User talk:P進大好きbot

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)
 * 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 me...here 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)

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 (talk • contribs) 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)
 * 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)
 * 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)
 * 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)
 * 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(Lσ≺1Lσ+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)
 * 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 \((M,\in)\)?


 * 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:
 * 3) 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).
 * 4) 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.
 * p-adic p進大好きbot

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)

MK+ vs ZFC+WA
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 computable 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)