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: The definition of the OCF works, but it is not what Buchman defined, is it? I guess that it is Rathjen's recast rather than Buchman's original ψ. Also, fundamental sequences are ill-defined by the same reason as fundamental sequences in the FGH section are ill-defined.
: The definition of the OCF works, but it is not what Buchman defined, is it? I guess that it is Rathjen's recast rather than Buchman's original ψ. Also, fundamental sequences are ill-defined by the same reason as fundamental sequences in the FGH section are ill-defined.
: [[User:p進大好きbot|p-adic]] 05:18, March 25, 2020 (UTC)
: [[User:p進大好きbot|p-adic]] 05:18, March 25, 2020 (UTC)
So, about these issues:
didn't i already say how to scan?
I will replace "can be expressed as" as simply "looks like". In fact, every non-standard expression can be expressed in stardard form. And I'll put a upper limit of the ordinals here.
[[User:12AbBa|Zongshu]] [[User talk:12AbBa|Wu]] 12:00, March 26, 2020 (UTC)

Revision as of 12:01, March 26, 2020

private contact

hi P進大好きbot, i need to tell you something in private if you don't mind. could you please create an account on googoldon and tell me your username? i'm @vel on there. -- vel! 18:59, July 31, 2018 (UTC)

Hi! Thank you for the message. I already have accounts on too many web services e.g. googology wiki, 巨大数研究wiki, twitter, pixiv, and so on, and hence I would not like to create a new account on another web service. I am sorry. If you have a private message, please send a DM on twitter or pixiv. -- p-adic 20:54, July 31, 2018 (UTC)
no worries, i completely understand. i'll get in touch via pixiv private message. -- vel! 22:50, August 4, 2018 (UTC)
Ok. I replied to the message. -- p-adic 05:50, August 5, 2018 (UTC)

Set of rules for BM4 pair sequences

I know you've proven that BM2/BM4 pair sequences always terminates.

I'm not particularly interested in the details of the proof itself, but I am very interested in knowing which rule-set you've used in that proof. PsiCubed2 (talk) 10:17, November 20, 2018 (UTC)

As I wrote in the comment of your blog post, I formulated a specific version of PSS in my own way in my blog post of the proof. I did not use any rule set of BM2/BM4. Remember that I also wrote the following:
> It works in the same way as BM1.1, 2, 2.3, 3.1, and 3.2 with respect to the 20/8/2018 version of koteitan's mathematical classification if I am correct. (A proof of the comparison to such existing versions is not verified, because I am not interested in such an unstable work.)
Therefore if you need to know the rule set of BM2 to which I am referring to, then read the one written in the corresponding version of that page. I emphasise that what I dealt with is not BM2 itself, but a specific version of PSS which I explicitly wrote the definition in my proof, and hence it does not matter if there is any trouble on BM2 itself.
p-adic 10:34, November 20, 2018 (UTC)
That page is in Japense. How do you expect me to read that? PsiCubed2 (talk) 10:53, November 20, 2018 (UTC)
There is an alternative translation in English, but it is not precisely what I referred to. Although the contents might be completely the same, I have never verified the coincidence. (At least, I know that the first edition of the English one contained many typos.) Therefore I need to write the link to the original Japanese page so that you can translate it with google by yourself.
In addition, I know nothing about BM4. If I had ever stated that my definition actually works in the same way as BM4, then it would be just a typo.
p-adic 11:03, November 20, 2018 (UTC)
Yeah, I already know that page. It contains only computer code for the relevant version, which is not what I'm looking for. It's absolutely amazing that people are actually writing analyses for this notation, yet nobody is able to provide an actual (non-computer-based) definition.
As for your own version of pair sequences, I'm pretty sure I know how it works since they are equivalent to Buchholz Hydras, and I already know how those work :-) PsiCubed2 (talk) 13:50, November 20, 2018 (UTC)

Are you a bot?

ReactorCoreZero (talk) 23:18, January 30, 2019 (UTC) Are u a bot?

Yes. p-adic 01:01, January 31, 2019 (UTC)

What does this mean?

So i was wondering what this meant and i would apreciate if anyone could explain it to it is \(\lbrack ()! \wedge \stackrel{\biguplus_{\hookrightarrow}}{{}_{\twoheadrightarrow}\textrm{よ}} \rbrack \left\langle \begin{array}{ccc} \surd & \leftrightarrow & \oplus \\ \swarrow & \models & \otimes \end{array} \right\rangle \underbrace{\vdash \rhd \vdash}_{\bot} \bowtie_{\smile \top}^{\wp} \Im_{\mho}\) it is from this post  Ynought (talk) 15:50, February 15, 2019 (UTC)

\lbrack ()! \wedge \stackrel{\biguplus_{\hookrightarrow}}{{}_{\twoheadrightarrow}\textrm{よ}} \rbrack \left\langle \begin{array}{ccc} \surd & \leftrightarrow & \oplus \\ \swarrow & \models & \otimes \end{array} \right\rangle \underbrace{\vdash \rhd \vdash}_{\bot} \bowtie_{\smile \top}^{\wp} \Im_{\mho}

Naruyoko (talk) 16:22, February 15, 2019 (UTC)

As I wrote " I emphasise that this table is not the analysis of my ordinal notation, but is a desired image.", it means nothing :D
p-adic 01:17, February 16, 2019 (UTC)

well i meant that it should be an ordinal and i meant if you ould explain that ordinal —Preceding unsigned comment added by Ynought (talkcontribs) 18:08, February 17, 2019 (UTC)

No. It is just a meaningless string, which just explain that the ordinals might not be what you know because they seem to be greater than the countable collapses of large cardinals.
p-adic 22:17, February 17, 2019 (UTC)
So it means the same as (0,0,0)(1,1,1)(2,2,2) in a non-specified version of BMS ;-) PsiCubed2 (talk) 15:02, February 20, 2019 (UTC)
Exactly. p-adic 22:34, February 20, 2019 (UTC)
Thanks Ynought (talk) 08:48, February 23, 2019 (UTC)

Is the limit of Stegerts ordinal notation known?

Because i couldn't seem to find anything. Ynought (talk) 20:43, April 6, 2019 (UTC)

Stegert introduced two ordinal notations in the thesis available in a repogitry of Munster university. Since I had never read it, I take a brief look at it. According to the explanation in the beginning of Section 2, the first ordinal notation \(T(\Xi)\) perhaps goes beyond the proof theoretic ordinal of \(\Pi_{\omega}-\textrm{Ref}\). I could not find a result on the limit the second ordinal notation \(T(\Upsilon)\) nalogous to \(T(\Xi)\). Maybe it is related to the theory called "Stability". In order to understand precise estimations, we need to read it more deeply.
p-adic 23:18, April 6, 2019 (UTC)

Thanks Ynought (talk) 16:37, April 10, 2019 (UTC)

Question about OCF and ordinal notation.

If I understand correctly (after reading this), the basic requirement for ordinal notation is that it is possible to write any ordinal in a finite way? Not all collapsing functions provide such an opportunity. For example: these and these functions make it possible to write any ordinal in a finite way; therefore, it meet the requirement ordinal notations. But these functions do not provide such an opportunity, so these are just OCFs and not ordinal notations. Scorcher007 (talk) 02:29, May 5, 2019 (UTC)

> the basic requirement for ordinal notation is that it is possible to write any ordinal in a finite way?
No. It is impossible to "write any ordinal in a finite way". Given a countable set of letters, any expression of ordinals using them just describes countably many ordinals, while there are uncountablly many ordinals.
I note that if you allow to use uncountably many letters, then the requirement is non-sense, because you can choose to use every ordinal as a letter. Therefore the countability is important.
> For example: these and these functions make it possible to write any ordinal in a finite way
It is wrong by the reason above. Moreover, they lack proimitive recursive intrepretations. Since one of the most difficult parts to define an ordinal notation is to define a primitive recursive intrepretation, any OCF without such explicit interpretation could not be regarded as an ordinal notation. It is just a set theoretic stuff irrelevant to computable googology.
> But these functions do not provide such an opportunity, so these are just OCFs and not ordinal notations.
Exactly. To be more precise, I think that almost all OCFs created by hyp cos are OCFs which are not equipped with a primitive recursive interpretation so that they yield ordinal notations. Hyp cos just did not know what ordinal notations were at the time, as you can see the fact in the comments to the blog post which you are referring to. Since it is very difficult to construct an ordinal notation, "ordinal notations" created in this wiki are not actual ordinal notations.
p-adic 03:43, May 5, 2019 (UTC)
> "write any ordinal in a finite way"
I meant that with the use of notation, we can express with a finite expression any countable ordinal up to limit of notation. Not all (!), but any successor or limit ordinal. Scorcher007 (talk) 06:25, May 5, 2019 (UTC)
Ok. I see what you meant. But my statement does not change. Even if you have an OCF such that you can express all countable ordinals up to limit in a finite expression, it is not an ordinal notation. You need an algorithm (which can be written in arithmetic without using set theory) iterpreting the \(\in\)-relation. If you do not have it, then the resulting notation does not satisfy the definition of an ordinal notation. As I wrote above, it is the most difficult part. Therefore claiming "I created an ordinal notation" by showing a notation without a primitive recursive interpretation of \(\in\)-relation is something like claiming "I created a computable large number" by showing how large it is without an explicit way to compute it.
p-adic 07:56, May 5, 2019 (UTC)
Oh, thank you. I finally got an understanding. We need arithmetic functions associated with notation expressions!
Then we can say that the 1st function from here (Bachmann's style) comes to the definition, because it contains in its definition an arithmetic function ωα.
But starting from the second function (Buchholz's style) and other function from here and here no arithmetic functions in the definition, only the definition of sets. Then OCF from here and here can express with a finite expression any countable ordinal up to limit of notation, but do not contain fundamental sequences, which determine arithmetic relation between this ordinals.
Then the functions from here don't even allow express with a finite expression any countable ordinal up to limit of notation, because HypCos uses sets of strings instead of sets in a function definition. But if we assume that the sets of strings will be infinite then not way express with a finite expression any countable ordinal.
 What about professional OCF? At Stegert about the ordinal notation system (which has a limit of KPi+∀n∃σ≥n(Lσ1Lσ+n) said only half a page (Stegert 12.4 p 113). But T.Arai wrote 30 pages about ordinal notation system (which has a limit of KP-Пn) with the description of arithmetic functions and without OCF. Finally, what about the TON? On the Taranovsky page I did not find anything about the description of arithmetic functions, although he states that he defined Ordinal Notation. Scorcher007 (talk) 09:07, May 5, 2019 (UTC)
> Then we can say that the 1st function from here (Bachmann's style) comes to the definition, because it contains in its definition an arithmetic function ω^α.
Maybe you have some misunderstanding again. You need a primitive recursive algorithm to determine \(\alpha \in \beta\) for given expressions \(\alpha\) and \(\beta\) only using the expression. It is not relevant to the fact that the notation only uses elementary functions such as \(\omega^{\alpha}\).
> What about professional OCF? At Stegert about the ordinal notation system (which has a limit of KPi+∀n∃σ≥n(Lσ≺1Lσ+n) said only half a page (Stegert 12.4 p 113).
Theorem 12.4.2 is the statement on the primitive recursive interpretation. Stegert omitted the proof because the proof is quite similar to that of Theorem 2.4.4.
Well, I think that it is not a usual paper, but a doctoral thesis, which surveys the author's results. Many of other papers contain sufficient proofs.
> Finally, what about the TON? On the Taranovsky page I did not find anything about the description of arithmetic functions, although he states that he defined Ordinal Notation.
As I wrote above, being an ordinal notation is not relevant to using elementary functions. Taranovsky defined a primitive recursive well-ordering \(<\) (without using an OCF), and hence it forms an ordinal notation.
Now please read back the begining of this section. Then you will find that you need no elementary ordinal function or ordinals in order to define an ordinal notation. What you need is just a primitive recursive well-ordering \(<\). We usually use OCFs and actual ordinals just because they help us to define an ordinal notation.
p-adic 09:48, May 5, 2019 (UTC)

Question about Contribution

What did you just contribute to the wiki? I am a Googologist at the extreme!!!! 08:54, May 13, 2019 (UTC)

I replaced a wrong explanation of Kleene's O by a correct one. I sometimes edit pages related to set theory when I found mistakes. Also, I updated my personal space so that others can refer to my recent googological stuffs.
p-adic 09:57, May 13, 2019 (UTC)
OK then. I am a Googologist at the extreme!!!! 11:30, May 13, 2019 (UTC)

Minor edits

Please avoid marking edits as minor when you significantly change the content of pages (such as changing their meaning). Edits should only be marked as minor when fixing spelling mistakes, other minor corrections, and reverting obvious vandalism. Thank you. -- ☁ I want more clouds! ⛅ 18:07, May 29, 2019 (UTC)

Oh, sorry. Ok, thank you.
p-adic 22:08, May 29, 2019 (UTC)

Big Foot

"This is my first attempt at using this notation, so I probably made many mistakes."

On your proof for the ill definedness of Big Foot, you say that if \(\varphi(\alpha)\) is a definition of \(\alpha_0\), \(\alpha_0\)<\(\alpha_0\), but did you consider that this could be expressed as a schema. For example, work in the language \(\Form_∈\+\alpha_0\), were \(\alpha_0\) is a constant symbol. Then define (\R\) as the schema defining \(\alpha_0\), and then the theory \(\ZFC+R\) proves the existence of \(\alpha_0\). MasterOfArda (talk) 02:14, May 30, 2019 (UTC)

We do not have to consider the possibility that the creator set contant term symbols and schemas, because it is clearly declared that the formal language of oodle set theory does not admit such symbols here.
p-adic 02:57, May 30, 2019 (UTC)
Yet you said any extension \(T\) of \(ZFC\) was inconsistent, not just those using the symbols from \(FOOT\). I am not arguing the original construction is not ill-defined, simply that it can be well-defined. Even \(Ord\) can be well defined, as \(Ord\) is simply the least correct cardinal, and if \(\kappa\) is stationarily superhuge, \(\kappa\) is correct and so \(ZFC\)+There exists a stationarily superhuge cardinal gives a basis for \(FOOT\). MasterOfArda (talk) 03:27, May 30, 2019 (UTC)
Yeah, I agree that there are many ways to define a large number which is very "similar" to BIG FOOT. However, such an interpretation is not BIG FOOT any more by definition. My statement in the article is the following:
"For any theory \(T\) extensing \(\textrm{ZFC}\) set theory, if the well-definedness, i.e. the unique existence, of BIG FOOT with respect to the original defining formula in \(\textrm{ZFC}\) set theory is provable, then \(T\) is inconsistent."
It does not mean that there is no alternative defining formula similar to the original one. You know that the natural number defined by the formula \(c \in \mathbb{N} \wedge \forall n \in \mathbb{N}, n < c\) with a free occurrence of a variable term \(c\) is ill-defined in any reasonable set theory, while \(d\) is well-defined in \(T\) given as \(\textrm{ZFC}\) set theory with an additional constant term symbol \(d\) and axiom schema \(d \in \mathbb{N}\), \(0 < d\), \(1 < d\), \(2 < d\), and so on. They just look similar to each other, but are completely different to each other. Also, the well-definedness of \(c\) (not \(d\)) contradicts the axiom of \(T\).
Actually, I personally use constant terms and schemas when I define uncomptable large numbers, and hence your reformulation is natural also for me although I have not precisely followed it because such reformulation is not unique. The point is just that it is not the orginal defining formula.
Do you think that my explanation in the article is ambiguous? If so, I will clarify that I am referring to the original defining formula. Or do you have any other suggestion?
p-adic 06:47, May 30, 2019 (UTC)
I say we write this: "Unfortunately, such a set theory is inconsistent. Namely, for any set theory \(T\) extending \(\textrm{ZFC}\) set theory such that \(T\) is in the langauge of \(Form_∈\) (I.e. No constant symbols), if \(FOOT\) is well-defined in \(T\), then \(T\) is inconsistent. The following proof is originally posted by the Googology Wiki user p進大好きbot:

Suppose that \(\alpha_0\) is formalised in \(T\) by a defining formula \(\varphi(\alpha)\) with a free occurence of a variable term \(\alpha\). Then the existence of \(\alpha_0\) satisfying \(\varphi(\alpha_0)\) ensures that the existence of \(\beta < \alpha_0\) satisfying \(\varphi(\beta)\) by the definition of \(\alpha_0\). By the minimality of \(\alpha_0\), it implies \(\alpha_0 = \beta < \alpha_0\), which contradicts the well-foundedness of \(\alpha_0\).

Yet, by adding a collection of constant symbols to \(Form_∈\), we can define \(\alpha_0\) by a schema, and so on for each \(\alpha_n\), and so define \(Ord\) and \(Ord_2\) and so o. Yet the original formulation includes no such constant symbols." MasterOfArda (talk) 17:07, May 30, 2019 (UTC)

Hmm, I think that the description is inaprropriate. As I wrote above, formalisation with coding (without constant terms) is the different defining formula from formalisation with constant terms and schemas. Your \(\alpha_0\) does not satisfy the universality of \(\alpha_0\). Read my example with \(c \in \mathbb{N} \land \forall n \in \mathbb{N}, n < c\) above. We should distinguish \(\alpha_0\) and your alternative \(\alpha_0\), as we distinguish Rayo's number (which is well-defined in second order logic but not in ZFC) and its ZFC variant using provability (which is well-defined in ZFC). The names of \(\alpha_0\), FOOT, and BIG FOOT in that context are given only for stuffs defined by the original defining formulae. We should not use the same names for other notions defined by different defining formulae even if they look similar.
p-adic 21:49, May 30, 2019 (UTC)

I think we should note that it is possible to create a similar construction, simply not the original. MasterOfArda (talk) 04:51, May 31, 2019 (UTC)

Yeah, it is good as long as there is no abuse of notation, which is ambiguous for wiki users who do not know set theory well but just want to know whether it is actually ill-defined or not. I added an additional information of an alternative formulation. Could you like it?
p-adic 05:54, May 31, 2019 (UTC)

I think your additions make sense to the Big Foot page. I do have some questions about Big Biggedon. You said "For example, \(R\) is defined after setting the condition "\((\bar\in,R,F)\vDash t\text{ is an ordinal}\)", and this causes circular logic." But isn't the bigger problem that the relation \((\bar\in,R,F)\vDash \phi\) is not defined, since relativazation is only defined in Set Theory for structures of form \((M,\in)\)? —Preceding unsigned comment added by MasterOfArda (talkcontribs) 16:36, May 31, 2019 (UTC)

Thanks. If I understand correctly, the statement "\((\bar\in,R,F) \models \phi(t)\)" on a parameter \(t \in V\) and a \(\in\)-formula \(\phi(\alpha)\) with a free occurrence of a variable term \(\alpha\) just means "\((V,\bar\in,t) \models \phi(\alpha)\)". There are several conventions for "\((M,\bar\in,t) \models \phi(\alpha)\)".
  1. When \(M\) is a small set, then it is the usual satisfaction.
  2. When \(M\) is a definable class, then there are two conventions:
    1. The base set theory proves \(\phi(\alpha)^{(M,\bar\in,t)}\). This convention makes sense only when \(t\) is a definable set and \(\phi\) is a specific formula (i.e. given as an explicit statement instead of an abstract Goedel number).
    2. The satisfaction of \(\phi(\alpha)\) at \((M,\bar\in,t)\). This convention makes sense only when the base set theory can formalise the satisfaction of \(\phi(\alpha)\) at a class.
Since \(V\) is not small, the convention 1 is not applicable. I guessed that the intended convention is 2-2, because it is more useful than 2-1 in order to define large numbers. A point is that the axiom of the base set theory is not fully specified in the definition of Big Bigeddon. At least, unlike BIG FOOT, there is a choice of suitable axioms which enables us to define "\((V,\bar\in,t) \models \phi(\alpha)\)". For example, \(\textrm{MK} + (V=L)\) works. (I note that \(V = L\) or a weaker axiom is originally assumed in order to define the well-ordering on \(V\).)
That is why I did not regard it as a serious problem compared to the circular logic. For me, circular logic is as serious as contradiction, because it allows us to state anything wrong. Of course, the lack of the clarification of the axiom is a problem, too.

Thank you for the assistance. So the problem is \(\phi(\alpha)\) is a formula in \(\{\in,R,F\}\), which is defined using \(\phi(\alpha)\). Yet \(\phi(\alpha)\) is here the statement "\(\alpha\ is\,an\,ordinal\)," and doesn't require using \(R\) or \(F\), so there is no circular logic because \(\phi(\alpha)\) doesn't use \(R\) or \(F\), unless I misunderstand? MasterOfArda (talk) 00:21, June 1, 2019 (UTC)

Right. Actually, the resulting "values" of the redefined \(R\) and \(F\) do not depend on the original \(R\) and \(F\). On the other hand, their definitions themselves include the appearrence of the original \(R\) and \(F\). Then it causes circular logic, strictly speaking. Now I guess that this is kind of typo, because the definition of Little Bigeddon also contains several typos.
p-adic 01:11, June 1, 2019 (UTC)

That's strange. Several times, with constants \(c\), they write \(c^\bar\in\) to denote replacing \(\in\) with \(\bar\in\). I wonder why they didn't do "\(\alpha\ is\,an\,ordinal^\bar\in\)." MasterOfArda (talk) 01:29, June 1, 2019 (UTC)

For each explicit \(\in\)-formula \(\phi\), it is verifiable in a suitable set theory that \((M,\bar\in) \models \phi\) is equivalent to \(\phi^{(M,\bar\in)}\) (not to \((M,\bar\in) \models \phi^{\bar\in}\)). Since I could not understand what you feel strange, I might misunderstand what you need to know.
By the way, could you mark "minour edit" when you just correct typos in your reply? Otherwise, whenever you make an edit, an alert comes to my e-mail address. Since I myself often forget to mark "minour edit", I have my default settings automatically mark "minour edit". (Then I often forget to remove the mark when I make a non-minour edit, though.) Thanks.
p-adic 01:43, June 1, 2019 (UTC)

Sorry about that. I am simply curious why the author did not write \(\alpha\ is\,an\,ordinal^\bar\in\) to denote saying \(\alpha\) is \(\bar\in-\)transtive and well-ordered by \(\bar\in\), when they did so in other places. It's not a question, just a thought. MasterOfArda (talk) 02:26, June 1, 2019 (UTC)

I see. In order to define a large number, \(\models\) is much useful and safer than the relativisation by the difference of their presentability. As I wrote above, they play the same role when we deal with a single (or finitely many) explicit formula. On the other hand, when we consider infinitely many formulae, \(\models\) plays a role which the relativisation does not. Also, when we consider a definable proper class, the relativisation plays a role which \(\models\) does not, but this merit rarely appears when we use set theory stronger than \(\textrm{ZFC}\). That is why we usually prefer \(\models\) to the relativisation even when the relativisation could work, I think.
p-adic 02:56, June 1, 2019 (UTC)

Thank you for your assistance. MasterOfArda (talk) 02:57, June 1, 2019 (UTC)

It is my pleasure. I am glad to talk about "mathematical" statements, because mathematical statements are very difficult for almost all googologists who are recently active here.
p-adic 06:04, June 1, 2019 (UTC)


In this article, you write about the function CoRayo ("is the greatest large function ever defined in ZFC set theory"). In this article, you write about the function defined in MK+ set theory. Classic MK theory as strong as ZFC+inaccessible cardinal. What is the strength of MK+ set theory? What about this article? It is about large function ever defined in ZFC+WA set theory. Which function will be stronger? Defined in MK+ set theory or defined in ZFC+WA or ZFC+rank-into-ranks? Scorcher007 (talk) 03:59, June 1, 2019 (UTC)

> Classic MK theory as strong as ZFC+inaccessible cardinal.
It is wrong. Maybe you are confounding the consitency strength with the strength. For example, \(\textrm{MK}+\) shares the consistency strength with \(\textrm{MK}\), but is much stronger than \(\textrm{MK}\). According to the definition originally described by Emlightened, \(\textrm{ZFC} + \textrm{WA}_0\) is not much stronger than \(\textrm{ZFC}\) because she does not require the non-triviality of \(j\). For example, the critical point \(\kappa_0\) is unique, but does not necessarily exist as she desired. (Therefore the explanation in her blog is wrong.)
Then \(\textrm{ZFC} + \textrm{WA}_0\) in her convention is much weaker than \(\textrm{NBG}\), which is weaker than \(\textrm{MK}\), which is much weaker than \(\textrm{MK}+\). Therefore a large function defined in \(\textrm{ZFC} + \textrm{WA}_0\) is usually much weaker than a large function defined in \(\textrm{MK}+\). Moreover, I explicitly constructed the defining formula of the uncomputable large number under \(\textrm{MK}+\) syntax-theoretically in the mata theory. Therefore my large number is not presented as the input of a large function defined in \(\textrm{MK}+\). Namely, it goes beyond such a function.
Of course, the question "which one is stronger?" makes sense only when we explicitly fix large numbers. I do not know a significant large number which needs \(\textrm{ZFC} + I0\) (rank-into-rank). If you just consider a naive extension of Rayo's number, it is not significantly large.
p-adic 05:59, June 1, 2019 (UTC)
What about computable functions? When f(n) eventually dominates every computable function in some theory. Like Friedman's FLCI(n) eventually dominating ZFC+"there exists n-Mahlo cardinal" or Friedman's FFT(n) eventually dominating ZFC+"there exists an n-subtle cardinal" or USGDCS(n) ZFC + "there exists an n-huge cardinal". Are these functions stronger than some f(n) eventually dominates every computable function in MK? Scorcher007 (talk) 06:38, June 1, 2019 (UTC)
I think that you are comfounding "a computable function in a theory" and "a provably total computable function in a theory", because a computable function in \(\textrm{ZFC}\) + some large cardinal axiom is actually a computable function in \(\textrm{ZFC}\) as long as the cord is given in an explicit natural number. Therefore I replace "computable" in your question by "provably total computable".
Although I do not know the precise comparison of the strength restricted to arithmetic sentences such as the termination, I guess that \(\textrm{MK}\) set theory is not stronger than \(\textrm{ZFC}\) + sufficiently strong large cardinal axiom. Since I have no proof or evidence on this, I might be wrong.
In my opinion, it is better to fix an axiom when we consider computable large numbers, because you can add the axiom "my function terminates" to the base theory as long as no one has a proof of the contradiction. For example, to create a computable large number in \(\textrm{ZFC}\) set theory is much significant. It is pity that gooogologists talking about transcendental integers usually do not understand them and have thoughts like "If we need to work in \(\textrm{ZFC}\) set theory, then it is impossible to create a computable large number beyond the least transcendental integer! It is not interesting!" It is completely wrong. For example, my greatest computable number is far beyond the least transcendental integer and is well-defined in \(\textrm{ZFC}\) set theory.
p-adic 07:13, June 1, 2019 (UTC)
You: It is pity that gooogologists talking about transcendental integers usually do not understand them
Also you:
"Although I do not know the precise comparison of the strength restricted to arithmetic sentences such as the termination, I guess that..."
"I have no proof or evidence on this, I might be wrong."
"In my opinion, it is better to..."
Then why are you still speaking like you know what you're talking about? MrSpacedralf (talk) 00:30, November 30, 2019 (UTC)
Oh, couldn't you even understand the difference between expressing "this is just a guess" and stating "this is a truth"?
p-adic 01:30, November 30, 2019 (UTC)

"Ill Defined"

What does Ill-Defined mean, and why are some numbers associated with it? (like BIG FOOT)

Luckyluxius (talk) 20:52, December 6, 2019 (UTC)

"Ill-defined" means "not well-defined", and "well-defined" means "uniquly characterised by a mathematical formula" (like equalities, inequalities, set thoeretic properties, and so on). See each article on an ill-defined notion (like BIG FOOT), because specific reasons are already specified in them. For example, it was verified by me that the existence of BIG FOOT contradicts set theory, i.e. it does not actually exist.
p-adic 23:51, December 6, 2019 (UTC)

Is it just me, but are you a little bit rude at defining numbers when people ask?

I have seen your comments and you seem a tiny bit rude when you call stuff ill-defined when people ask why. —Preceding unsigned comment added by Luckyluxiuz (talkcontribs) 17:47, December 30, 2019 (UTC)

If you feel my reply above uncomfortable, then I am sorry about it. I tried to sincerely explain why BIG FOOT is ill-defined in a precise manner. But I am not understanding what phrases above make it look rude. Could you tell me which ones you feel uncomfortable?
p-adic 23:20, December 30, 2019 (UTC)


you know im trying to make all 0s like in the millillion page if we can help together on doing thisFleetave1 (talk) 01:00, January 17, 2020 (UTC)

It is not helpful for us to understand the number. Say, writing down 1+…+1 (one thoudand 1s) is not helpful to understand 1000.
p-adic 03:25, January 17, 2020 (UTC)
Writing out 3 septillion zeros isn't even possible. Plain'N'Simple (talk) 04:07, January 17, 2020 (UTC)
p-adic 04:26, January 17, 2020 (UTC)

About User talk:FundamentalSeq

There's no need to be so harsh about a relatively new user's edit. The user probably did that edit in good faith, and might genuinely thought the edit was just a simple rephrasing. Instead of being so harsh, tell them what they did wrong, and talk with them to find a better solution. -- ☁ I want more clouds! ⛅ 15:57, January 27, 2020 (UTC)

Excuse me, but am I so harsh...? Literally, you can immediately see that it is not rephrasing, because it refers to an OCF, which does npt appear in the original description. Believing rare possibility unreasonably does not help us to tell others how to improve the way to edit articles. You know that we can make mistakes when we do not know well about the topics. Therefore I honestly show the user a simple guideline, i.e. not to add new information about what he or she does not know. Do you have a better solution? At least, I did my best in good faith.
p-adic 23:19, January 27, 2020 (UTC)
I have added instructions for the case which you care very much about, i.e. the case where the user somewhy did not notice the fact that he or she wrote a new explanation. Is it sufficient?
p-adic 02:05, January 28, 2020 (UTC)
I don't think it would be good to tell a user to "not to add new information about that the user doesn't know". Wikipedia has a "be bold" guideline, and I think we should follow that too. -- ☁ I want more clouds! ⛅ 16:43, January 29, 2020 (UTC)
It does not mean "feel free not to be responsible".
p-adic| 04:24, January 30, 2020 (UTC)


Mango523WNR (talk) 13:36, February 2, 2020 (UTC),2020#WikiaArticleComments

I'm mango523wnr, how about my new function?

Nihao. I wrote an estimation in a comment. (Sorry if I am wrong.)
p-adic 13:39, February 2, 2020 (UTC)

Requests by Mango523WNR

Mango523WNR (talk) 12:20, February 6, 2020 (UTC)Something,_2020#WikiaArticleComments

I think that your local setting does not allow other users to comment on your latest blog post. By the way, could you understand what I wrote in my comment to your previous blog post? In order to improve your numbers, it is good to study from feedbacks, and hence I hope that you did.
p-adic 12:42, February 6, 2020 (UTC)
Hmm, I actually can post blog comment under her post... Triakula (talk) 12:47, February 6, 2020 (UTC)
Oops. But I am certain that I saw the warning saying that I am not allowed to submit comments... (Now I can submit comments, and hence it might be a temporary error on wikia.)
p-adic 12:55, February 6, 2020 (UTC)

Mango523WNR (talk) 23:15, February 9, 2020 (UTC)

It looks interesting, but I advice you not to comment to talk pages in that way. Putting only url + your signature is usualy regarded as an impolite attitude. (Since you seem to be young, I guess that you do not intend to be rude and are just unaware of how your attitude looks.) For example, if you want a feedback on a url, then it is good to add "Could you read this blog post? I appreciate any feedbacks." or somthing like that.
p-adic 23:32, February 9, 2020 (UTC)

What Happened?

I saw that will be deleted just now, WHAT HAPPENED? COULD YOU TELL ME???

Read the description in the deletion tag.
p-adic 09:46, February 13, 2020 (UTC)


There is a typo in your Twitter. It should be "non_archimedean". — Best regards, Triakula 13:54, February 13, 2020 (UTC)

Oops, thank you. Now I corrected the link to my twitter account.
p-adic 14:26, February 13, 2020 (UTC)

w^^^w=BHO? (correction: w^^^w=LVO?)

GoogolFan2000 (talk) 23:20, February 14, 2020 (UTC) When you replied to a comment I made on a blog post explaining the standard and climbing methods of ordinal hyperoperators, you mentioned that there was another theory on ordinal hyperoperators where w^^^w is the Bachmann-Howard ordinal. Could you please explain to me where you found this mythical theory and how it works?

Hi.You are referring to this page, right? Then there seems to be nobody who is talking about ω^^^ω or BHO. If you are referring to another page, then I would like to remember the context. Could you tell me the page?
p-adic 23:46, February 14, 2020 (UTC)
My memory seems to have failed me a bit on the specifics, but yeah, it was the page you linked. GoogolFan2000 (talk) 23:51, February 14, 2020 (UTC)
I see. If I am correct, I did not say "ω^^^ω = BHO", and hence I have no source on it.
p-adic 00:01, February 15, 2020 (UTC)
In that case, could you explain the one where w^^(w+1)=phi(w,0), the one where w^^^w supposedly equals the LVO? GoogolFan2000 (talk) 00:04, February 15, 2020 (UTC)
As I wrote in the comment, I have never seen a definition for the equality. I searched it now, but I could not find a related argument... Therefore it might be my miunderstanding. Sorry for not being helpful.
p-adic 01:48, February 15, 2020 (UTC)

My web book

Please check my web book and see if it has any mistakes.Zongshu Wu 12:10, March 19, 2020 (UTC)

  1. In section 1.2, the rule 1↑^c b = 1 is redundant because it follows from other rules.
  2. In 1.3, the phrase "There can numbers" seems to be a typo of "There can be numbers".
  3. In Case 1.1 in 1.3, the 0's in the expression 10R10R10 are not zero but are digits in dicimal expansions of tens. Therefore the explanation seems to be wrong.
  4. In Case 1.2 in 1.3, the written rule is not valid because it is applicable to an expression in a non-unique way. For example, the rule is applicable to 1R{0}{0} in two ways. You need to specify which 0 you are referring to by saying "the leftmost 0 right to R". More precisely, it is better to clarify that you are writing a rule to compute a valid expression of the form nRb, where n and b include no R. Then you can write "Case 1: 0 occurs in b. Case 1.1: The rightmost entry of b is 0 and no other entry is 0. Case 1.2: b is of the form c{0}d, where c is a formal string in which 0 does not occur..." or something like that. If you are only considering the case where b includes at most one number, then you need to clarify it. Otherwise, the written Case 1 should be applicable to any case in which b includes 0.
  5. In Case 2 in 1.3, the case division is overlapping, and hence is invalid. For example, the written Case 1 and 2 are both applicable to 1R2{0}, because there are both 0 and non-zero number 2.
  6. In Case 3 in 1.3, the case division is overlapping, too, and hence is invalid. So if your explanations are correct, R function is just ill-defined. If your explanations are incorrect, then you need to fix them.
  7. In section 2.1, the phrase "Most numbers are outputted by functions" does not make sense, because every number is an output of a function such as a constant function. Given two numbers a and b, there always is a function f such that f(0) = a and f(1) = b. Therefore the "approximation" due to the distance of the input of an arbitrary function is meaningless. In the googological sense, the "distance" of two numbers is essentially scaled by a more intuitive (unformalisable) measure such as "the number of known numbers between them". (Here, "known numbers" roughly means numbers admitting googological names uniquely characterising them and consisting of 100 or less letters or something informal like that.)
  8. In section 2.1, the similarity of the growth rates of functions is ill-defined because we are not allowed to quantify the intuitive statement "f(x) ≈ g(h(x))". We can only quantify mathematical statements. Qunatification is quite sensitive.
  9. In section 2.1, the "infinite block of fractions" is displayed as "1/11/21/31" without spaces in my environment, and hence might be a typo if it is not a local problem on my computer.
  10. In section 2.1, the proof of "there are possible 2^ℵ_0 real numbers" is wrong, because you just showed a way to expand a real number into ℵ_0 digits. It just means that "there are at most 2^ℵ_0 real numbers". In order to complete the proof, you need to show that there are actually at least 2^ℵ_0 numbers.
  11. In section 2.1, "Strangely, the Continuum hypothesis hasn't been proven or disproven yet, but it has been proven that you CAN'T prove or disprove it!" is not precise. The precise statement is "As long as ZFC is consistent, CH is neither provable nor disprovable." The assumption on the consistency itself is not provable (if ZFC is actually consistent), and hence cannot be removed from the statement.
  12. In section 2.3, the phrase "For example, the sequence {ω,ω+1,ω+2,ω+3,...} is the fundamental sequence of ω2." seems to be a typo of "For example, the sequence {ω,ω+1,ω+2,ω+3,...} is a fundamental sequence of ω2."
  13. In the definition of a fundamental sequence in section 2.3, the α in ω^α[n] and the β in (α+β)[n] should be of cofinality ω. Even if we add the conditions, the fundamental sequences are ill-defined because those rules are not uniquely applicable to a single ordinal. For example, rule 4 is applicable to ω+ω^2+1+ω^2 in so many ways. Moreover, when we apply rule 3 to ζ_2, it causes a circular reference.
Since there are the same issues on R functions in latter sections, I stop here. The contents look quite interesting and instructive, and hence I hope that you will fix all errors.
p-adic 14:11, March 19, 2020 (UTC)

Hmmmmm... Mistake #4 can be removed, because in nR{0}{0} we stop when we scan the first {0}, and we don't get to the second {0} in a long time. Zongshu Wu 03:10, March 21, 2020 (UTC)

Also #5 is good. When we scan nR2{0}, we first scan the 2. Then apply Case 2. Then Return. So we'll never get to scanning and decomposing the {0} until later. Zongshu Wu 03:16, March 21, 2020 (UTC)

So how do I define "approximation"? Also, #13 works like this:

What is (ω+ω^2+1+ω^2)[3]?

it is ω+(ω^2+1+ω^2)[3]?



Then finally ω+ω^2+1+ω3. 

Also #9 works for me.

Zongshu Wu 05:41, March 21, 2020 (UTC)

> 4,5
No. As I wrote, I referred to the written rules. You can change the rules so that they work as you intend, but the writtent rules currently do not work as you intend. It means that if you do not change the rules, then the resulting system is just ill-defined.
> So how do I define "approximation"?
There is not formal definition of "approximation". I can say that at least what you defined is not what we call "approximation". Isn't my formulation of the approximation in 7 for you? It is not an almighty formuation, but works better.
> Also, #13 works like this:
No. You "defined" the value of (ω+ω^2+1+ω^2)[3] using the expression. When you define α[n], you can just use α, n, objects which have already been defined, and quantified objects. For example, when α = ω+ω^2+1+ω^2 and n = 3, then the expression "ω+ω^2+1+ω^2" of α has not been defined, and hence your system of fundamental sequences is ill-defined. Should I show a more elementary example? Compute α[3] in the case α = ω^2+ω+1+ω. Then you will find that the value is actually ill-defined.
> Also #9 works for me.
So the remaining issues are 1,2,3,4,5,6,7,8,10,11,12, and 13.
p-adic 06:41, March 21, 2020 (UTC)

OK. So in the next update I will fix issues 1 2 3 7 8 10 11 12, and release Chapter 4 (OCFs). I still do not understand how I would fix the other issues, could you please explain more clearly, issues 4 5 6 13?Zongshu Wu 10:38, March 21, 2020 (UTC)

> 4
The issue is that Case 1 just refers to "the existence of 0" right to R. In that case, an expression can contain two or more 0s right to R. Then "the 0" in Case 1.1, Case 1.2, and Case 1.3 does not make sense, because such an occurrence of a 0 is not necessarily unique. For example, you can replace Case 1 by "There is precisely one 0" or "The leftmost number occurring right to R is 0" so that such a 0 is unique.
Also, you are simultaneously trying to define valid expressions by saying "In this case, there must be an R before the 0" and something like that. But the definition is incomplete, because you have not defined or quantified n. For example, you can clarify that the current exprresion is of the form nRb, where n is a positive integer and b is formal strings including no R, as I recommended in the comment above. Then you can more clearly replace Case 1 by "There is precisely one 0 in b" or "The leftmost number in b is 0", and replace the restriction in Case 1.1 by "there must be an R before the 0" by "b = 0". In that case, the rule nRb = 10^n is not ambiguous because you have quantified n as any positive integer. See this blog post on the issue on the lack of appropriate quantification of symbols.
> 5,6
The issue is that Case 1 just refers to "the existence of 0" right to R, Case 2 just refers to "the existence of non-0 number", and Case 3 refers to "the existence of an enbrace". For example, consider the expression 1R{1}{0}{1}{0}. Then there are 0 (1R{1}{0}{1}{0}), a non-0 number (1R{1}{0}{1}{0}), and also an enbrace (1R{1}{0}{1}{0}) right to R, and hence all of the three rules are applicable. It means that the case division is overlapping.
Also, unlike Case 1, you have not defined a valid expression for these cases. Nevertheless, you wrote nR(a+1) = …. Does it mean that you restrict b to be expressions of the form a+1? How do you solve nRb when b is not of the form, say nR1{2}? It has non-0 numbers 1 and 2, and hence Case 2 should be applicable because there is no written restriction to Case 2 other than that b includes a non-0 number.

> 5
The phrase "the number" in Case 2.1 and Case 2.2 does not make sense, because such a non-0 number is not necessarily unique. You need to solve this issue by a similar way to the solution for Case 1.
> 6
The phrase "continue scanning" does not make sense because you have not specified the location to scan. For example, let us consider 1R{1}{0}. Then Case 3 is applicable. But there is no written rule to characterise the substring of 1R{1}{0} to scan. 0? 1? How do you scan it? You have not written those rules.
> 13
To begin with, could you tell me the definition of α[3] in the case α = ω^2+ω+1+ω?
p-adic 11:33, March 21, 2020 (UTC)

> the definition of α[3] in the case α = ω^2+ω+1+ω?


Also, for the R function issues, I realized that I forgot to say "scan from left to right".Zongshu Wu 12:48, March 21, 2020 (UTC)

> α[3]
Then could you tell me the definition of α[3] in the case α=ω^2+ω2 ?
> Also, for the R function issues, I realized that I forgot to say "scan from left to right".
It is still ambiguous. What does "scan from left to right" precisely mean? For example, when you want to evaluate nRb such that b includes at least one enbrace, what do you precisely do? To search the leftmost occurrence of a substring in b which is also a valid expression? To search the leftmost occurrence of {? After searching such an occurrence, what do you do? Replace it by something?
p-adic 12:56, March 21, 2020 (UTC)

> α[3]

So (ω^2+ω2)[3]=ω^2+(ω2)[3]=ω^2+(ω+ω)[3]=ω^2+ω+(ω)[3]=ω^2+ω+3

> still ambiguous

"Scan" doesn't mean to search for anything, it is just looking one symbol at a time. for example when you scan nR{0}{1} First you see the leftmost {. So according to case 3, move on, and then you see the 0. Then apply Case 1.2 to change the {0} to n. Then return. When you scan again, you find the n. So apply case 2.1. ...Zongshu Wu 13:22, March 21, 2020 (UTC)

> (ω^2+ω2)[3]=ω^2+(ω2)[3]=ω^2+(ω+ω)[3]=ω^2+ω+(ω)[3]=ω^2+ω+3
Now it is time to show contradiction. We have ω^2+ω2 = ω^2+ω+1+ω. Then we have α[3] = ω^2+ω+3 ≠ ω^2+ω+4 = α[3] in the case α = ω^2+ω2 (= ω^2+ω+1+ω). This means that α[3] is ill-defined. The ill-definedness is mainly due to the use of an expression of α rather than α itself, although it is not unique.
> scan
I could not understand what you mean, maybe because you have not solved the issues on the overlapping case division. Anyway, could you solve the issues, and clarify what you do in the scanning step in your site?
p-adic 13:28, March 21, 2020 (UTC)

> Now it is time to show contradiction.

would it work if I said that ω^α1+ω^α2+...+ω^αk where α1≥α2≥...≥αk? Also, I have given an example of R function on my site now. Zongshu Wu 14:28, March 21, 2020 (UTC)

> would it work
It would not work because α[3] is still ill-defined for the case α = ω^{ζ_2}. The issue is that such an expression is not unique.
> Also, I have given an example of R function on my site now.
Giving an example is good, but does never give a definition. You need to fix a precise definition in order to solve the ill-definedness issue.
p-adic 14:36, March 21, 2020 (UTC)

OK, I fixed 13 by forcing a "standard form" to apply the rules. Zongshu Wu 18:23, March 21, 2020 (UTC)

It sounds good, although I have not checked the updated contents. I am glad if you tell me when you finish all the revision. Then I will check the contents again.
p-adic 22:13, March 21, 2020 (UTC)

Now, how should I define fundamentalseqs for Bachmann's psi?Zongshu Wu 02:24, March 22, 2020 (UTC)

It is the same as you do when you define fundamental sequences for other systems such as Wainer hierarchy, Veblen hierachy, or Buchholz's hierarchy. Namely, define strictly increasing sequences which converge to the given limit ordinals of cofinality ω only using the ordinals themselves and other objects which have already been defined. (It means that you are not allowed to use non-unique unquantified objects such as arbitrary "expressions of ordinals".)
p-adic 02:51, March 22, 2020 (UTC)

Also I don't quite understand cofinality. What is cof(ω+3)?Zongshu Wu 11:34, March 23, 2020 (UTC)

Nevermind. its 1.Zongshu Wu 11:43, March 23, 2020 (UTC)

Actually, cofinality is one of a difficult notion which is required to understand an OCF. Remember that when we define a system of fundamental sequences for ordinals in an OCF, we directly use the cofinality. It means that when we define a recursive system of fundamental sequences for terms in an ordinal notation associated to an OCF, we need to define a recursive analogue of the cofinality. For example, see the article on Buchholz's function if you do not know how to define a recursive analogue of the cofinality for this purpose.
p-adic 11:50, March 23, 2020 (UTC)

The update is out. :) Zongshu Wu 13:01, March 23, 2020 (UTC)

OK. I will check it later.
p-adic 21:56, March 23, 2020 (UTC)
> 4,5,6,11
The issues have not been solved... Please read my comments above on those issue again. If there are uncertain points, please ask me.
> 10
> This is because there are already 2^ℵ0 reals between 0 and 1. There are ℵ0 digits after the decimal point, and for each combination there is a real number corresponding to it. So there are at least 2^ℵ0 reals.
The reasoning is still wrong, because two distinct decimal expansions do not necessarily correspond to two disctinct real numbers.
> 13
> An ordinal here is in standard form if it can be expressed as ... where all of the α ordinals are in standard form and α1≥α2≥...≥αk.
It does not work because it is based on a circular logic. For example, is ε_0 an ordinal in standard form? In order to check this, we need to consider whether it is expressed as ω^α where α is in standard form. However, α should be ε_0 itself. Therefore the definition is not inductive.
Moreover, the notion of a standard form is basically a predicate on an expression but not on an ordinal. Therefore if you fix the issue, your predicate does not work as you intend. For example, is 1+ω in standard form? In order to check it, it suffices to check 1 is in standard form, because we have an expression of 1+ω given as ω^1. I guess that you intend that 1 is in standard form, and hence 1+ω is in standard form in your convention although you might not intend so.
p-adic 12:36, March 24, 2020 (UTC)

o dear...

also, is the ocf section ok?Zongshu Wu 04:41, March 25, 2020 (UTC)

There are several errors. Since they might be due to some misunderstandings based on previous sections, I think that it is better for you to fix errors which have already been pointed out above. For example, I list portion of errors in the OCF section.
  1. > An uncountable ordinal is the order type of a set: The notion of "the order type of a set" is ill-defined.
  2. > A set that has order type: It is ill-defined by the same reason.
  3. > Note that here we cannot write ℵ_{ℵ_0} here, since the subscript must be an ordinal (it expresses order).: It is wrong. We can write it because the cardinal ℵ_0 is an ordinal by definition.
  4. > Though it is not the ℵ fixed point since it makes no sense to say ℵ_ℵ_ℵ_ℵ_.... When you have to express it as a cardinal, you do ℵ_Ω_Ω_Ω_....: It is wrong by the same reason.
  5. > For example the cofinality of any countable limit ordinal is ω. ... The "fundamental sequence" of ω+2 is {ω+1}) and the cofinality of 0 is 0: It conflicts your explanation "An ordinal is a successor when it is something plus 1. Otherwise it is a limit ordinal. Only limit ordinals have fundamental sequences." in the FGH section. The explanation implies that the notion of a fundamental sequence for a successor ordinal is ill-defined, and 0 is a countable limit ordinal. On the other hand, you now write that the singleton of the predeccessor forms a fundamental sequence of a successor ordinal, and every countable limit ordinal has cofinality ω. You need to fix the definitions.
  6. > Tho, cofinality is sometimes hard to see directly: What is "Tho"?
I stop here, because it is better to fix errors in previous sections.
p-adic 05:01, March 25, 2020 (UTC)

Does that mean that the actual OCF is ok, just the explanation?Zongshu Wu 05:10, March 25, 2020 (UTC)

The definition of the OCF works, but it is not what Buchman defined, is it? I guess that it is Rathjen's recast rather than Buchman's original ψ. Also, fundamental sequences are ill-defined by the same reason as fundamental sequences in the FGH section are ill-defined.
p-adic 05:18, March 25, 2020 (UTC)

So, about these issues:


didn't i already say how to scan?




I will replace "can be expressed as" as simply "looks like". In fact, every non-standard expression can be expressed in stardard form. And I'll put a upper limit of the ordinals here.

Zongshu Wu 12:00, March 26, 2020 (UTC)

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