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## Result on Σ_2-stability

Let $$n\in\omega$$. Let $$\chi_n$$ denote a sentence formalizing "for any $$\Pi_n$$ sentence $$\phi$$, then $$\phi\rightarrow\exists\beta((L_\beta,\in)\models\phi)$$".

Then, let $$\alpha$$ denote the least countable ordinal where $$\exists\xi(\alpha\in\xi\land L_\alpha\prec_{\Sigma_2}L_\xi)$$, and let $$\beta=\textrm{min}\{\xi:L_\alpha\prec_{\Sigma_2}L_\xi\}$$. Applying User:C7X/Stability#3.2 to each formula $$\chi_n$$ yields that for all $$n\in\omega$$, each set of $$\Pi_n$$-reflecting ordinals $$<\beta$$ is unbounded in $$\beta$$. However, this result relies on the given that each set of $$\Pi_n$$-reflecting ordinals $$<\alpha$$ is unbounded in $$\alpha$$, which I believe can be shown from previous results in [RichterAczel 1973, Inductive Definitions and Reflecting Properties of Admissible Ordinals], along with the fact that $$\exists(\alpha')(\alpha\in\alpha'\land L_\alpha\prec_{\Sigma_1}L_{\alpha'})$$ (e.g. this holds when $$\alpha'=\beta$$). Is this a correct result? C7X (talk) 00:00, 1 February 2021 (UTC)

First, you cannot drop "∈On" from "∃ξ∈On" and "min {ξ∈On : ～}". In mathematical papers, we omit "∈On" after clarifying something like "In the following in this paper, ξ always stands for an ordinal".
> Theorem 3,2
There are several errors:
1. The statement is not written appropriately, as α appears outside the scope of the the quantification "∃α".
2. Let T denote the theory on which you are working on, MT denote the meta theory of T, and φ(χ) denote Theorem 3.2 applied to a sentence χ. Then φ(χ) is a formula in MT on provability of a statement in T parametrised by a sentence χ in T. Therefore you cannot bound it in T like "∀n∈ω, (statement using φ(χ_n))". A formula in T cannot include a formura in MT. (Even if you set both of T and MT as ZFC set theory, it does not solve the problem.Indeed, φ(χ) formalised in T is a statement of the provability of a statement in ZFC set theory formalised in T, and hence does not give a property of V in T itself.)
3. If you intend "for each n∈ω" to be the quantification in MT, please distinguish from the quantification in T, because it is quite ambiguous. In order to do so, please clarify a meta theory, and write "Let n be a meta natural number" or something like that. Mixing the two layer should be avoided clearly.
In addition, χ_n is always false for any n≧1, as the Π_n sentence ∀x(x≠x) cannot be true in L_β. (I assumed the convention in which a model cannot be empty. If you assume the convention in which a model can be empty, then please replace "n≧1" by "n>1" and "∀x(x≠x)" by "∃x(x≠x)". Could you tell me which convention you are assuming?) Therefore if you apply φ(χ_2), which is valid because I am not quantifying the subscript of χ_n it in T and hence χ_2 can be regarded as a formula in T rather than the Goedel number of a formula in a ZFC set theory formalised in T, then Theorem 3.2 says nothing, as L_α|=χ_2 is false. Therefore your conclusion, which I cannot understand how you derived because of the errors in Theorem 3.2, does not follow from φ(χ_2).
p-adic 01:13, 1 February 2021 (UTC)
This is what I understand so far, and some of what I've done, along with fixing the ∈On:
1. I tried to fix the quantification here
2. I don't understand the meta-theory well. If instead of accepting a formula $$\chi$$, $$\phi$$ accepts a Godel-number of a formula $$\ulcorner\chi\urcorner$$ (denote it as $$k_n$$ and define it in T), and the satisfaction predicate in the statement of Theorem 3.2 is changed to accept Godel-codings in T, is the problem solved?
3. Similarly to the above, am I able to let $$k_n$$ denote a Godel-number characterized by a natural number $$n$$, and define this in T?
I don't understand why the Π_n sentence ∀x(x≠x) cannot be true in L_β. Also I will allow models to be empty C7X (talk) 02:22, 1 February 2021 (UTC)

C7X (talk) 02:22, 1 February 2021 (UTC)

> I tried to fix the quantification here
Now you are doubly quantifying α. (You first quantified it by "Let α be any ordinal", and secondly quantified it by "∃α".)
> is the problem solved?
No, because {η∈On|χ^η} is ill-defined when χ is just given as a Goedel number. If you want to regard {η∈On|χ^η} as a valid expression of a term in T, χ should be a formula in T instead of a Goedel number in T. You need to formalise {η∈On|χ^η}.
> Similarly to the above
You can, as long as you distinguish T and MT. The point is that "n∈ω" in your context is ambiguous. Could you clarify whether n is a natural number in T or not?
> I don't understand why the Π_n sentence ∀x(x≠x) cannot be true in L_β.
If you allow a model to be empty, then it is true in L_0 = ∅. As I clarified, if you allow a model to be empty, please replace replace "n≧1" by "n>1" and "∀x(x≠x)" by "∃x(x≠x)".
p-adic 03:01, 1 February 2021 (UTC)
OK (I forgot to delete the ∃α)
1. I need to fix this, and I will try again soon
2. n is in T, as well as each $$k_n$$. I don't understand meta-theories vs. theories well, so I don't know why the theory that n is quantified in matters in this context
3. > it is true in L_0 = ∅
True, although I thought that the right-hand side of the implication (i.e. $$\exists\beta((L_\beta,\in)\models\phi)$$) would be false at $$L_0=\varnothing$$, so overall $$L_0=\varnothing$$ wouldn't satisfy $$\chi_n$$ for $$n\ge 1$$ C7X (talk) 06:33, 1 February 2021 (UTC)
> 2
If n is a natural number in T, the statement "φ(χ_n)" does not make sense, because "φ(χ)" is defined only for a formula χ in T, which cannot be parametrised by a term in T such as n.
> 3
I meant (L_0,∈)|=∀x(x≠x), but not (L_0,∈)|=(∃β∈On((L_β,∈)|=∀x(x≠x))). My logic was that if you did not allow a model to be empty, then "for any Π_n formula φ, ∃β((L_β,∈)|=φ)" would be false because no β satisfying L_β≠∅ and (L_β,∈)|=∀x(x≠x). However, since you allow a model to be empty, this counterexample does not work, as (L_0,∈)|=∀x(x≠x) holds. I mentioned that in that case, if n>1, then "for any Π_n formula φ, ∃β((L_β,∈)|=φ)" is false, too, because no β∈On satisfies (L_β,∈)|=∃x(x≠x).
p-adic 07:42, 1 February 2021 (UTC)
EDIT: Oops, I did not notice that there is "φ→" before "∃β(～)". Therefore please do not mind 3.
p-adic 07:45, 1 February 2021 (UTC)
If χ is a formula in T, then is it an object defined in MT? I guessed that "χ is a formula in T" meant "χ is defined in T" C7X (talk) 16:36, 1 February 2021 (UTC)
Sorry for the ambiguity. I meant a formula in the language of T, which is a term (a Goedel number) in MT.
p-adic 23:05, 1 February 2021 (UTC)
OK. I will try to define a predicate defining my intention of $$(\ulcorner\chi\urcorner)^x$$ for a set $$x$$, however because I've never defined truth predicates before and only have some knowledge of Tarski's T-schema, it will likely have errors C7X (talk) 02:36, 2 February 2021 (UTC)
Isn't it good to use (x,∈)|=χ instead of the relativisation? In many cases, the satisfaction plays a role similar to the relativisation. One problem is that an ordinal η rarely satisfies a theorem of ZFC set theory, as η is not a model of a set theory. (Even the axiom pair does not hold.)
p-adic 02:49, 2 February 2021 (UTC)
AFAIK (x,∈)|=χ is no longer necessarily Δ0, so then ∃γ∀δ(((x,∈)|=χ)&rightarrow;δ∈γ) isn't necessarily Σ2, and this prevents the theorem from being proven. C7X (talk) 03:33, 2 February 2021 (UTC)
I see. (I did not read further, because I stopped at the point which you had not formalised.) But why do you consider χ^η? The relativisation of χ at η has few information on the satisfaction at segments of L.
p-adic 04:02, 2 February 2021 (UTC)
Some past results have characterized properties of a set by truth values of formulae relativized to them. For example, [RichterAczel, "Inductive Definitions and Reflecting Properties of Admissible Ordinals" (1973) (Theorem 2.4)] asserts existence of a $$\Pi_3$$-formula $$\sigma_0$$ such that $$x\vDash\sigma_0$$ implies admissibility of $$x$$, and then [RichterAczel, "Inductive Definitions and Reflecting Properties of Admissible Ordinals" (1973) (p.45)] applies Theorem 2.4, except using the relativization $$(\sigma_0)^x$$ instead (the relativization of $$\sigma_0$$ is done by letting $$\varphi_1$$ be a separate formula that is a conjunction including $$\sigma_0$$, then relativizing $$\varphi_1$$ with respect to $$x$$), implying that the characterization also holds when restated in terms of relativization. Also I was under the impression that usually the truth values of $$x\vDash\chi$$ and $$\chi^x$$ coincide, with pathological exceptions such as a very weak coded theory, etc. C7X (talk) 04:20, 2 February 2021 (UTC)
I am still confused. Are you confounding χ^η with χ^{L_η} or something like that? You are using χ^η, and the point is χ^η is irrelevant to reflection principle for L.
p-adic 05:45, 2 February 2021 (UTC)
I did, I apologize and will try to sort it out. Also I might remove the usage of the class "{η:χLη}" and instead use a different formulation C7X (talk) 18:56, 2 February 2021 (UTC)
I see.
p-adic 22:49, 2 February 2021 (UTC)
There should no longer be χη where η is (explicitly) an ordinal (in the case that η∈2, then Lη=η is an ordinal) C7X (talk) 05:23, 3 February 2021 (UTC)

There are still issues on "∈On". For example, "∃(β>α)" and "∃γ∀δ" should be appropriately bounded by "∈On". Otherwise, the L-hierarchy is ill-defined for them. Also, the statement "χ^{L_β} is Δ_0" is quite ambiguous, as you have not specified the language. Are you considering the language of first order set theory with parameters from V? Or are you considering a parameter-free language? If you allow parameters from V, then L_β is not a structure for the language, and hence "L_β|=φ" does not make sense.

If you do not allow parameters, then you do not have the existence of L_δ in V^{L_β} = L_β, and hence the L_β|=φ is quite ambiguous. (Since L_β might not satisfy sufficiently strong segment of ZFC, the notion of L in L_β is quite ambiguous.) Even if I assume the existence of L_δ in L_β, I do not know whether χ^{L_δ} is actually Δ_0 or not. Since L_δ is not in the language, χ^{L_δ} should be the abbreviation of something like "∀x((x satisfies the defining formula of L_δ) → χ^x)" or "∃x((x satisfies the defining formula of L_δ) → χ^x)". However, they include an unbounded quantifier. What is the definition of χ^{L_β}?

Also, you have never formalised {η|χ^{L_η}} for a Goedel number χ in T, as the convention is just defined for a formula in the language of T (which is a Goedel number in MT instead of T).

p-adic 06:07, 3 February 2021 (UTC)

> I do not know whether χ^{L_δ} is actually Δ_0 or not.
You're right, I think it is Σ1. So ∃γ∀δ(χLδ→δ∈γ) is not Σ2 like I thought, so I'm not sure how to prove the theorem, or if the theorem even holds C7X (talk) 03:13, 4 February 2021 (UTC)
You mean that you have an explicit Σ_1 formalisation of L_δ, right? (Since L_β does not necessarily satisfy segments of ZFC which is used in the definition of L hierarchy, the quantifier rank makes sense only when you fix a formalisation of L_δ.)
p-adic 03:22, 4 February 2021 (UTC)
The $$\Sigma_1$$-definability of Lδ is mentioned in p.61 of [Barwise, Admissible Sets and Structures], although in KPU, however I don't understand the definition given by Barwise. Several Discord users have claimed that it's $$\Sigma_1$$-definable as well C7X (talk) 03:58, 4 February 2021 (UTC)
But you are referring to the definability under ZFC, aren't you? As I wrote several times, L_β is not necessarily a model of a sufficiently strong segment of ZFC which is used in the definition of L.
p-adic 05:17, 4 February 2021 (UTC)
I guess there are these possibilities (I hadn't thought about most of these until your explanation, thank you):
1. $$L_\beta$$ is a model of that fragment of ZFC, and $$L_\gamma$$ (with $$\gamma\in\beta$$) is $$\Sigma_1$$-definable, and T can prove that $$L_\gamma$$ is $$\Sigma_1$$-definable
2. $$L_\beta$$ is not a model of that fragment of ZFC, and $$L_\gamma$$ (with $$\gamma\in\beta$$) is $$\Sigma_1$$-definable, and T can't prove that $$L_\gamma$$ is $$\Sigma_1$$-definable
3. $$L_\beta$$ is not a model of that fragment of ZFC, and $$L_\gamma$$ (with $$\gamma\in\beta$$) is not $$\Sigma_1$$-definable, and T can't prove that $$L_\gamma$$ is $$\Sigma_1$$-definable
4. $$L_\beta$$ is not a model of that fragment of ZFC, and $$L_\gamma$$ (with $$\gamma\in\beta$$) is not definable in it
If none except #4 are plausible for (arbitrary) admissible $$\beta$$, then it could be definitely not considered to be Σ1. I think that if L is Σ1-definable in x under KPU, then it would also be Σ1-definable in x under ZFC for set x, but I don't know if Lβ=x has the left-hand side of the implication hold. C7X (talk) 17:11, 4 February 2021 (UTC)
Edit: I will try to prove a different theorem instead, avoiding Lβ and using Rathjen-like reflection properties
> I think that if L is Σ1-definable in x under KPU, then it would also be Σ1-definable in x under ZFC for set x,
What does "L is Σ_1-definable in x under T" mean?
p-adic 05:02, 5 February 2021 (UTC)
$$x$$ satisfies the statement "$$L_\gamma$$ is defined by a $$\Sigma_1$$-formula in $$T$$" (I meant to say $$L_\gamma$$ instead of $$L$$) C7X (talk) 05:15, 5 February 2021 (UTC)
I do not understand what you mean. L_γ is not a subset of x. Are you confounding the definability of a subset with the definabilty of a definable class? The issue in your formulation was that given a definable set X in ZFC set theory with parameters from V, the relativisation "X^x" is ill-defined unless x includes the parameters in the definition of X and x satisfies a sufficiently strong segment of ZFC which ensures the well-definedness of X. Since you are assuming that X is already well-defined, it does not solve the issue.
For example, let φ(a,y) be a Σ_1-defining formula of L_a in ZFC set theory with a parameter a∈V. Then L_a^x is ill-defined if x does not satsisfy ∃!y(φ(a,y)). If x satisfies a sufficiently strong segment of ZFC which imply ∃!y(φ(a,y)), then you can define L_a^x by the defining formula φ(a,y)^x.
p-adic 06:30, 5 February 2021 (UTC)
> L_γ is not a subset of x.
I should have quantified x, is it true that "L_γ isn't necessarily a subset of x"? Also if I can ask about convention in math, why aren't contractions such as "isn't" allowed in papers (StackExchange had differing answers)?
> Then L_a^x is ill-defined if x does not satsisfy ∃!y(φ(a,y)).
I see. Also I think there exists such x where x does not satisfy ∃!y(φ(a,y)), but the segment of ZFC can't prove that "x does not satisfy ∃!y(φ(a,y))". C7X (talk) 19:58, 5 February 2021 (UTC)
> I should have quantified x, is it true that "L_γ isn't necessarily a subset of x"?
Right. Sorry for the ambiguity.
> why aren't contractions such as "isn't" allowed in papers (StackExchange had differing answers)?
I do not know, because I just have taught so. (As I clarified, mathematicians sometimes ignore the restrictions.) It might be partially because the abbreviations are informal. If we freely allow abbreviations in papers, a sentence will be like "The top spc X isn't loc cpt at the base pt, so a cont fct mightn't be bdd", because mathematicians prefer abbreviations very much when we write sentences outside paper.
> Also I think there exists such x where x does not satisfy ∃!y(φ(a,y)), but the segment of ZFC can't prove that "x does not satisfy ∃!y(φ(a,y))"-->.
Right. Sorry for the ambiguity.
By the way, I created a guideline page following the proposal at the talk page of the policy. Could you check/correct it? If it is ok, I will add a link to it to the policy. Also, I appreciate if you add a link to it to the main page, which is protected.
p-adic 22:49, 5 February 2021 (UTC)

It seems OK C7X (talk) 23:24, 8 February 2021 (UTC)

Thank you. Then could you add a link to the guideline to the main page, which is protected?
p-adic 23:58, 8 February 2021 (UTC)
OK C7X (talk) 00:04, 9 February 2021 (UTC)
Thank you.
p-adic 00:34, 9 February 2021 (UTC)
No problem. Also, something I don't understand about formal languages is the unions of them with other sets. For example, here on page 16, Arai mentions the language "$$\mathcal L_0\cup\{St\}$$" where $$\mathcal L_0$$ is a first-order language and $$St$$ a predicate. Does this mean that in Arai's convention a language is a set of predicates appearing in wffs, or is this an informal notion? I was under the impression (from this source) that a formal language had to also contain a formal grammar, and is different than a set of predicates C7X (talk) 00:42, 9 February 2021 (UTC)
The convention of a formal language depends on an author. It is roughly a set of formal strings equipped with a (recursiv) way, i.e. a formal syntax, to classify elements as "constant term symbol (i.e. symbols for constants)", "relation symbols (i.e. symbols for predicates)", and "function symbols (i.e. symbols for functions)" togerther with a way to determine their arities. For example, you can define a language L as the singleton {"∈"}of a formal string equipped with the algorithm "Given a formal string s in L, s is a relation symbol of arity 2". Namely, it is a formal language consisting of (a symbol for) a predicate of arity 2.
In addition, there are natural ways to enlarge a formal language. One typical way is to use parameters. Given a language L and an L-structure M, we frequently "add" M to L. More precisely, we consider L_M := L ∪ {c_m|m∈M}. For each m∈M, c_m (defined in any reasonable way, e.g. (L,m) or something like that) is a new constant term symbol. Then M again forms an L_M-structure, as the interpretation of L can be extended by setting the interpretation of c_m as m. Similarly, you can add predicates and functions on an L-structure.
p-adic 03:27, 9 February 2021 (UTC)

## Talk pages

In the case that a user doesn't have an alert or popup when a message is left on their talk page, it could be possible that a very new user never checks it or doesn't know that talk pages exist, resulting in continued multiple good-faith violations. Should policy reflect this to ensure we inform the user of rules, and if so, how should warnings be left? C7X (talk) 02:32, 22 February 2021 (UTC)

Exactly. Therefore I usually try to ask a new user to stop violation and to check rules at both of the summary box and the use's talk page without using Template:Warning. (For example, I prepared Template:Unsourced for this purpose.) But if any notifications at the summary boxed and the talk pages are checked, then it is unreasonable to believe good faith, because the user is continuing to ignore actual rules. (Generally speaking, website users are required to check rules. If they intentionally skip to check rules and any notifications, then it is not good faith.)
p-adic 03:37, 22 February 2021 (UTC)
OK C7X (talk) 04:30, 22 February 2021 (UTC)

## Unrelated blog posts

A user has given concern that this blog post and some others by the same user are unrelated to googology, so they shouldn't be posted here, or deleted. Should there be a policy on enforcing relevancy of blog posts? (Unless I misunderstand, the current policy only talks about counting blogs) C7X (talk) 18:53, 23 February 2021 (UTC) Also I recommend archiving your talk page so far

> Should there be a policy on enforcing relevancy of blog posts?
It is good to have such a rule in order to keep the soundness of this community. I prefer that we are allowed to create a blog post on general mathematics, because it will help us to invent new googology. (For example, pure study on large cardinals which have never appeared googology might be currently irrelevant to googology itself, but can be a part of googology in the future)
> archiving
OK.
By the way, I ask you to check the last proposal in the policy and a related issue. I would like to hear your opinion.
p-adic 23:10, 23 February 2021 (UTC)
OK C7X (talk) 23:51, 23 February 2021 (UTC)

## My web book

(Continuation of User talk:P進大好きbot/Archive1#My web book on )

1. The title contains Ordinal Analysis, but since it is such an advanced topic, it must be placed somewhere around Chapter 20 in the book.

3.1~4. Fixed.

I also clarified the definition of a hypercube.

3.5 I got rid of it :)

Also please refrain from using complex blog posts to solve simple problems such as clarifying the domain, not everybody is an expert lol :)

Zongshu Wu 10:19, 27 February 2021 (UTC)

> Also please refrain from using complex blog posts to solve simple problems such as clarifying the domain, not everybody is an expert lol :)
If it were simple, few members here repeat the same failure. Could you imagine how many times I repeated to explain to members here "Please add the precise definition of the domain" but they reply irrelevant answers like "The output is an integer!", "a, b, and c are integers (but the tuple (a,b,c) is assumed to satisfy unwritten secret conditions!)", and so on? I am not referring to my blog posts under an assumption that you are a professional mathematician.
Well, if you feel that the issue on the domain is simple, please do not repeat it. Anyway, I try Chpater 1.3.
1. "pseudonym": I do not know whether hyp cos is pseudonym or not.
2. "sloppy mathematician": Is he a mathematician?
3. "However, it is very easy to clarify things up and make it well-defined": If I correctly remember, you tried so hard to fix the issues with my help. Then it is unreasonable to conclude that it is easy, isn't it?
4. "The number before the "R" is known as the base, and the stuff after the "R" is known as the recursive part.": It might be better to replace the "known as" by "called".
5. "When a {1} appears right after the R, it splits into n copies of {0}, where n is the base": It imples 2R1{0} = 2R{0}{0}{0}, which contradicts what you wrote.
6. "If no rules can be applied, then the expression is illegal.": It is good to define precisely when an expression is legal.
7. "1.2: 2R{0}{0}{2}R0{0}{2}": It should be "1.2: 2R{0}{0}{2}R0{1}{2}".
p-adic 14:16, 27 February 2021 (UTC)

1. Hyp cos is definitely not a real name. What are the chances that someone's first name is "Hyp" and last name is "Cos"?!

5. In 2R1{1}, the {1} is not right after the R.

All the rest are fixed. I also tried to replace "legal expression" with "legal recursive part", and make lots of other changes.Zongshu Wu 13:37, 28 February 2021 (UTC)

> 5
Oops. Sorry. I missed the {}.
> All the rest are fixed.
OK. I will check them later.
p-adic 14:17, 28 February 2021 (UTC)

Would you please check as soon as possible? I need the first two chapters to be done as soon as possible :) Zongshu Wu 13:50, 1 March 2021 (UTC)

Sorry, but I am busy these days. Please do not expect that I will do "as soon as possible".
p-adic 15:36, 1 March 2021 (UTC)

So.. is Chapter 1 finished, or does it still contain unformalized stuff? Zongshu Wu 07:19, 8 March 2021 (UTC)

Sorry, but I am still busy... I have not checked it, and I cannot promise that I will read it in a week. Well, since I have adviced you so many times, I guess that you can now judge by yourself whether you wrote an unformalised stuff or not. We agree that you know how to define notations, don't we?
p-adic 08:32, 8 March 2021 (UTC)

So, first I need to clearly state the domain. Then, I need to properly define all terms I mean to use in the definition of the notation. Third, I must avoid ambiguity (such as referring to just "1" when there could be multiple 1's). Lastly, I must make sure that there are no mistakes that prevent the notation from being evaluated properly. Am I right? Zongshu Wu 11:01, 8 March 2021 (UTC)

Right. In order to do what you said, you need to avoid several traps explained here. For this purpose, please check this guideline. Especially, people tend to drop quantifiers ("for any ～" and "there exists ～") and to use undefined symbol like "(a,b+1,@) = (a,b,…,b,@) with n b's (without defining n)".
p-adic 12:06, 8 March 2021 (UTC)

I asked if someone could teach me set theory symbols and they said to contact a professional like you. Can you teach me?

It is good for you to know that it is bad to command others when you ask something.
p-adic 23:27, 7 March 2021 (UTC)

## Set theory

I don't understand how set theory can make a really large number like Rayo's number. LuckyLuxius (talk) 15:57, 10 March 2021 (UTC)

To my knowledge first we need to fix a langauge like $$\mathscr L_\in$$ (think similar to "Rayo's language"of $$\exists,\in,\textrm{etc.}$$ informally, but it's more sophisticated in terms of a formal definition). Then we need to fix a truth predicate. Because of Tarski's undefinability theorem, if we try to make a truth predicate using universe we're working in as the model, we get paradoxes. So we need a smaller set to work in, called the model. We also "encode" the nature of truth itself into a truth predicate, which can be defined using something like Tarski's T-schema.
This next bit is sort of tautological. We need the definition of "definition" in math. PBot has made an explanation of this, see the section on definitions near the top of "list of common mistakes on formal logic". For example, if we need to define what "5" means, we can define it using a formula $$\Phi(x)$$ that's true if and only if $$x=4+1$$, for example (formally we do this using von Neumann ordinals and a function such as $$\lambda a.a\cup\{a\}$$, but I've included this for formality). If we need to define what "Turing machine" means, we can input the tuple of functions, sets of strings, etc. that characterize a Turing machine.
So since the model's language (usually a limitation of $$\mathscr L_\in$$) is so expressive at defining things, we can define all sorts of things within this smaller model of set theory. For example, EmK made a blog post defining the usy Beaver function in around 10,000 symbols. Even more powerful concepts like oracle TMs, ITTMs, etc., and even things that haven't been discovered yet, are likely definable within $$10^{100}$$ symbols, so Rayo's function will begin to overtake functions based on those, as it can utilize their definition in its set of numbers definable in $$<10^{100}$$ symbols.
I'm not an expert though, PBot knows more accurately than me about this C7X (talk) 17:20, 10 March 2021 (UTC)
@XiTetrateiX
If you want to know a theoretic framework, please read C7X's explanation above.
@C7X
Thank you. One point to correct: Rayo has updated his webpage to clarify that he is not using a specific model. Therefore he is not using a satisfaction at a model. (Also, using a model cannot be a reasonable solution as the original definition does not exlpain how to fix a model. I explained the ill-definedness of the first order alternative definition using of a model here.) Rayo's original explanation does not give a valid definition in terms of mathematics, because Rayo clarified (in the update) that he used a philosophic hypothesis which is unformalised in mathematics. One reasonable alternative formulatio, which is partially accepted by googologists who knew second order sematincs, is to define the truth predicate on the universe in second order logic, by fixing a sufficiently strong second order axiom. (The reason why Rayo updated the website is perhaps due to the issue pointed by me: Rayo have never clarified a second order axiom which justifies the formalisability of the truth predicate.) I note that Tarski's paradox is not applicable here, because the universe itself is not the domain of the second order logic. The formalisation works even if we use first order class theory or other first order theory instead of second order set theory. Indeed, emplighted tried (but failed) to formalise the truth predicate in the definition of Geddons, and I extended the truth predicate in MK for the universe to that in a stronger set theory MK_+ by restricting the length of formulae here.
p-adic 23:12, 10 March 2021 (UTC)

## Cheatsheet on properties of OCFs

I was thinking about writing a draft for the "Rathjen's weakly compact Ψ" section to save time, although since some of what I assumed I had known about the OCF turned out to be misunderstandings, I'm not sure if it would end up taking more time fixing my draft's errors than writing it from scratch C7X (talk) 22:53, 20 March 2021 (UTC)

Right, writing a cheatsheet costs much more time than other guess. It requires to study correctly by reading the corresponding paper with understanding proofs instead of guessing "what should be a correct behaviour". Even for Buchholz's OCF, it is true. People just worked on it with the reasonless "what it should be a correct behaviour", and spread wrong informations. Although those common misconceptions might be currently considered as obvious fakes, but it was not the case before I pointed out them. Indeed, when I pointed out them, all reactions by "experts" told me that I was misunderstanding OCFs, even though I repeated to explain true sttaements precisely citing theorem numbers in Buchholz's paper. In this way, people strongly believed their intuition rather than actual mathematical papers. In order not to repeat the awful failure, which perhaps delayed googology by years,
p-adic 23:38, 20 March 2021 (UTC)

## Hey

Can you analyze My $$\theta$$

Sure. It is ill-defined because of the ambiguous use of ellipses, the lack of the definitions of the domains, the lack of the quantifications and the range of variables. Please study what a function means. Then please check "General setting" section in this introduction, because your failures are listed in the post.
p-adic 23:54, 23 March 2021 (UTC)

## Well-definedness of "denoted on paper"

Define a set of formal strings $$T$$ in the following way:

• For all ordinals $$\alpha$$, if there exists a formal string $$a$$ of symbols of the way $$\alpha$$ is denoted on paper, then $$a\in T$$

Is $$T$$ well-defined?

In the event that shorthands are used:

A problem I thought of is that there is no formalization in mathematical logic of "$$a$$ denotes $$\alpha$$ on paper". However, could each string that denotes an ordinal on paper, such as $$\varepsilon_0\!"$$, be considered a shorthand of a term in a very large ordinal notation encompassing every ordinal ever denoted on paper, or a similar scheme? Or does mathematical convention forbid defining transfinitely many shorthands?

C7X (talk) 01:25, 30 March 2021 (UTC)

Of course, it is ill-defined, as there is no mathematical formalisation of the definition. The well-definedness makes sense only when we have a defintiion, which should be a mathematical formula by the definition of a "definition".
Also, we are allowed to introduce transfinitely many shorthand, as long as it is precisely defined in terms of transfinite induction. (It is justified by the formalisability theorem of the transfinite inductive definition, which is non-trivial.)
p-adic 13:05, 30 March 2021 (UTC)

## Hello!

Are you interested at the Extensible-E System, hyper-operators, and BEAF? I would love all of those things. —Preceding unsigned comment added by Stevenalexwcr1 (talkcontribs)

p-adic 04:16, 8 April 2021 (UTC)

## Hi!

Hey there. Please view the source carefully before editing. No offense, this is just a reminder. You have to find it before editing. Also i may cause an edit war with you, because you sdone that. But i dont want to break rules. I know how to cite source. Thank you. Hypertetrakulus44 (talk)

What source are you talking about?　I reverted your edit because you broke the rule for the citation, which is strictly given in the policy page. If you know that it is insufficient just to add a link, why did you do it? It cannot be an edit war, because I clarified that you violated the current policy. I did not put warning because I guessed that you did not check the policy on how to precisely cite a source. But if you insist that you have already checked it and are intentionally breaking a rule, it is not the case. Have you read the policy?
p-adic 09:35, 24 April 2021 (UTC)

Uhh, P-adic, the source was valid, i saw it on Cookiefonster's site. If you finded it in [1], you would already know.

Why don't you read what I wrote above? I clarified that the way you cite a source, i.e. just putting a link, is wrong. Please read Googology Wiki:Policy#How to cite a source before strongly believing that you are correct. It is really rude to insist your correctness by ignoring the requirement to check the policy.
p-adic 04:46, 25 April 2021 (UTC)

## Hey there.

Hi there P-adic. I just needed a redirect for corporatrix. Please move the corporatrix page I just created and please leave a redirect. Thank you. 2x2 guy 23:56, 12 May 2021 (UTC)

Hello. I think that you are requesting two things: (1) Moving the contents of corporatrix to some page, and (2) replacing the contens of corporatrix by a redirect to that page. But the policy does not allow us to create an article without a valid source. Therefore if we moved it to some page, then the resulting page violates the policy due to the lack of a valid source. Am I missing a point? Also, please tell me what page do you want me to move corporatrix to? Thank you.
p-adic 00:40, 13 May 2021 (UTC)

## Growth rate of my notation

Hi, can you tell me how fast my notation grows? (here) DRC 15:20, 27 May 2021 (UTC)

Even when the domain is clarified, the definition unfortunately seems to miss some cases, such as B(a\<b,b,b>) C7X (talk) 16:30, 27 May 2021 (UTC)
Oops, fixed it. B(a\<b,c,d>) = B(a\<b,B(a\<b,...B(a\<b,c>)...>)>) DRC 17:23, 27 May 2021 (UTC)
OK. Before that, please check the well-definedness by yourself first. At least, could you check "General setting" section in this guide line to ensure the non-existence of obvious errors?
p-adic 23:06, 27 May 2021 (UTC)

I seem to distinctly remember you saying that the regiment project isn't possible because the pages' creators own each page's rights, but can't find the link. Is this the copyright interpretation this wiki uses? C7X (talk) 20:57, 22 June 2021 (UTC)

What I said is that a regiment project (usually causinging removals and copies of many articles) is not allowed unless we discuss it. That was a problem, because it started without permissions by copyrigt holders. On the other hand, if we have discussed it in this wiki and nobody disagrees with the project, there is no problem. Here, copyright holders can insist that they do not permit the usual execution of a regiment project. In that case, we need to consider an alterative way without violating their copyright. For example, we will not remove the articles, and just replace them by redirects without moving. (Then the history page is kept, and hence we do not violate the copyright.) Then we will copy (if necessary) the contents of the redirected articles and paste them to the target of the redirects with the credit exoression like "Copied from (the original page title)" at the summary box. I note that even if we agreed some regiment project, it does not mean that we agree all other regiment projects. Therefore such decisions do not violate copyrights of future users.
p-adic 23:08, 22 June 2021 (UTC)
I still don't understand why copyright restrictions on a page created to describe someone else's googolism are either so strict, or enforced quite strictly. On GWiki, if the credit to the page's creator is the issue, wouldn't it be possible to add comments to the regiment's page with the username of the page's creator, avoiding the need for thousands of redirects?
Or if permission is the issue (at least mostly unless Denis agrees), users on other Wikimedia sites are often allowed to heavily modify (e.g. move, delete, etc.) other users' pages despite copyright, so I don't think Wikimedia's copyright is so strict for text (or in my example, article) content. There are deleted pages on Wikipedia whose creators may own copyright on them, that are deleted (almost certainly) without permission, although usually for breaking a rule. Here is an example of one such non-vandalism page. C7X (talk) 00:42, 23 June 2021 (UTC)
Both of credit and permission are the issues.
> On GWiki, if the credit to the page's creator is the issue, wouldn't it be possible to add comments to the regiment's page with the username of the page's creator, avoiding the need for thousands of redirects?
Yes, you can clarify the credit by adding comments. (One more accurate way is to clarify that "those contents are copied from the individual articles for the numbers.")
> users on other Wikimedia sites are often allowed to heavily modify (e.g. move, delete, etc.) other users' pages despite copyright,
In wikis, we have history pages. We have copyright on what we wrote. Even if someone changed the content, it does not reflect the copyright of the past version. The only exceptions are moving and deleting. For example, please consider wikipedia. (I expect that you are not referring to quite specific young communities like GFE, because such violations are due to careless managements.)
In wikipedia, it is not allowed to move or delete a page in a way ignoring the credit or violating copyright, as is clarified here and here. You can see explicit rules on the reuse of wikipedia contents here.
As I have proposed several times, we can have a disclaimer to ask users to agree on a helpful promise: "When we decide to start a regiment project, please allow movings without leaving copyright notice and credit clarification of the moved contents. Of course, if you can ask admins to rollback the moving when you find something unintentional. This is just a conventional promise to confortably manage this wiki."
I think that this proposal is reasonable, but somewhy nobody has agreed with it, because of wrong logics like "We do not have copyright for articles in this wiki!" especially by the one who really hates copyright, who we know well. That is why this rule has never been validated. So, this is due to the one who hates copyright.
> that are deleted (almost certainly) without permission, although usually for breaking a rule.
They have an explicit rule for speedy deletion, as is written here.
> Here is an example of one such non-vandalism page.
Although I do not know whether it is valid or invalid, users can make mistakes. Namely, even if rules forbid something, users might do something. It does not mean that rules have been invalidated. Indeed, users can discuss the validity of the deletion.
p-adic 01:21, 23 June 2021 (UTC)
You're right about edit history, Wikipedia does explicitly require copyright as you pointed out. I suppose the connection to CC-BY-SA (although so far I've only seen a copy of the 2.0 license, I'll take your word for it) is if the page could be considered to have multiple authors, all need to be visible, hence the requirement of a label such as an HTML comment. Or could merging edit history be more feasible? C7X (talk) 01:33, 23 June 2021 (UTC)
> I suppose the connection to CC-BY-SA (although so far I've only seen a copy of the 2.0 license, I'll take your word for it) is if the page could be considered to have multiple authors, all need to be visible, hence the requirement of a label such as an HTML comment.
> Or could merging edit history be more feasible?
In a big project like a regiment project, I think that it is better to choose as easy a way as possible. We have several easier choices than writing HTML comments on the credit or merging edit history:
1. The one which I wrote above, i.e. having a policy to ask users to agree with the promise "When we decide to start a regiment project, please allow movings without leaving copyright notice and credit clarification of the moved contents. Of course, if you can ask admins to rollback the moving when you find something unintentional. This is just a conventional promise to confortably manage this wiki." In that case, users automatically agree with the usual way of regiment projects, i.e. moving and copying contents without adding new credit notifications.
2. Having a policy to ask users to agree with the promise ""When we decide to start a regiment project, please accept the way to express the credit by writing "Contents on this article are copied from the articles on indivisual works". This is just a conventional promise to confortably manage this wiki."" In that case, users automatically agree with the modified way of regiment projects, i.e. redirecting (without deletion) and copying contents without adding new credit notifications.
But it is good to ask ARsygo's opinion rather than mine, because ARsygo is working much harder than me on managing articles. After then, we can discuss whether ARsygo's suggestion is valid or not. Of course, I respect your opinion if you have easier solutions.
p-adic 01:50, 23 June 2021 (UTC)

I don't think facts or information (such as the FGH approximation of a given number) or the basic structure of a page (such as the structure of regiment articles) can be copyrighted, as copyright covers expression, not ideas.

Also, having thousands of redirects is actually a good thing, as not only they preserve attribution and history, but also allows internal and external links to those pages still work, by redirecting them to the regiment articles where the number can be found. -- ☁ I want more clouds! ⛅ 08:18, 27 June 2021 (UTC)

We are not talking about "copyright for facts or information", but "copyright of descriptions of facts or information". Say, when we write a mathematical paper/preprint, then it is usually copyrighted even if it only includes facts. If you disagree with the existence of the copyright of a mathematical paper/preprint, please tell me the reason.
p-adic 08:42, 27 June 2021 (UTC)

## Talk:55,440 and Talk:Quinquagintacentillion

This log keeps track of what has happened.

Normally, when we move page A to B, we move Talk:A to Talk:B. 55,440 was moved to 55440, and Quinquagintacentillion was moved to Quinquaginticentillion. So Talk:55,440 should be moved to Talk:55440 and Talk:Quinquagintacentillion should be moved to Talk:Quinquaginticentillion.

Hexirp (talk) 02:00, 23 June 2021 (UTC)

Oops. We simultaneously write comments on each talk page. Right, I agree that the moving from A to B usually causes the moving of the talk pages. But as I wrote in the summaries, the original talk page only has topics on "moving" itself. In that case, it is more convenient to keep the talk pages, isn't it? I think that the current version is quite misleading, unless we add some comments like "The proposal above is on the original articles."
p-adic 02:03, 23 June 2021 (UTC)
IMO it's fine to move them since it's clear what happened whether we leave a notice or not, e.g. the 55440 talk page says "Can I move this page to 55440?" (with that explicit title) with approval as well. So a future user would understand that it did get moved. The Quinquaginticentillion page is longer but I believe it's a similar scenario C7X (talk) 02:38, 23 June 2021 (UTC)
I see. Then I agree with both of you. (This was based on my estimation of how the future user will guess, but perhaps I was underestimating.) Thank you.
p-adic 02:45, 23 June 2021 (UTC)
• There is a principle: In MediaWiki, Talk:A is strongly tied to A. Therefore, when A is moved, Talk:A should also be moved.
• If we don't move the talk page, when a new article about A is created, the talk about it will be mixed with the talk about B. That's why the principle was created.
• Deviating from the principle can easily lead to misunderstandings. This is the reason for my actions.
• Hexirp (talk) 02:48, 23 June 2021 (UTC) Hexirp (talk) 02:50, 23 June 2021 (UTC) Hexirp (talk) 02:51, 23 June 2021 (UTC) Hexirp (talk) 02:53, 23 June 2021 (UTC)
I agree with the moving now, but I think that expecting the new creation of A is too special. At least, I do not know an example (except for vandalisms) within two years. In that case, we can simply solve the issue by archiving the talk page, or clarifying "this section is referring to an old version". Moreover, even if we move the talk page of A, a similar confusion occurs. The future reader will guess that the section is referring to a moved page, but there is an actual page which is not moved. Therefore if we really matter your special case, I think that it is better to keep the talk page. (But I think that the special case does not occur so often. Therefore I can agre with the move. This comment is just intended to be an opinion that your reasoning looks weird for me.)
p-adic 02:57, 23 June 2021 (UTC)
• I do not agree that it is a special case.
• Anyway, MediaWiki moves A and Talk:A together by default. We should follow that unless we have a specific reason not to.
• Hexirp (talk) 07:22, 23 June 2021 (UTC)

## Where did your photo come from?

• Where did File:Tetramur's harassment log.png, File:Ytosk's harassment log.png, File:Ytosk's harassment log 2.png, and File:Ytosk's harassment log 3.png come from? It was a surprise to me that you uploaded these pictures, since we knew very well that you are not on Discord.
• Hexirp (talk) 08:12, 23 June 2021 (UTC)
> since we knew very well that you are not on Discord.
Oh, how could you prove that I was not on discord? I think that you just knew that I did not want to use discord.
p-adic 09:39, 23 June 2021 (UTC)
• You have been refusing to talk on Discord. That is why I have reasoned that way.
• Hexirp (talk) 09:59, 23 June 2021 (UTC)
• In addition, I've seen a few of your tweets about not being able to see inside the Discord.
• Hexirp (talk) 10:07, 23 June 2021 (UTC)
In that case, what you can deduce is that I could not see it at the time.
p-adic 10:09, 23 June 2021 (UTC)
• Thank you for your answer. And my concern have not been addressed. I mean, ...... what would you do if you were asked to prove that these screenshots are real?
• In Discord, deleted messages can't be recovered, even by administrators. I know they are real, but there is a possibility of future complications.
• Hexirp (talk) 10:36, 23 June 2021 (UTC)
That is why I asked the admin to check them. Since the admin can check both discord and this wiki, it has no problem. After the admin confirmed the validity, the deletion of the messages does not make sense. In order not to make the future user confused, we can just clarify that the admin actually checked it.
By the way, are you seriously interested in the future confusion rather than the current harassment?
p-adic 11:29, 23 June 2021 (UTC)
• I don't think it's enough for the administrator to have temporarily confirmed it. Evidence needs to be available for future users to check. Sometimes, administrators are not trustworthy.
• In this case, I fear that the rules will be abused in the future.
• Hexirp (talk) 14:24, 23 June 2021 (UTC)
In that case, how do you confirm that vandals vandalised deleted pages? Only admins can check deleted pages. If we insist that confirmations by admins are not enough, we should unblock so many vandals.
Also, there are many living evidences, including you. Please imagine how Japanese courts work. You know that they are mainly based on evidence given by materials, but frequently use living evidences. You can estimate living evidences more positively as usual.
At least, this is not an imaginary experiment, but is an actual incident causing harassments. I also think that it is important to make this wiki better for the future user, but it is also important to currently keep this wiki good. Even if the victim were you, I insist that we should save you by blocking the user whose existence in this wiki as an "accepted" user would shrink your activity. What we should save are both of the future user and the current user.
p-adic 14:46, 23 June 2021 (UTC)
By the way, if you just want to personally share your opinion with me rather than discuss this topic here with other users, isn't it better to talk in Japanese? (Perhaps people do not like to see Japanese conversation, and hence it is good to use other place.) If you want others to pay attention to this conversation, then it is ok to continue this here.
p-adic 14:54, 23 June 2021 (UTC)
• Articles in this wiki can be restored. There is still an opportunity to have them re-verified in the future.
• In a space of anonymity, the witness system is not so meaningful.
• I'm trying to make the rules of this wiki good. That is one way to make this wiki good. Bad rules also discourage participants.
• Let's continue here.
• Hexirp (talk) 03:02, 24 June 2021 (UTC)
> I'm trying to make the rules of this wiki good.
Then could you summarise the discussion to the talk page of the policy after the conclusion, because talking at the talk page of a normal user is not easy to find.
> Articles in this wiki can be restored. There is still an opportunity to have them re-verified in the future.
How do you check the deleted page now, for example? Will you ask admins to revert all the deletion related to blocked users? If we need to do so whenever you (or another new alternative account) request it, then it is easy to abuse the rule to vandalise this community. Therefore requiring "always accessible" evidence is not a good solution.
Perhaps you are only allowing evidences which can be always accessible. I agree that such evidences are great. But please seriously remember my comment above mentioning courts. In a standard managing of societies and communities, we do not require so strict stability of evidences. This is not "0 or 1" problem. Therefore in my opinion, if the actual harassment, death threatening, and doxing are evident although the "proofs" will be expired in the future, we should recognise the incidents.
This topic balances "the confusion of the future user" and "the safeness of the current user who is a target of such a personal attack". We should care about both, and should not ignore one. When the actual personal attack prevents a sound activity of a user in the community, we need to help the user. Through the process, we should choose a better way, but might not be able to the "desired best" way. This case, you are seeking for the best solution, which does not exist if we want to help the user. Should we give up to help the user in order to only consider the "desired best" way? I do not think so, and I think that the strict requirement does not follow the purpose to maintain this community.
p-adic 03:20, 24 June 2021 (UTC)
• I'll do that.
• I'm not worried about future users getting confused. I am afraid that future users will be harmed by perjury.
• Hexirp (talk) 05:46, 24 June 2021 (UTC)

This contradicts your statements "I don't think it's enough for the administrator to have temporarily confirmed it" and "Sometimes, administrators are not trustworthy."

> I'm not worried about future users getting confused. I am afraid that future users will be harmed by perjury.

I am explaining the importance of the fact that the current user is actually harmed, and we should not ignore the actual victim in order to only help futue users.

Also, please do not skip my point. I explained that the desired way is too restrictive. Of course, we can make it more restrictive. For example, should we only allow evidences with confirmations by third companies and lawyers? Those evidences are better, but then we essentially cannot help users. Restrictions should be given only within the range where we can help users. At any rate, you re ignoring the fact that your too restrictive requirement causes a problem.

p-adic 06:03, 24 June 2021 (UTC)

• I agree that my explanation was inadequate. It's the new administrators we've hired. After they are replaced, we should have a chance to check the past.
• In this case, I am more concerned with the negative effects of the rules. Unfortunately, you and I seem to feel differently on this point.
• Hexirp (talk) 11:54, 24 June 2021 (UTC)
> It's the new administrators we've hired.
I do not understand what "it" means here, because we have hired no administrator.
> In this case, I am more concerned with the negative effects of the rules. Unfortunately, you and I seem to feel differently on this point.
I see. Since this is also a matter of extent, such a diffence can occur.
By the way, this is just my interest. Could you tell me whether you think that the current court system in Japan is inappropriate? (You do not have to answer it, because it is just my off-topic question.)
p-adic 12:20, 24 June 2021 (UTC)
• The wording of it was incorrect. We, or our next generation, can ask them newly trusted admins for verification.
• I think that Japanese laws are designed for the physical world and do not fit this wiki.
• Hexirp (talk) 15:16, 24 June 2021 (UTC)
I am not asking whether the Japanese law fits this wiki or not. I am curious about it, because if you care the future trouble to that extent, you might not feel OK due to the fact that Japanese law does not satisfy your request.
p-adic 16:27, 24 June 2021 (UTC)
• I'm just weighing the risks of admitting that against the risks of not admitting that. The fact that this is on the Internet makes the risk of admitting that greater.
• Hexirp (talk) 10:32, 25 June 2021 (UTC)
Of course, I know it. (And also I guess that "risks and actual harms" is more appripriate than "risks" in this context, because we were discussing on both of risks and actual harms.)
p-adic 10:54, 25 June 2021 (UTC)
• Hmm, let's end this topic.
• Hexirp (talk) 11:16, 25 June 2021 (UTC)
p-adic 11:29, 25 June 2021 (UTC)
• 確かに、そうだったかもしれません。しかし、私が現実の法体系に対してどのように考えているかを私がインターネットの上に公開することに私が不愉快さを感じていることに私は気付いたので、この話は終わりにしたいと思います。
• Hexirp (talk) 11:39, 25 June 2021 (UTC)
OK.
p-adic 12:28, 25 June 2021 (UTC)

## Regarding Omegaggol

This regards the recent page creation about my omegaggol(https://googology.wikia.org/wiki/Omegaggol). I apologize for forgetting about the non negative whole integers part, but since I only use a whole number for it, it shouldn't be a huge deal. In the issues section you say that it lacks a bound for n, as in creating an infinite loop, but in the definition it says "b(n,m) = b(n-1,b(n-1,b(n-1,b(n-1,m)))) if n > 0" and the if n >0 grounds the recursion. If it is a mistake on my part, I apologize for my mistake. However, the source does say if n > 0. Mumuji Penguin (talk) 13:18, 25 June 2021 (UTC)

You do not have to apologise! We can coin numbers even if they are ill-defined. Please feel free to coin numbers.
> However, the source does say if n > 0.
So, you just dropped the condition when you wrote the article? I see the situation. Since what matters is the original source, I correct it. Thank you. (By the way, it is good to write such a comment to the talk page of the number rather than here.)
p-adic 13:24, 25 June 2021 (UTC)

## Growth rate (again)

Alright P進大好きbot, does my notation finally work this time? DRC 15:55, 29 June 2021 (UTC)

OK. Whenever you ask the growth rate, please check the well-definedness by yourself first, because almost all the time when people ask the growth rate, the notation is ill-defined. (This is natural, because people who tend to check the well-definedness from the beginning is usually able to analyse notations by themselves.) Therefore I would like to ask you the well-definedness firsy. (For example, it is good to read "General setting" section in this guide line to ensure the non-existence of obvious errors. If you have something that is not easy to understand, please do not hesitate to ask it.)
p-adic 22:45, 29 June 2021 (UTC)
Hi, sorry for the late response, for some reason I didn't get a notification. Anyways, I've done best to see how the notation works and if it follows the rules you've supplied in your list of common failures, but if I've missed something please tell me. DRC 20:36, 1 July 2021 (UTC)
Thank you for the trial to check the well-definedness by yourself! Then ok, I will try to check the growth rate when I have time. If you require detailed proofs, please tell me the request. Otherwise specified, I will simply write my expectation of the growth rate.
p-adic 01:28, 2 July 2021 (UTC)
Now I checked it. Unfortunately, it includes the following errors:
1. Common failure 2.2: Lack of the Clarification of the Domain
2. Common failure 2.3: Lack of the Quantification of Variables
Please check the two sections. Then please refer to User blog:P進大好きbot/How to Create a Recursive Notation#General Setting to understand what you can use in the definition of a notation. For example, you cannot use an undefined variable. Therefore it is good to quantify all variables in the definition. Similarly, you need to clarify the domain in a prcise manner. (Say, an "array" is just an array, but I think that you intend more restrictions.) Also, it is good to avoid partial specialisation, because it can be cause of massive errors. Detailed explanation is available in User blog:P進大好きbot/How to Create a Recursive Notation#Partial Specialisation.
If you feel that those guidelines are not helpful after you read it, please tell me so. Then I will try to show an alternative formulation of your notation, and analyse it, although the definition might nont match your intention. Please imitate it when you create another notation.
p-adic 14:54, 2 July 2021 (UTC)

## Accidental rollback

I'm sorry for rollbacking your edit on the voting page, because I actually meant to click the difference between revisions of the Voting page, but my mouse accidentally clicked the "rollback" button. Do you accept the apology? ARsygo (talk) 02:26, 6 July 2021 (UTC)

Sure. Please never mind it :D
p-adic 03:13, 6 July 2021 (UTC)

## Reasons

@C7X

Have you ever checked the actual discussion? Putting the warning templates for Plain'N'Simple's excessive harassments is allowed by ARsygo, and ARsygo recommended me to do so in User talk:ARsygo.

I am afraid that you have misunderstood the situation. Plain'N'Simple harassed me so many times by labeling like saying "You cannot invent a new number larger than Rayo's number" and "You have never listened to others". If Plain'N'Simple apologised for those harassments, I could accept the user. However, as I wrote in the warning section, Plain'N'Simple refused the proposal. We are not claiming for the mathematical errors, as all of the warning templates just refer to harassments. Due to your explanation, I guess that you misunderstand the situation, perhaps because of the private communication with Plain'N'Simple. Has Plain'N'Simple given one evidence of your claims? Please share them, because Plain'N'Simple refused to clarify the evidences of his claims. Please check User talk:ARsygo and the warning section, because you can find the incident with full sources.

@ARsygo

Could you explain to C7X what happened? I guess that C7X has been given wrong information from Plain'N'Simple, as I have requested to apologise for Plain'N'Simple's excessive harassments but never for Plain'N'Simple's mathematical errors. Thank you.

p-adic 23:05, 14 July 2021 (UTC)

My account's getting deleted soon, so I won't have much time.

> Have you ever checked the actual discussion?

I did, and most of the messages on the original comments by Psi appear to be suggestions or recommendations - put a bit harshly, I will admit - but still recommendations. See the below:

> Plain'N'Simple harassed me so many times by labeling like saying "You cannot invent a new number larger than Rayo's number"

> If Plain'N'Simple apologised for those harassments, I could accept the user

Psi seems rightfully indignant in this scenario, since you haven't apologized for any offensive comments towards Psi. Psi has indeed made some rude comments, but I honestly agree with Psi that it seems like this is being made out to be one-sided.

> Please check User talk:ARsygo and the warning section, because you can find the incident with full sources.

Here we find some links to reiterated quotes and points. By no means am I saying that Psi is an angel who did everything right, some of Psi's statements are rude as I have said. (See the above for some examples of similar points.) But still, this is quite an overreaction.

> he is the only one person who has been given a specific issue to apologise

> I am asking him to cite the exact sentence which I inappropriately wrote, but he does not listen to it. Even then, he insists that I should apologise for it. It is impossible, because I am not an esper.

For example, these quotes are given as one of the reasons that Psi is immature and unreasonable for not following request, and elaboration, respectively. Even if it's the case that apologies must be explicitly requested to be warranted (which is the original goalpost, which I still disagree with), there's still a power imbalance as you had threatened to get Psi blocked if he didn't apologize immediately ("So, please strictly judge if his next first action is not actual apology to ALL of his bad attitudes.") And also, it is not impossible to apologize for an attitude, instead of a literal exact sentence pointed out by the person asking for an apology. So I judge that this request is an overreaction
Edit: About citing "the exact sentence which I inappropriately wrote", I would like to add that much of communication is nonverbal, and this carries over to writing as well. What's not written is just as important as what's written. Psi has apologized for even just a few comments that PBot found scathing, but PBot's comments about apologies consist of those asking Psi to apologize, never apologizing for any remarks that Psi found scathing. It's this imbalance in what's not written that implies that PBot believes all the blame is on Psi's side. C7X (talk) 01:26, 15 July 2021 (UTC)

@C7X

> If we take the interpretation that this is labelling since it's assuming that Psi doesn't care about your numbers, then what about numerous other comments left by others that deride sizes of others users' numbers? Such as 1 2 3 4 5 6 7.

First, the point is that Plain'N'Simple is labeling others' ability without any evidences. Say, if people freqently made elementary mistakes on formal logic, then it is reasonable for Plain'N'Simple to insist that they cannot currently create a new number larger than Rayo's number. If Plain'N'Simple had ever seen my failed trials to invent new numbers, then it might be justifiable. However, Plain'N'Simple did not. It is really inappropriate to label others' ability without giving any evidences. Even worse thing is that Plain'N'Simple labeled that we cannot create such a number in the future.

Therefore your examples are not analogues of the point. You can see that they made mistakes on what they are using, and I honestly tried to explain their errors. Unless they understand their mistakes, it is really hard to use those difficult notions. Such a suggestion is helpful for them, because they gain oppotunity to notice what they do not understand and learn them to invent new large numbers. It is irrelevant to Plain'N'Simple's labeling that I cannot create a sufficiently large number even in the future without showing evidences such as poor misunderstandings.

Also, could you block me only after the discussion? It prevents me to add comments on related pages like User talk:ARsygo. Since my point is that your points are incorrect in my point of view and I am sincerely trying to solve the issues, it is not good to make me silent before asking the situation.

EDIT: By the way, 7 is irrelevant, and hence is perhaps a typo.

> Psi seems rightfully indignant in this scenario, since you haven't apologized for any offensive comments towards Psi. Psi has indeed made some rude comments, but I honestly agree with Psi that it seems like this is being made out to be one-sided.

As I explained in the warning section, I have asked Plain'N'Simple so many times to precisely cite what I made inappropriate comments in, and clarify that I will justify or apologise for each comment. The reason why we need such a precise citation is because Plain'N'Simple reported ARsygo a wrong statement on me in order to block me. This is precisely explained in Warning 6.

Plain'N'Simple blamed me for not listening to others and asked ARsygo to block me by saying "This all started in a year-old discussion, where I was talking about philosophical things like "the nature mathematical truth" and P-bot, some how, misunderstood non-exact statements in english to mean (wrong) statements in mathematics." However, as I precisely cited in Warning 2, I have already pointed out the possibility that Plain'N'Simple was talking about philosophy at leat 5 times, and Plain'N'Simple clarified that Philosophy is completely irrelevant at least 8 times. In that case, the claim that I am just labeling that Plain'N'Simple was not talking about philosophy is obviously incorrect.

Since Plain'N'Simple tried to make ARsygo block me by the incorrect reasoning, it is quite natural to specify the precise sentences which I should justify or apologise. I sincerly asked to cite them, but Plain'N'Simple refused and insisted that I am insiting that I am 100% good, although I have never insisted so.

So, the one who is trying to make the situation one-sided in some purpose is Plain'N'Simple. What is the benefit? There are several possibilities. Say, if the situation looks "one-sides", then Plain'N'Simple can blame me for the situation. That is why I sincerely asked Plain'N'Simple to precisely cite what I wrote in order to rrecognise what to justify or apologise.

> Here we find some links to reiterated quotes and points. By no means am I saying that Psi is an angel who did everything right, some of Psi's statements are rude as I have said. (See the above for some examples of similar points.) But still, this is quite an overreaction.

Please imagine why I needed to do so. If I simply wrote warnings, then I need to cost my time because Plain'N'Simple can repeat the same attitude requiring days. Therefore, I listed our past discusssions in order to make them easier for admins and future users to check. Say, Plain'N'Simple required us to list actual mathematical errors, even though I have ever mentioned twice in these days. It actually wasted ARsygo's time to judge the situation. Since this suspension is harmful, I should have listed all of my points with full sources in order to avoid the further suspension.

> For example, these quotes are given as one of the reasons that Psi is immature and unreasonable for not following request, and elaboration, respectively.

You mean, my quotes of Plain'N'Simple's sentences, right? But it is actually needed, because Plain'N'Simple is not admitting the existence of harassments. Generally speaking, we need to judge everything by using accessible evidences. If I abstractly blamed Plain'N'Simple for ambiguous statements, then Plain'N'Simple would not admit the fact. Indeed, Plain'N'Simple was denying the fact that Plain'N'Simple wrote rude comments until ARsygo sincerely tried to be involved. In that case, citing what Plain'N'Simple actually wrote is quite important, in order to avoid further suspension. If Plain'N'Simple had not done such negations, then the situation would be much simpler. Could you assign a better way when a user who actually harassed others denied what he or she did? I honestly thought that it is the best to cite the precise sentences.

p-adic 01:47, 15 July 2021 (UTC)

> About citing "the exact sentence which I inappropriately wrote", I would like to add that much of communication is nonverbal, and this carries over to writing as well.

Right, but you know that such claims based on unwritten sentences are quite dangerous. Indeed, Plain'N'Simple reported to ARsygo a wrong statement on me, and I clearly disproved it using "what are actually written". If problems on Plain'N'Simple's way to blame others for unwritten sentences are known, isn't it quite natural to require to precisely cite what I wrote? At least, I clarified that I would justify or apologise for cited comments, but Plain'N'Simple refused the proposal. If Plain'N'Simple had sincerely tried to finish the issue, Plain'N'Simple could have simply cited what I wrote. Also, the "one-sided" claim is obviously made-up, because I clarified that I will justify or apologise for what I wrote. So, respecting Plain'N'Simple's opinion on the "one-sided" issue sounds strange for me.

Also, please keep in mind that blaming actual harassments should not "judged" more seriously than the actual harassments. When you think that I should change a way to blame the actual harassments, you can tell me instead of making me silent by blocking. The reason why I put the warning templates is because he expressed the denial of the apology, which means that he will not be better, and because ARsygo suggested to do so. Therefore I do not think that it is appropriate to pick blaming for the actual harassments as a reason of the blocking. Please tell me what I should have done instead.

p-adic 02:21, 15 July 2021 (UTC)

@ARsygo

Could you persuade BlankEntity to come back to this community after the incident is solved? I know how BlankEntity felt for the incident, but we do not want to reduce honest active users who are trying to make this community better.

Also, could you please explain to C7X what occurred? I guess that C7X thought that I put the warning template as a portion of some personal attacks, but you know the whole story: Plain'N'Simple tried to make you block me by using an incorrect statement on me without evidences, I disproved it and requested Plain'N'Simple to precisely cite what I should justify or apologise for, I proposed Plain'N'Simple to apologise all harassments many times, Plain'N'Simple refused the proposal, you suggested to use the warning templates, and I actually did so following the policy with precise citation of what Plain'N'Simple wrote. I know that you feel bad to be involved in this, but this community will be worse and worse...

p-adic 03:47, 15 July 2021 (UTC)

you just read the policy, right? So, the block is quite harsh to me, but I really hoped that Plain'N'Simple should apologise the mistakes that the user has been done, because the community here is getting worse if not resolved quickly, and I, well, am the only person to deal with it, since other users wanted to leave the wiki, eventually the community is dying, and finally, closed by Fandom. ARsygo (talk) 04:03, 15 July 2021 (UTC)
> you just read the policy, right?
Perhaps I am misunderstanding this question, because we are discussion on the current incident rather than polocy. Could you please tell me more details on this question? I appreciate your feedback to help my understanding.
> So, the block is quite harsh to me, ... and finally, closed by Fandom.
It is really bad... That is why I, you, C7X, BlankEntity, Hexirp, kyodaisuu, and many other members sincerely shared their opinions. We are not willing to make this community die. I am not understanding why blaming Plain'N'Simple's harassment can be a target of blocking while Plain'N'Simple is allowed. Please discuss here what occured. (I requested C7X to unblock me because keeping me silent does not give a solution, but this request is not accepted. Therefore what I can do is only to explain to C7X here what happened. I appreciate if you help me to explain to C7X the incident.)
p-adic 04:22, 15 July 2021 (UTC)
Hello P-bot, please read the new blog post on why I quit. Lilypad Conduct 04:58, 15 July 2021 (UTC)
Hello. I feel sad to hear your leaving, and I am sorry that I am involved in the incident, which made you decide the leaving. I tried to make this community better by strictly forbidding harassments, but it costs further time. I hope that this community will be better, after community members agree that we should not harass others. When the community becomes better, i.e. members are respecting each other and harassments are strictly forbidden, I wish that you will come back.
p-adic 05:18, 15 July 2021 (UTC)
@ARsygo
Perhaps you did not notice my reply to you above. I think that you are pointing out that I am missing some rule in policy. Could you tell me the rule? Also, I request you to explain to C7X what actually occurred. I hope you to solve this incident before maintaining articles, because it will make discussions flow from Recent Changes.
Now you see another violation by Plain'N'Simple: the removal of the warning templates. Could you revert the removal? It includes evidences of what Plain'N'Simple actually did and proofs that Plain'N'Simple's claims on me are incorrect. Allowing harassments and repeated violation of the policy makes this community worse. This time, I followed both the policy and your suggestion to put warnings. Warnings on actual harassments should not be blamed more seriously than actual harassments, because then users will hesitate to report harassments when they suffer from harassments by a user who does not agree with them unless clear citations are put. This time, Plain'N'Simple denied the existence of harassments, and hence we actually needed th precise citations to what he wrote. Therefore C7X's point that I should not cite the precise sentences sounds strange for me.
p-adic 07:31, 15 July 2021 (UTC)
I'll just wait for the follow up by Plain'N'Simple, because sometimes the user blames me for reverting. ARsygo (talk) 08:07, 15 July 2021 (UTC)
I understand that Plain'N'Simple's blaming is quite unpleasant, as we are being patient for the rude attitudes for years. This is not the first time... We were just keeping patient because we believed that Plain'N'Simple would become matured. Perhaps this incident is the result of mildly accepting the behaviour. In that sense, we are responsible for Plain'N'Simple's current attitude. ...But I do not think that Plain'N'Simple also harasses you because you are an admin. This case, Plain'N'Simple obviously violated the rule which forbids normal users to remove the warning templates, and hence you can safely revert it and add a warning for the violation. Please do not keep the violation allowed, because the change of the border will be a bad example referred to in the future. Please be confident of your guiding this community. We trust you.
p-adic 08:49, 15 July 2021 (UTC)
ARsygo, could you please check the warning templates again? You can see that I wrote nothing wrong. You actually saw Plain'N'Simple's excessible harassments. You do not have to wait for Plain'N'Simple's permission, because you know that Plain'N'Simple will never admit the fact of harassments. Please imagine if somebody else is harassed by Plain'N'Simple in the future. What will you do...? If you are afraid of Plain'N'Simple's blaming, then there will be more victims. Since there are actual evidences, please sincerely face up to this problem. As a beaurocrat, please look at this situation. Plain'N'Simple excessible harassed users, I tried to stop all harassments, and Plain'N'Simple make me blocked. It is quite unusual.
Also, please talk to C7X, because C7X seems to be considering that I am using the warning templates as a personal attack. You know that we have patient discussion with Plain'N'Simple, and you proposed to put the warning templates. Therefore you know that the warning is not any personal attack. It just includes evidences of Plain'N'Simple's harassments. Moreover, C7X's logic that a personal who has a doubt of a violation cannot blame for others' violation is wrong. Everybody has right to blame for others' harassments. In addition, C7X confounded labeling (harassmental statements o ability or intention not based on any evidences) with advices based on evidences such as mistakes. When we do not have evidences that a user has no ability, then we should not label that the user has no ability. Please persuade C7X, and deal with Plain'N'Simple. Please be confident, because Plain'N'Simple perhaps does not harass you, as Plain'N'Simple clarified to respect your decision.
p-adic 22:20, 15 July 2021 (UTC)
I'll leave a message to them via talk page soon, or even Discord, because C7X says that the user's account (C7X) will be disabled soon. ARsygo (talk) 22:38, 15 July 2021 (UTC)
Thank you. Then please revert Plain'N'Simple's removal of warnings, which violates the policy, and add a warning for the removal. It is quite important. This year, FANDOM announced us to clarify the border of blocks, and precisely follow specific rules. Therefore allowing harassments, which are forbidded in the policy, and blocking users which do not violate the policy (and even precisely follow the policy) violate FANDOM's announcement. Plain'N'Simple does not harass you, as long as you follow thepolicy and FANDOM's guideline, as Plain'N'Simple clarified to respect your decision as an admin. You can just follow the policy, and FANDON's term of uses.
p-adic 00:30, 16 July 2021 (UTC)

## I wrote an article about you

I wrote an article about you, User_blog:Hexirp/(2021-07-20)_p進大好きbot_and_Plain'N'Simple. Hexirp (talk) 15:43, 20 July 2021 (UTC)

## Attention

I think it is better to refrain from agitating the other party like the comment. -- Hexirp (talk) 09:47, 22 July 2021 (UTC)

Thank you for the suggestion. I would like to follow your advice. The comment includes four sentences. Could you tell me which sentence you are referring to?
p-adic 10:02, 22 July 2021 (UTC)
I am not concerned with your single expression, but with the effect of the combination of multiple ways of expression. -- Hexirp (talk) 04:26, 23 July 2021 (UTC)
Thank you! Perhaps I should have separated the comment into two: one for just praising the improvement, and the other one for clarifying that we do not allow such a rude labeling. (Or it might be better to choose only one for them. Perhaps the former one?) Since I combined them in a single comment, it might look something like irony. It is my failure.
p-adic 05:00, 23 July 2021 (UTC)
If you praise it in Japanse, すばらしい！ついに投稿者に対する失礼な態度を謝罪できるようになったんだね！ Maybe some people may feel irony from this. I know that you have no intention of irony, but it is just a tone. And if you don't feel any ironical tone from this sentense, it is difficult to explain why this sentense may sound ironical... It is not something like a mathematical proof. 🐟 Fish fish fish ... 🐠 05:07, 23 July 2021 (UTC)
Thank you. I rather meant "それでいいんです！　投稿者に対する失礼な態度を謝罪しなければいけないことがもう分かったでしょう！", because Plain'N'Simple persisted to refuse to apologise for the rude labeling toward the OP. I know how people feel irony from your Japanese translation, and I guess that you know how my Japanese translation directly (unambiguously? "ぼかさず") teaches what we need to keep in mind without irony, although they share the original English sentences. But my intention might not be an excuse here, because the problem is my careless tone which led confusion. I am very sorry about it. Well, if Plain'N'Simple had pointed out what comments I was blamed for among so many my comments in this one year, I could have improved my poor phrasing (or started to completely use google translation in order to avoid further confusion)... Do you guess why Plain'N'Simple thoroughly refused to tell me what comments I was blamed for even though I asked it so many times...? Perhaps I might make another mistakes. For example, is there a common belief (in English conversation) like that we should not directly ask others what phrased we are blamed for?
p-adic 05:25, 23 July 2021 (UTC)
I don't know about the intention of Plain'N'Simple so I don't comment about it. I think you have a right to blame Plain'N'Simple for rude attitude, when Plain'N'Simple directly insult you or whatsoever. However as this was a conversation between other people, not involving you, you don't need to comment about the attitude of Plain'N'Simple here (of course mathematical discussion is welcome). So you don't need to praise here. That is all I can comment regarding this comment that Hexirp felt agitating tone. 🐟 Fish fish fish ... 🐠 05:36, 23 July 2021 (UTC)
It makes sense. I thought that we should blame a user when we see him or her do something inappropriate against a new user, but perhaps I am wrong due to this result. Thank you both for many suggestions. I will make me less active here, in order to avoid further troubles.
p-adic 06:55, 23 July 2021 (UTC)

## Attention 2 (provocative)

1. Target
1. the comment
2. Reason
1. provocative

Hexirp (talk) 04:09, 1 August 2021 (UTC)

You mean the first sentence, right? Or the whole comment? (I appreciate specifying the exact sentence if there is, or clarifying that the whole comment is bad if so.) Thank you.
p-adic 04:50, 1 August 2021 (UTC)

## A Variant of Defining Epsilon Null

/(ε_0/) is also /({}^ω ω/) —Preceding unsigned comment added by RudyStaysForever (talkcontribs)

Hello. Thank you for a comment! I think that "/" is a typo of "\". And yes, it can be defined in that way as long as you fix the deinition of the transfinite tetration (cf. Common failure 4.7: Undefined Transfinite Extension).
p-adic 12:32, 9 August 2021 (UTC)

Based on the recent changes given, it looks like the contributions of this user actually "spammed" the recent changes until I cannot see the other contributions, can you warn the user? ARsygo (talk) 08:47, 12 September 2021 (UTC)

> until I cannot see the other contributions
You mean that you cannot see contributions by other uses in the recent changes, right? If so, you can solve the problem by clicking the button displaying "X CHANGES, Y DAYS". Indeed, I am choosing "250 CHANGES, 7 DAYS", and I can easily check contributions by other users. If you mean that you cannot see contributions by the user which are not minor corrections of articles, then I personally do not regard the situation as a bad thing, as we are usually encourging new users to contribute articles.
> can you warn the user?
So, I cannot warn the user because I currently do not find any violation, but one problem is that the user is not marking "minor edit" when he or she give minor corrections. Since we are supposed to mark "minor edit" for such edits in order to help others to recognise non-trivial edits, I ask the user to mark it. If I am misunderstanding your request, then please ask me again with more detailed explanations about what the user should stop.
p-adic 09:08, 12 September 2021 (UTC)
OK, so because, sometimes, the user tends to make unconstructive edits before slowly turning into constructive edits, actually. Because the other users in the wiki, added descriptions in a single edit, like myself, it's quite hard for me to patrol other new pages created by the other users as well. Even if I changed to "250 CHANGES, 7 DAYS", there's still "spam edits" by the user, actually. Because, it seems annoying, to me. ARsygo (talk) 09:14, 12 September 2021 (UTC)
Thank you for the trial and for the detailed explanation. Then I asked the user to use "minor edit" box to appropriately mark minor edits and to use "preview" command to reduce repeatitions of minor edits of a signle article. The first request helps you to find significant changes. The second request helps the user to reduce the number of edits displayed in "Recent Changes". They might not be complete solutions, but let us check the result. If there will be still a problem, then please notify me again. In that case, let us discuss another solution, because it is not good for us to increase your cost for maintaining this wiki.
p-adic 09:33, 12 September 2021 (UTC)
It looks like the user continued to "spam" multiple edits based on Recent Changes, actually, until I can't see the past contributions, actually. Can you deal it with the user ASAP? ARsygo (talk) 22:14, 13 September 2021 (UTC)
I see. Since I have already asked the user to improve the way to edit articles and notified the user that the current way is preventing others' contributions, we unfortunately need to deak with the user in some more effective way. On the other hand, the use of the warning rule is still risky for me, because there are several members who try to violate FANDOM's terms of use by playing "sanction games" against innocent users who follow the warning rule. So, I try to ask Fish to request the user to improve the way again. Since Fish is also an admin, this is more effective than my request, I expect.
p-adic 22:50, 13 September 2021 (UTC)
Alright, anyway, the user still didn't listen to us, and continued to flood recent changes with the user's edits again, but the article is different this time. ARsygo (talk) 22:29, 14 September 2021 (UTC)
OK. I will check the edits later.
p-adic 23:08, 14 September 2021 (UTC)