 2020 (UTC)
- Is there any evidence? If so, please tell us the evidence, because it is obviously hard to guess the comparison.
- p-adic 23:10, July 31, 2020 (UTC)
- I see. There might be some comparison in Stegert's thesis, but I am too lazy to check the issue. I note that Arai's OCF in p. 2, which you referred to in the article, does not use indescribability, although the context states that the OCF is compatible with the OCF in latter pages based on the indescribability. This is also an evidence of your statement that what ordinals is used in the construction of an OCF does not characterise the strength.
- p-adic 00:26, August 1, 2020 (UTC)
Rapidly computing PTOs[]
Here Rathjen lists off a large number of equalities of PTOs of various theories (along with some inequalities). Some of these PTOs have had bounds shown earlier in the paper, such as the bound from theorem 14.4. However, AFAIK there is no published justification for these equalities other than the remark "these bounds turned out to be sharp". Assuming that there are no unpublished results on these PTOs, how did Rathjen compute them? C7X (talk) 20:05, 4 April 2021 (UTC)
- I have heard that ordinal analysis has so many unpublished works which are well-known by experts now because those have been spread in conferences.
- p-adic 22:53, 4 April 2021 (UTC)
\(\psi_\Omega(\varepsilon_{\textrm I+1})\) for \(\Delta_2^1\textrm{-CA+BI}\)[]
An ordinal called "\(\psi_\Omega(\varepsilon_{\textrm I+1})\)" is already on this list for the theory \(\textrm{KPi}\), but the one on the list uses Buchholz's OCF using a Mahlo cardinal (whose definition I don't have), while Rathjen's uses an OCF based on \(\textrm I\) without \(\textrm M\). Not assuming the two are equal, where should it be added? C7X (talk) 17:42, 22 May 2021 (UTC)