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## First section

Any values of the first few terms? Also, can anyone program it at Turing machine? It would be a big breakthrough in googology, because with this function we can easily beat even the number proposed by Chris Bird at the end of his paper. Ikosarakt1 (talk ^ contribs) 13:49, April 10, 2013 (UTC)

Well, (+,0) dies after one cut, so BH(0) = 1. (+,0,$$\omega$$) goes to (+, 0, 2), which goes to (+, 0, 1, 0), which goes to (+, 0, (1, 1, 1)). It gets quite complicated after that, so even BH(1) is hard to determine, although it doesn't surpass the Bachmann-Howard ordinal. Deedlit11 (talk) 14:13, April 10, 2013 (UTC)

I thought TFB was greater than Bachmann-Howard? FB100Ztalkcontribs 01:17, April 11, 2013 (UTC)
It is. The strength of trees labelled with 0's and 1's is the Bachmann-Howard ordinal, so BH(1) can be at most f_BHO(n) for a small n. Deedlit11 (talk) 01:41, April 11, 2013 (UTC)

I haven't studied this much, but what if we allow other ordinals?

1. The first rule is unchanged.
2. If node we chose is a successor ordinal $$\upsilon + 1$$, we go down the tree looking for a node $$b$$ with label $$\phi \leq \upsilon$$, etc.
3. If the node we chose has a limit ordinal $$\upsilon$$ for its label, we relabel it with $$\upsilon[n + 1]$$.

How much would this benefit us? FB100Ztalkcontribs 19:53, April 10, 2013 (UTC)

I believe that, using ordinals up to $$\alpha$$, we can get a growth rate of $$f_{\psi_0(\varepsilon_{\Omega_{\alpha}+1})}(n)$$. If we use trees as labels for trees, then use those trees as labels for even larger trees, etc., we reach a limit of $$f_{\psi_0(\Omega_{\Omega})}$$. Deedlit11 (talk) 20:01, April 10, 2013 (UTC)
Actually, if I recall correctly, limit of $$\psi$$ function is $$\psi_0(\varepsilon_{\Omega_\omega+1})$$, TFB ordinal, so we probably need stronger notation for that. I also had this idea, although I have no idea about its strength. LittlePeng9 (talk) 20:11, April 10, 2013 (UTC)
Yes. TFB marks the limit of the psi-0 function. We need a new function to go any further. FB100Ztalkcontribs 01:01, April 11, 2013 (UTC)
Again, why TFB ordinal is limit for $$\psi_0(\alpha)$$ function? Is it not possible to have ordinal, say, $$\psi_0(\zeta_{\Omega_\omega+1})$$? Ikosarakt1 (talk ^ contribs) 10:16, April 11, 2013 (UTC)
$$\psi_0$$ function is said to be eventually constant. It means that ordinal you wrote is no greater than TFB ordinal. I think every collapsing function asks for smallest not in a given set. TFB isn't in such set for any ordinal, and this is connected to fact that $$\varepsilon_{\Omega_\omega +1}=\Omega_\omega^{\Omega_\omega^{...}}$$, so using exponentiation we can't reach it and then apply $$\psi$$ function to it. LittlePeng9 (talk) 15:47, April 11, 2013 (UTC)
It's straightforward to extend the notation - just add more cardinals of the form $$\Omega_{\alpha}$$ (and add corresponding functions $$\psi_{\alpha}(\beta)$$). But the stronger notations that I have seen use cardinals as subscripts for $$\psi$$, so I guess I should say that the Buchholz hydras indexed with ordinals up to $$\alpha$$ have a growth rate of $$f_{\psi_{\Omega_1}(\varepsilon_{\Omega_{\alpha}+1})}(n)$$. Deedlit11 (talk) 13:24, April 12, 2013 (UTC)
Given that it has three rules — zero, successors, and limits — it looks suspiciously like an ordinal hierarchy... FB100Ztalkcontribs 01:01, April 11, 2013 (UTC)
It sure does! The Buchholz hydras closely match an ordinal notation for ordinals up to $$\psi_0(\varepsilon_{\Omega_{\omega}+1})$$. In fact, if you read Buchholz's paper, he exploits this relationship to prove that the termination of Buchholz hydras cannot be proven in $$\Pi^1_1 \text{- CA + BI}$$. Note that the reduction of Buchholz hydras closely match up with taking fundamental sequences of ordinals in the ordinal notation, except that the rule for when you chop off a head with label u+1 isn't quite right - the rule only appends one subtree, whereas to be a proper fundamental sequence it should append n subtrees, one on top of another. But Buchholz shows it doesn't fundamentally affect the growth rate. Deedlit11 (talk) 13:38, April 12, 2013 (UTC)

## Implementation

I finally got around to implementing this beast (pun intended). Are these first few steps of BH(3) correct?

(+(0(ω)))
(+(0(2)))
(+(0(1(0))))
(+(0(1)(1)(1)(1)))
(+(0(1)(1)(1)(0(1)(1)(1)(0))))
(+(0(1)(1)(1)(0(1)(1)(1))(0(1)(1)(1))(0(1)(1)(1))(0(1)(1)(1))(0(1)(1)(1))(0(1)(1)(1))))
(+(0(1)(1)(1)(0(1)(1)(1))(0(1)(1)(1))(0(1)(1)(1))(0(1)(1)(1))(0(1)(1)(1))(0(1)(1)(0(1)(1)(0)))))
(+(0(1)(1)(1)(0(1)(1)(1))(0(1)(1)(1))(0(1)(1)(1))(0(1)(1)(1))(0(1)(1)(1))(0(1)(1)(0(1)(1))(0(1)(1))(0(1)(1))(0(1)(1))(0(1)(1))(0(1)(1))(0(1)(1))(0(1)(1)))))
(+(0(1)(1)(1)(0(1)(1)(1))(0(1)(1)(1))(0(1)(1)(1))(0(1)(1)(1))(0(1)(1)(1))(0(1)(1)(0(1)(1))(0(1)(1))(0(1)(1))(0(1)(1))(0(1)(1))(0(1)(1))(0(1)(1))(0(1)(0(1)(0))))))
(+(0(1)(1)(1)(0(1)(1)(1))(0(1)(1)(1))(0(1)(1)(1))(0(1)(1)(1))(0(1)(1)(1))(0(1)(1)(0(1)(1))(0(1)(1))(0(1)(1))(0(1)(1))(0(1)(1))(0(1)(1))(0(1)(1))(0(1)(0(1))(0(1))(0(1))(0(1))(0(1))(0(1))(0(1))(0(1))(0(1))(0(1))))))


FB100Ztalkcontribs 20:45, April 16, 2013 (UTC)

Looks right to me. Deedlit11 (talk) 20:50, April 16, 2013 (UTC)

I'm trying to implement these hydras in Python but I'm having a hard time figuring out how to make labeled trees. They're a little overwhelming tbh. QuasarBooster (talk) 01:30, December 1, 2018 (UTC)

Nevermind. I managed to implement BH in Python. There's a post on my blog with the code. QuasarBooster (talk) 17:28, December 8, 2018 (UTC)

## Larger transfinite labels

If we can have label $$\omega$$, why not have labels $$\omega+1,\omega+2,\omega 2,\omega 3,\omega^2,\cdots,\varepsilon_0,\Gamma_0$$, etc? Ikosarakt1 (talk ^ contribs) 10:03, April 22, 2013 (UTC)

Of course we can have larger labels. Rules are simple to extend, it'd be awesome if anyone found what ordinals are limits of these extended hydras. LittlePeng9 (talk) 10:23, April 22, 2013 (UTC)

I proposed this a couple sections up on this page. Let $$\text{BH}_\alpha(n)$$ be the BH function starting with a chain of $$\alpha$$ ordinals, so $$\text{BH}(n) = \text{BH}_\omega(n)$$. If I have it correctly, this function is infinitely extensible a la FGH. FB100Ztalkcontribs 21:55, April 22, 2013 (UTC)

I wonder how large catch-up ordinal i. For every function g(n) there is $$\alpha$$ so that $$f_\alpha(n)$$ outgrows g(n). So if we take $$BH_\omega$$ to be g, we can find $$\alpha$$ which outgrows it. Then we can find $$\beta$$ which outgrows $$BH_\alpha(n)$$. Then we can find $$\gamma$$ outgrowing $$BH_\beta(n)$$, etc. If I'm not mistaken, limit of this sequence marks catch-up ordinal. LittlePeng9 (talk) 06:00, April 23, 2013 (UTC)

Yikes! That sounds incredibly remote.
Consider the general case of $$\text{BH}_{\alpha}(n)$$. What system analogous to $$\Pi_1^1-\textrm{CA}+\textrm{BI}$$ is the $$\alpha$$-level hydra theorem unprovable in? (yes, that's a very awkward sentence) FB100Ztalkcontribs 17:04, April 23, 2013 (UTC)

## Most powerful recursive function

I believe that D(n) grows faster asymptotically, since we have notation for TFB ordinal, but ordinal for D(n) is so large that we can't define it using canonical ordinal collapsing functions. Ikosarakt1 (talk ^ contribs) 13:13, April 25, 2013 (UTC)

Almost surely you are right. BH(n) is said to be one of the fastest growing functions defined, not fastest one. D(n) is waaay above most ordinal notations. But this one may be a bit closer to it. LittlePeng9 (talk) 13:38, April 25, 2013 (UTC)

Maybe that page is useful for high-frequency algorithmic trading. [1] It's quite likely a joke. I want more clouds! 14:17, April 25, 2013 (UTC)

I believe so. Collapsings to nonrecursive ordinals aren't of that much interest. LittlePeng9 (talk) 14:11, April 25, 2013 (UTC)

## Technical questions

Can we transform hydras to ordinals in Hardy hierarchy as easy as it has been done for Goodstein function? It would be helpful, because it allows to determine the number of steps for each hydra without actual simulating all steps. For example, if $$([])\Rightarrow\varepsilon_0$$, then, say ([]) would require $$H_{\varepsilon_0}(3)$$ steps to resolve.

Also, in section "rules", why the number of steps (n) doesn't play the role? If I get it correctly, ([]) should be always transformed to ((())), regardless to the number of steps. It is very logical that ([]) must be (((...((()))...))) (with n ()'s). Ikosarakt1 (talk ^ contribs) 13:44, April 25, 2013 (UTC)

Once again, choice of adding of only one copy was made by Buchholz himself. He didn't want to create fast growing function or representing ordinals exactly. He wanted just that independence result. LittlePeng9 (talk) 14:17, April 25, 2013 (UTC)

## Another question

What happens if we start with $$(0,+,\text{TFB})$$? FB100Ztalkcontribs 18:50, April 25, 2013 (UTC)

Well, as $$\text{TFB}=\psi_0(\varepsilon_{\Omega_\omega+1})=\psi_0(\Omega_\omega^{\Omega_\omega^{...}})$$ its fundamental sequence is $$\psi_0(0),\psi_0(1),\psi_0(\Omega_\omega),\psi_0(\Omega_\omega^{\Omega_\omega}),...$$. TFB label reduces to some smaller label, and then it continues in pretty obvious way. LittlePeng9 (talk) 18:54, April 25, 2013 (UTC)

## BH(3)

How big is BH(3)? All I know is that it's over 100. FB100Ztalkcontribs 14:20, May 3, 2013 (UTC)

It almost certainly is not expressible in many strong notation systems, like chained arrows. I can't be sure, though, but when I think about Hydra(5) I'm pretty sure BH(3) outgrows it largely. So yeah, above 100. LittlePeng9 (talk) 15:05, May 3, 2013 (UTC)

It expands to into a fairly large tree with 0 and 1 labels, so I believe you need a notation system that goes all the way up to the Bachmann-Howard ordinal to express it. Deedlit11 (talk) 00:35, May 4, 2013 (UTC)

I'm guessing that means greater than TREE(3)? FB100Ztalkcontribs 04:29, May 10, 2013 (UTC)
I believe so, however there is the annoying fact that we don't have a good upper bound for TREE(n). Deedlit11 (talk) 11:11, May 10, 2013 (UTC)
I have a proof BH(3) >> Hydra(6):
(+(0(1)(1)(1)(0(1)(1)(1))(0(1)(1)(1))(0(1)(1)(1))(0(1)(1)(1))(0(1)(1)(1))(0(1)(1)(0(1)(1))(0(1)(1))(0(1)(1))(0(1)(1))(0(1)(1))(0(1)(1))(0(1)(1))(0(1)(0(1))(0(1))(0(1))(0(1))(0(1))(0(1))(0(1))(0(1))(0(1))(0(0(0)))))))
is a step of BH(3). If we remove some things, we get (+(0(0(0(0(0(0))))))),it is the start thing for Hydra(6) so BH(3) >> Hydra(6). Wythagoras (talk) 06:11, July 12, 2013 (UTC)

Are you sure that removing these things can lead to smaller hydra? Ikosarakt1 (talk ^ contribs) 06:55, July 12, 2013 (UTC)

## http://googology.wikia.com/wiki/Buchholz_hydra?diff=next&oldid=36344

lol. ${\displaystyle Jiawhein}$$$a$$$$l$$$$t$$ 02:51, May 10, 2013 (UTC)

please be nice to your fellow gwikians. nothing wrong with serious people. 'k?
I propose an alternative name for TFB: the "Holy–Shit ordinal." FB100Ztalkcontribs 04:28, May 10, 2013 (UTC)

## ordinal trees?

I wonder if these have anything to do with hydras. --Ixfd64 (talk) 18:07, June 5, 2013 (UTC)

I think my brain just went numb FB100Ztalkcontribs 18:14, June 5, 2013 (UTC)
These have no direct connection to hydras. This is standard well-ordering of finite trees, which is linear extension of homeomorphic embedding relation. By using more complicated concepts for labelled trees however we reach well-order of $$\psi_0(\Omega_\omega)$$ if I'm not mistaken. LittlePeng9 (talk) 18:39, June 5, 2013 (UTC)
Is there a standard well ordering for infinite trees? Tomtom2357 (talk) 07:22, July 10, 2013 (UTC)
There is continuum of different infinite trees. Existence of well ordering would imply well ordering of continuum, which is independent of ZF theory. Even with choice we only know ordering exists, without knowledge on how it works. LittlePeng9 (talk) 08:53, July 10, 2013 (UTC)

It looks pretty cool and intriguing, but I don't know how these ordinals have been obtained. Ikosarakt1 (talk ^ contribs) 19:00, June 5, 2013 (UTC)

## Restrictions for number "n"

From the article :"At each step we select a leaf node a on the hydra together with a number n". On what depends this "n"? Ikosarakt1 (talk ^ contribs) 18:40, June 30, 2013 (UTC)

For the general theorem, there is no restriction on n; you could choose Rayo(i) at step i, for instance. For BHydra(n), we choose the number i at step i. Deedlit11 (talk) 18:52, June 30, 2013 (UTC)

## Unexplained expression

In "there exists $$k$$ such that $$R^n(1)(2)(3)...(k)$$ is root tree." what (1)(2)(3)...(k) is supposed to mean? Ikosarakt1 (talk ^ contribs) 21:58, September 16, 2013 (UTC)

From the previous section, T(n) is the result of taking the tree T and applying the process using n. So $$R^n(1)(2) \ldots (k)$$ is the tree you get by starting from $$R^n$$ and applying the process with the numbers $$1,2, \ldots$$ and so on up to $$k$$. Deedlit11 (talk) 00:18, September 17, 2013 (UTC)

The notation is a bit confusing. I will edit to clarify. FB100Ztalkcontribs 05:43, September 17, 2013 (UTC)
Okay, thanks. Ikosarakt1 (talk ^ contribs) 07:55, September 17, 2013 (UTC)

## Buchholz Hydras with braces

Can you show me the rulesets of Buchholz Hydras with (), [], {} and ()_n where ()_1 = (), ()_2 = [] and ()_3 = {}, etc?

-- A Large Number Googologist -- 21:43, October 18, 2014 (UTC)

No wait, i think that was another thing.

Can you give me an easier ruleset of this hydra? -- A Large Number Googologist -- 21:24, October 19, 2014 (UTC)

figure it out yourself it's vel time 21:42, October 19, 2014 (UTC)

really.

maybe i will try again -- A Large Number Googologist -- 00:50, October 20, 2014 (UTC)

i can't figure it with trees

please use symbols and/or braces -- A Large Number Googologist -- 00:52, October 20, 2014 (UTC)

## Inappropriate citation

A source should be cited explicitly, and the current list of sources does not work.

In addition, the last listed source "K. Amiko, Amiko Hydras, 2020" is not appropriately mentioned, because it lacks a citation infomration. Is it an iaccessible website? Is it a book without a publisher information? Is it a paper without a journal information? In order to make a source accessible, please add sufficient information.

p-adic 11:23, 2 October 2021 (UTC)

If nobody corrects the last listed source and cite it in an appropriate way, I will remove the corresponding section.
p-adic 00:37, 3 October 2021 (UTC)
I assume that the description is from https://komiamiko.me/math/ordinals/2020/06/07/higher-order-hydras.html , but as I am not confident and the data was not updated for about a month, I deleted. If someone wants to describe Amiko hydra, I suggest making a new page by correctly citing the source. 🐟 Fish fish fish ... 🐠 14:42, 1 November 2021 (UTC)

## Rewriting "Hydra Theorem" Section

This is the current content of "Hydra theorem" section as I am writing this:

It was believed in this community that Wilfried Buchholz showed that there are no losing strategies for any hydra, but the source of this article does not include such a result. A similar made-up description is found in Kirby-Paris hydra#Hydra Theorem.

Call this statement the hydra theorem. What he actually showed is the canonical correspondence from a hydra to an infnitary well-founded tree (or the corresponding term in the notation system $$T$$ associated to Buchholz's function, which does not necessarily belong to the ordinal notation system $$OT \subset T$$), preserves fundamental sequences, i.e. the strategy to choose the rightmost leaves and the $$(n)$$ operation on an infinitary well-founded tree (or the $$[n]$$ operation on the corresponding term in $$T$$). Although it might fortunately imply the hydra theorem under some weak set theory, the statement that he showed the hydra theorem is wrong because he even does not state the hydra theorem. He only refered to the fact that the sequence of the rightmost leaves is a winning strategy. The hydra theorem is unprovable in $$\Pi_1^1-\textrm{CA}+\textrm{BI}$$, but for individual hydras it was believed to be provable in this community without a source.

I have seen some criticize this and the similar statements on the mainspace pages to be overly critical of the community. I agree that the section contains some inappropriate writing styles. I, therefore, propose a clean-up to make the writing more neutral while still maintaining the purpose. Upon the successful rewrite, I hope that the discussion serves as a guide for other existing and future problematic sections that seek to warn readers of similar inaccuracies.

The following is my initial proposal:

It was believed by some that there are no losing strategies for any hydra. Call this statement the hydra theorem. Wilfried Buchholz was cited for showing this, but the source of this article does not include such a result. A similar description is found in Kirby-Paris hydra#Hydra Theorem.

What Buchholz showed is the canonical correspondence from a hydra to an infinitary well-founded tree (or the corresponding term in the notation system $$T$$ associated to Buchholz's function, which does not necessarily belong to the ordinal notation system $$OT \subset T$$), preserves fundamental sequences, i.e. the strategy to choose the rightmost leaves and the $$(n)$$ operation on an infinitary well-founded tree (or the $$[n]$$ operation on the corresponding term in $$T$$). He neither states nor implies the hydra theorem. Instead, he only referred to the fact that the sequence of the rightmost leaves is a winning strategy. The hydra theorem is unprovable in $$\Pi_1^1-\textrm{CA}+\textrm{BI}$$. It is unknown if it is provable for individual hydras.

Here are the notable changes:

1. Some emphasis on wrongness using bold characters was removed as they were unnecessary.
2. Some words were removed for similar reasons.
3. "Although . . . state the hydra theorem" was replaced with a simpler statement "This neither states nor implies the hydra theorem."
4. The statement on the hydra theorem on individual hydras was altered to simply state that its provableness is unknown.

Some things were not changed:

1. The hydra theorem is used later in the article, so it can not be removed.
2. The description of Buchholz's result is important, so it can not be removed either.

I welcome criticisms and suggestions from all groups of people with all opinions on the matter. I will replace the section in the article after 7 days of no unresolved issues.

Naruyoko (Talk) 06:03, 1 November 2021 (UTC)

> It was believed by some that
In my opinion, it is good to keep the fact that the "source" of the fake is this wiki itself. So, the updated description does not look good. It even looks like that there are somebody stating it outside this wiki. The problem is that the fake arose from this wiki. Such a clarification is necessary in order to be responsible for what we wrote. For example, if we talk about the fake as if it were a fake made by "some", readers cannot reach the actual fact that it arose from this wiki.
> It is unknown if it is provable for individual hydras.
By a similar reason, we should not hide the fact that this wiki spread a fake. When we spread a fake, others might believe it, and cite it. After then, should we silently delete it? I do not think so, because it leads a confusion of the "source" of the fake. Do you remember that there are many videos with unsourced descriptions due to the misfortune that the creator just believed wrong descriptions arose in this wiki? If we silently remove the "source" of the fakes, then the creator will be considered as a "source" of the fake. It is really irresponsible.
> I, therefore, propose a clean-up to make the writing more neutral while still maintaining the purpose
Do you mean that the current description is less neutral? Then please clarify that what description is biased, and is possibly wrong. If we need a source of the fact that there were fakes, then we can simply cite old versions of this article. I think that hiding the "source" of the fakes is not neutral.
p-adic 07:11, 1 November 2021 (UTC)
I like the updated description better. What do you think, other users? 🐟 Fish fish fish ... 🐠 07:15, 1 November 2021 (UTC)
I appreciate if you tell your opinion with reasons, rather than preference. Since it is a matter of neutrality and responsibility, a sort of preference is not the main issue.
p-adic 07:21, 1 November 2021 (UTC)
The tone of the current page is too bad. History of the wiki can be tracked easily, and therefore updating to new information is not blamed for hiding the page history. That is different point from other websites which does not have tracking feature. 🐟 Fish fish fish ... 🐠 07:31, 1 November 2021 (UTC)
It sounds weird. If we do not suggest the fact that the "source" of the fake is this wiki, nobody will check the history page to find that the "source" of the fake is this wiki. At any rate, describing the "source" as "some" is irresponsible. For example, please consider the case one watches the video with a fake from this wiki and searched the fake in google. If the article just says "by some" as the proposal says, then the one does not think that the actual "source" is this wiki. More awfully, the one would think that the "source" of the fake is the uploader of the video. It is quite irresponsible.
p-adic 07:42, 1 November 2021 (UTC)
So maybe changing the "It was believed by some that" to "It was described in the old version of this wiki that" and refer to old version of this wiki, and accept all the other update? 🐟 Fish fish fish ... 🐠 07:48, 1 November 2021 (UTC)
I pointed two points. The first one is "It was believed by some that", and the second one is "It is unknown if it is provable for individual hydras.". Other changes look good for me. (But Naruyoko is further stating that the current version is not neutral. Therefore we need to share what description in he current version is considered to be biased and hence is doubghtful before updating it, because it is a very serious issue.)
p-adic 07:58, 1 November 2021 (UTC)
So we can change "It is unknown if it is provable for individual hydras." to "It is unknown if it is provable for individual hydras, although it was described in the old version of this wiki that it is provable." and refer to the old version. 🐟 Fish fish fish ... 🐠 08:10, 1 November 2021 (UTC)
I agree with the proposal. Another choice is to create a separated section like "Fakes in old versions" in order to collect such descriptions, if you do not like the recalling.
p-adic 08:18, 1 November 2021 (UTC)
> I pointed two points.
> The first one
I wrote the initial revision thinking (somehow) writing "some" implied a part of the community. Now, I am rethinking this assumption.
> the second one
Whether or not to incorporate the suggestions depends on the purpose of the wiki. Should it be purely informative of the googology-related content, or should it also contain the past mistakes within the wiki (you seem to prefer this one)? The argument applies to both two suggestions.
This goes outside of the scope of the discussion, but it is important. It affects virtually all pages longer than a few paragraphs.
> what description in the current version is considered to be biased
I found the description "was . . . in this community" in the original text to be providing an unfair image. It implies that all of the community assumed the statements. Those removed with the notable changes 1 and 2 did not help the image of the image either. The statements may have been a reasonable conclusion, but I could not decipher it especially looking into other articles for the future.
Perhaps, I did not use the term "[(not)] neutral" correctly. If so, that is my fault.
Naruyoko (Talk) 09:10, 1 November 2021 (UTC)

> Should it be purely informative of the googology-related content, or should it also contain the past mistakes within the wiki (you seem to prefer this one)?
The question is seriously loaded, because it leads a wrong conclusion that "past mistakes within the wiki" is not googology-related. Since past mistakes within this wiki caused serious disrupting in googology, those are actually very notable. I am not understanding why you asked it, even after I explained the actual incident that the fakes are spread seriously. Say, could you please remember the fact that many of wrong descriptions in ordinal notations and OCFs from old members in this wiki prevented sincere studies by new people? Could you please remember the fact that a Japanese professor expressed disappointment against the fact that this wiki was spreading fakes on TREE function, which is part of the branch of mathematics which he is working on? It even prevents innocent mathematical students to study what have already proved in mathematics, because old wiki members wrote unproved facts as if they were already proved. Obviously, they are googology-related incidents.
> I found the description "was . . . in this community" in the original text to be providing an unfair image.
Could you explain how unfair it is (compared to whom)?
> It implies that all of the community assumed the statements.
What we can write in encyclopedias are sourced descriptions, descriptions which are considered to be obvious facts, and descriptions with sufficient evidences or proofs. If a description in an article frequently edited by many users remains to be unsourced, unproved, and free from warning on the accuracy like {{cite}}, it is reasonable to guess that nobody doubted the description before the issue was pointed out. Or have you (or other users) personally ever doubted it and made some action against it?
> Those removed with the notable changes 1 and 2 did not help the image of the image either. The statements may have been a reasonable conclusion, but I could not decipher it especially looking into other articles for the future.
I could not parse these sentences. What does the image of the image specificially stand for? Since we both know Japanese, I appreciate if you add a translation of these sentences into Japanese. (And if my English is awfully bad for you, I also write Japanese sentences and google translations.)
p-adic 10:11, 1 November 2021 (UTC)
Yes, they are googology-related incidents. As you suggested, one way is to describe it in a separate section. How about making the section as "History of discussion". If the section gets too long, we can divide it into subpage and write a brief summary in the main page. In this way, I think it is worth recording the incident objectively. Alternative way is to just adding brief sentences, as I suggested above. This does not require much time to revise. 🐟 Fish fish fish ... 🐠 11:05, 1 November 2021 (UTC)
> Could you please remember the fact that a Japanese professor expressed disappointment against . . . spreading fakes on TREE function . . . ?
While I am aware of the comment, I likely failed to consider it. I agree that we should correct any inaccuracies and it is regretful that it was left for so long.
> Say, could you please remember the fact that many of wrong descriptions in ordinal notations and OCFs . . . prevented sincere studies by new people?
Unfortunately, I am not aware of the extent of the effect of the wrong descriptions, especially when including those that were on the main articles. From the past comments of people who found those mistakes, I understand that weird interpretations were prominent. Since as far as I am aware, I never got to learn using them, I can not imagine how large or small each error had spread. One time I said while in a discussion over Discord: "I definitely see this [the misconception 'the existence of the correspondence to ordinals automatically ensures the comparability to the fast-growing hierarchy'] somewhat often, particularly on the 'edges' of the community."
> Could you explain how unfair it is (compared to whom)?
> Or have you (or other users) personally ever doubted it and made some action against it?
I have not. I have never personally seen others correct the statement (or state the wrong statement). That is why I wanted to add that the wrong statement was thought by some but not all.
Again, perhaps I do not understand the concept of fairness and biases well. My statement undoubtedly relies on my understanding of the community. I confess, it is limited and has only existed for a few years. You are much more experienced as an active member of the wiki, even if it may disagree with mine.
> What does the image of the image specificially stand for?
JA: "The image[1] of the image[2]"とは私が受け取った節の件の問題に対しての姿勢をあらわしています。上で記したように、私は構成や言葉遣いがコミュニティ全体に対して過度に批判的であると感じました(「恩着せがましい」や「圧制的」といった思いも含まれるかもしれませんが、そうである可能性は低いと思います)。実際そのような目的があって書かれたとは言おうとしていないことを理解してもらいたいです。まとめると、image2は著者が文を通してコミュニティの何をどのように見せたかったかの意図を指し、対してimage1はその文を読みimage2が何であるかを私が推測したものを指します。
EN: "The image[1] of the image[2]" referred to my perception of the connotations and the overall attitude, that is being overly critical ("condescending" might be included, but I do not feel that is true). I want to clarify that it is not a statement about the true intent of the writer(s). In conclusion, image2 = the image the authors of the passage intended to give, while image1 = what I thought is image2 when reading it.
P.S. I do not think your English is terrible. Simple grammar mistakes are easy to correct and small enough to not disrupt the message. You seem to use certain words like "wrong" and "fake" often, but I assume the reason is that you are taking a more systematic approach to the problems. They do not need to be fixed in my opinion. You can use either Japanese or English for me.
Naruyoko (Talk) 22:04, 1 November 2021 (UTC)

### Update of suggestion

Following some discussion, I propose the following revision to the initial proposal by me, 06:03, 1 November 2021 (UTC).

It was described in the old version of this wiki that there are no losing strategies for any hydra. Call this statement the hydra theorem. Wilfried Buchholz was cited for showing this, but the source of this article does not include such a result. A similar description is found in Kirby-Paris hydra#Hydra Theorem.

What Buchholz showed is the canonical correspondence $$|\cdot|$$<ref name="BH" /> p. 136 from a hydra to an infnitary well-founded tree (or the corresponding term in the notation system $$T$$ associated to Buchholz's function, which does not necessarily belong to the ordinal notation system $$OT \subset T$$), preserves fundamental sequences,<ref name="BH" /> 2.1 Theorem (b) i.e. the strategy to choose the rightmost leaves and the $$(n)$$ operation on an infinitary well-founded tree (or the $$[n]$$ operation on the corresponding term in $$T$$). He neither states nor implies the hydra theorem. Instead, he only referred to the fact that the sequence of the rightmost leaves is a winning strategy.<ref name="BH" /> Theorem I A statement weaker than the hydra theorem is unprovable in $$\Pi_1^1-\textrm{CA}+\textrm{BI}$$.<ref name="BH" /> Theorem I It is unknown if it is provable for individual hydras, although it was described in the old version of this wiki that it is provable.

Here are the notable changes:

1. The content surrounding Buchholz's theorem was updated to match the current revision of the article.
2. Altered "It was believed by some that" and "It is unknown if it is provable for individual hydras" to clarify that the description was part of the older version of the article. See p-adic, 07:11, 1 November 2021 (UTC); p-adic, 07:58, 1 November 2021 (UTC); 🐟 Fish fish fish ... 🐠, 07:48, 1 November 2021 (UTC); and 🐟 Fish fish fish ... 🐠, 08:10, 1 November 2021 (UTC).

The following were not changed/applied:

1. Fish's suggestion was not applied since it is still new and requires a restructuring extending beyond this section.

If you are viewing this discussion, please take note of each time this revision and the comments were posted.

Naruyoko (Talk) 06:19, 2 November 2021 (UTC)

When you quote a signature and a timestamp, could you put them into double quation mark or something like that? The currect reference is really confusing.
The updated version is good for me. I still prefer to add "without a proof or a source" for the description of the old version, because the description does not literally tell the reader anything on whether it is actually proved or sourced. I guess that you and Fish did not add it due to the belief that the reader can easily assume that any description which is not followed by citations is not proved or sourced, but it is a little hrsh in my opinion. But if you two still believe that we should not insert "without a proof or a source" under the belief, then you two might be correct, i.e. unlike those who believed unsourced and unproved statements, the reader is usually sufficiently educated to notice the possibility that it is just a fake.
p-adic 06:50, 2 November 2021 (UTC)
Original suggestion by Naruyoko was to keep the "Hydra Theorem" section and update the description, because the hydra theorem is used later in the article. After the suggestion, the hydra theorem in the later section was replaced by Theorem I of Buchholz, and the hydra thoerem is no more used elsewhere in the article. Therefore, now the whole section of the Hydra theorem can be renamed to "History of discussion" and the section can be moved to the last section of the article, as I suggested, and then update the description as Naruyoko suggested (the new suggestion). If we require further correction of the article, it can be added in the "history of discussion" section. However, if you prefer to update as you suggested, that is fine, because I will do this further update after your update (it was already approved by some members here). 🐟 Fish fish fish ... 🐠 08:21, 2 November 2021 (UTC)
I guess that this is a reply to me because it includes "Naruyoko" and "you".
> However, if you prefer to update as you suggested, that is fine, because I will do this further update after your update
Does "this further update" mean the moving of the corrections to "History of discussion" section? Then ok.
> (it was already approved by some members here).
Do you mean some indefinite member other than I, you, and Naruyoko? If so, I do not understand why you hide a link to the approval. If you simply mean I, you, or Naruyoko, then I do not understand why you mention us in an indefinite way here.
p-adic 09:32, 2 November 2021 (UTC)
It was a reply to Naruyoko and "you" means Naruyoko. I mean approved by this thread. At first I wrote to this thread, and then noticed that a new suggestion was made and moved to this thread. It made my suggestion unclear. Sorry for confusion. 🐟 Fish fish fish ... 🐠 09:44, 2 November 2021 (UTC)
I see.
p-adic 09:49, 2 November 2021 (UTC)
As no one disagreed to Naruyoko's suggestion for 2 weeks, I updated as Naruyoko suggested, and then move it to the history of discussion section as I suggested in this thread. 🐟 Fish fish fish ... 🐠 03:50, 16 November 2021 (UTC)
I completely forgot to reflect the suggestion, thank you. I will write up a summary in Googology Wiki:Discussion as suggested. --Naruyoko (Talk) 04:39, 16 November 2021 (UTC)

### Fish' suggestion

Yes, they are googology-related incidents. As you suggested, one way is to describe it in a separate section. How about making the section as "History of discussion". If the section gets too long, we can divide it into subpage and write a brief summary in the main page. In this way, I think it is worth recording the incident objectively. Alternative way is to just adding brief sentences, as I suggested above. This does not require much time to revise. 🐟 Fish fish fish ... 🐠 11:05, 1 November 2021 (UTC)

• Support. It is better to separate them than to mix them. -- Hexirp (talk) 11:27, 1 November 2021 (UTC)
• I agree. In addition, I follow Fish' suggestion here to insert {{cite}} for each description in this article which is not followed by a link to a source. (But what is the scroll bar suddenly appearing on the right side of this page?)
• p-adic 12:37, 1 November 2021 (UTC)
• I agree. The quoted comment contains two alternatives: separating the acknowledgements to its own section or keep them brief. I am fine with taking either direction. --Naruyoko (Talk) 22:09, 1 November 2021 (UTC)

## Amiko hydra

The first problem of Amiko hydra is that the site https://komiamiko.me/math/ordinals/2020/06/07/higher-order-hydras.html is not referred, which could be corrected easily. But still, should we put it in the same page of Buchholz hydra? I think making an independent page is better. 🐟 Fish fish fish ... 🐠 14:18, 1 November 2021 (UTC)

I suggest to combine this topic to #Inappropriate citation, because I proposed there to delete the section if nobody disagrees. The current descriptions are highly doubtful, because they are not appropriately followed by citations.
p-adic 14:25, 1 November 2021 (UTC)
I didn't notice that it was proposed a month ago. So I delete the section based on the proposal. 🐟 Fish fish fish ... 🐠 14:38, 1 November 2021 (UTC)
Then you can now see how I put the template.
1. The section was added on 10/2.
2. Right after 1, I added the template because it included so many unsourced descriptions. (I was sincerely costed time to check descriptions in the version before adding the section, but the additional massive unsourced descriptins in the new section made me giving up to continue it.)
3. Right after 2, I proposed the deletion if nobody precisely added citations.
By the way, some discord user (according to C7X) asks users in this community to remove such warnings so many times. Do you know who is repeating this? Although I am not confident that this discussion is suggested by the same user, if it is true, then it is really dishonest for the user to repeat it without clarifying who the user is. Indeed, hiding who the user is causes a problem that the repeated discussions look like new discussions which arose from distinct people. It is actually problematic, because the user can eternally repeats the same discussion until everything goes as the user desires.
So, I would like to ask you and Naruyoko "Did this discussion arise personally from Naruyoko without any connection to discord?" Or from discord gain? I wonder how many times we should cost time to give the same discussions on "hiding the history of fakes". Although the user might be the one who spread the fakes, it cannot be a reason why the user is allowed to make others cost time for the same discussion again and again without costing the user's own time itself. I have explained why simple removal is irresponsible and how fakes in this wiki effected googology so many times, and the user should understand what have been already discussed. Generally speaking, it is not good to repeat the same discussion again and again if there is no novel point of views. Therefore I personally felt bad when I saw Naruyoko gave the same statement again. Well, if this discussion has nothing to do with discord (i.e. nobody mentions to this topic), then it is ok. But if it is based on a suggestion from discord, I require not to repeat the same discussion.
p-adic 15:37, 1 November 2021 (UTC)
I understand that discussion about the wiki should take place on wiki. If user A reads discord and suggests something here, that is done with the decision of user A. I do not mention anything which happens in discord in this wiki. Any discord user, who is also gwiki user, can participate on the discussion here. That's as simple as that. 🐟 Fish fish fish ... 🐠 16:03, 1 November 2021 (UTC)
> So, I would like to ask . . .
I happened to make a discussion on this page first because it happened to appear in a discussion on Discord. I made my opinion over these writings for a couple of months on my own. No one directed me or even suggested that I create the discussion. I suggested that the pages be cleaned up by reorganizing the sentences before anyone commented the same. As far as I know, the closest idea was to remove the wrong description altogether. So while the idea of creating discussion may have been partially dependent, I believe my argument was made independently of the opinions on Discord.
Naruyoko (Talk) 22:28, 1 November 2021 (UTC)

@kyodaisuu
> Any discord user, who is also gwiki user, can participate on the discussion here. That's as simple as that.
The point is that it is not sound to repeat the same discussion. Since I am the one who corrected many errors in main space articles, I am the one who should be the most responsible for discussion on hiding the fact of the fakes But if people frequently try to repeat this discussion even after conclusions, it just wastes my time. I do not think that it is sound. I do not insist that they do not have right to participate in discussions here, but at least they should honestly accept the results of past discussions.
@Naruyoko
> I happened to make a discussion on this page first because it happened to appear in a discussion on Discord.
> So while the idea of creating discussion may have been partially dependent, I believe my argument was made independently of the opinions on Discord.
Then as the one who independently decided to repeat this discussion again, could you summarise the conclusion of this discussion that we should not hide the fact of spreading fakes at Googology Wiki:Discussion (including typical but denied opinions "those are off-topic to googology", "those are biased/non-neutral/unfair", "people never believe unsourced information", and so on)? Thank you.
p-adic 23:12, 1 November 2021 (UTC)
I understand that we do not want to repeat same discussion. That is why discussion in talk page in the respective article is preferred, so that future user can easily find the past discussion. If the discussion was on the talk page of another article, then pointing to the discussion can be useful. 🐟 Fish fish fish ... 🐠 02:02, 2 November 2021 (UTC)
Right. Also, it is good to summarise the conclusion of the repeated discussion at Googology Wiki:Discussion, because (you know) there are several people who just repeat to insist the same statements without listening to others or breaking promises to listen to others.
p-adic 02:17, 2 November 2021 (UTC)

Following Fish's proposal, I inserted {{cite}} for each unsourced description. So, the remaining tasks are Naruyoko's revision and Naruyoko's summary of the repeated discussion.

By the way, I have another issue: Should we keep the uncited references? In mathematical journals, it is usually forbidden to list uncited references, because purposeless lists do not help readers to understand the article. If it was listed only for making this article look "sufficiently sourced", it is better to remove it.

p-adic 05:25, 2 November 2021 (UTC)

When creating the topic, I stated that "I will replace the section in the article after 7 days of no unresolved issues." I wish to wait until the period ends before persuing either task to close the discussion with confidence. It is not fair to end it abruptly.
> Should we keep the uncited references?
No, if it provides useful information for the article, then it should either be incorporated or at the very least be used as a further reading, not a reference.
Naruyoko (Talk) 05:44, 2 November 2021 (UTC)
> I wish to wait until the period ends before persuing either task to close the discussion with confidence. It is not fair to end it abruptly.
Well, I think that I have never requested you to finish the discussion here and enjoy the tasks right now. I confirmed (mainly for the sake of you) the remaining two tasks.
p-adic 05:57, 2 November 2021 (UTC)
Okay. Good that it is a common understanding.
Naruyoko (Talk) 06:27, 2 November 2021 (UTC)
I agree to delete uncited references, after waiting for a few days, because it is not clear why they are cited. Actually I have no access to the remaining 2 reference. For giving further reading, showing this result would be better. 🐟 Fish fish fish ... 🐠 06:25, 2 November 2021 (UTC)
So I may update to
* Papers which may be related to Buchholz hydra
* Papers which cite Buchholz (1987)
🐟 Fish fish fish ... 🐠 09:23, 2 November 2021 (UTC)
Done. 🐟 Fish fish fish ... 🐠 09:14, 4 November 2021 (UTC)
Danke.
p-adic 09:33, 4 November 2021 (UTC)

## Citation for strategy

"The sequence of leaves is called a strategy" is tagged as "citation needed". Hamano and Okada (1997) uses the term "strategy" many times, mostly for KP hydra. In p.106, the term "winning strategy" is used for Buchholz hydra. However, the authors change the rule of the Buchholz game slightly. Can we use this paper as the source for the term "strategy"? I think that although there is a slight change in the rule of the game from the original, the term "strategy" is used in general meaning and could be used as a source, but I am not sure how strict we should be in this case. 🐟 Fish fish fish ... 🐠 10:17, 2 November 2021 (UTC)

Couldn't we simply explain what you wrote above and clarify that "Following the terminology, we abuse the term strategy to mean blah-blah"? The actual problem of the current version is that the article claims as if it is a academically common terminology actually in some sources.
p-adic 10:57, 2 November 2021 (UTC)
I see. 🐟 Fish fish fish ... 🐠 11:02, 2 November 2021 (UTC)

## Notation of hydra

Where is the notation of hydra, used in the table of this article, explained? As it is used in the article, we need to write an explanation or cite the explanation, but I'm not sure where it is written. 🐟 Fish fish fish ... 🐠 12:34, 3 November 2021 (UTC)

This article originally used an onion-like notation written by Ikosarakt1 at this edit without a link to the first source.
The current article uses the notation of labeled tree written by Komi Amiko (the author of Komi Amiko's hydra) at this edit without a link to the first source but following the formatting in the attached image, which was originally uploaded by Vel! from the public domain (according to Vel!).
Both edits use their notations as if they are commonly used (because of the lack of a link to the first source and the hypothesis that the information that the image was from the public domain), but it is generally difficult for us to check whether they appear in published works. So, it is reasonable to just clarify that they are added without links to first sources. (In my opinion, Vel!'s uploading the image was a little problematic. If it was derived from the public domain, it was better for Vel! to clarify where to access to obtain it. If the precise location would have been clarified, then we might find the source of the notation.)
p-adic 13:57, 3 November 2021 (UTC)
Hmm... It is also used in this page #Implementation, FB100Z (former Vel!) uses it and Deedlit11 says "Looks right to me.", just after the first version of this page was published. It seems like it is commonly used in this community and people somehow understand the meaning. It is just strange why people can understand something without explanation. 🐟 Fish fish fish ... 🐠 14:14, 3 November 2021 (UTC)
I added a note and wait for someone to explain. 🐟 Fish fish fish ... 🐠 14:27, 3 November 2021 (UTC)
> It is just strange why people can understand something without explanation.
Well, I think that you know the answer, after replacing the phrase "people can understand something" by "people tend to believe that the common definitions of something are perfectly shared in their mind". When people read something intuitive, they tend to react as if they knew the definition even when the actual definition is not given. This phenomenon frequently occurs when people use undefined googological notations like UNOCF, catching function, BEAF, SAN, and so on. Even though those do not have precise dfinitions, people talk as if they fully understood the definitions. Asking the precise definition is not even welcomed for such cases, because experts in some sense are assumed to understand everything. The attitude is a sort of wabisabi if we choose a positive description, although it is not good at all.
p-adic 16:16, 3 November 2021 (UTC)