## Contents

- 1 Welcome
- 2 wikiballs
- 3 Transfinite
- 4 Template:wedges
- 5 Congrats
- 6 Please, stop undo edits
- 7 name
- 8 prime counting function in FOA
- 9 check it
- 10 surreal fundseqs
- 11 Continued fractions: Copeland-Erdos vs. Champernowne
- 12 TM specialist
- 13 Bachmann OCFs
- 14 Friedman
- 15 stackoverflow
- 16 Editing blogs.
- 17 dude
- 18 w_1^ck = w_1
- 19 IRC
- 20 Problem 1
- 21 BIG HUGE REMINDER READ NOW SUPER IMPORTANT
- 22 Labelled graph minor
- 23 Birthday
- 24 Wiki Hero
- 25 Couldn't you solve this?
- 26 Milestone
- 27 Vandalism
- 28 A few questions about uncomputable functions
- 29 Anti-Nazi policy
- 30 BIG FOOT source
- 31 Turing machines
- 32 IRC
- 33 Relation of admissiblity and stablity
- 34 a completely random question
- 35 Judge the rate of growth
- 36 Your lower bound for tree (3)
- 37 Doubts

## Welcome

Hi, welcome to Googology Wiki! Thanks for your edit to the **Talk:Subcubic graph number** page.

Please leave a message on my talk page if I can help with anything! -- Ikosarakt1 (Talk) 16:18, January 28, 2013

## wikiballs

try this out. ^{\(a\)\(l\)\(t\)} 09:28, March 16, 2013 (UTC)

## Transfinite

Note: omega and Epsilon-zero is not infinite. ^{\(a\)\(l\)\(t\)} 11:48, March 29, 2013 (UTC)

- Depends on how one defines infinite. If we mean with it that something is larger than anything finite then transfinite -> infinite. LittlePeng9 (
**talk**) 12:03, March 29, 2013 (UTC)

## Template:wedges

Vote now! ^{\(a\)\(l\)\(t\)} 12:11, April 26, 2013 (UTC)

## Congrats

Congrats you with birthday and age \(2 \uparrow\uparrow 3\). Ikosarakt1 (talk ^ contribs) 11:47, May 23, 2013 (UTC)

Actually, today I hit 15. I earlier said I'm 15 because of rounding error. So yeah, smallest odd semiprime. LittlePeng9 (**talk**) 13:43, May 23, 2013 (UTC)

By the way, soon I shall reach really important milestone: July 8, 2013 I shall live my half-billionth second. Ikosarakt1 (talk ^ contribs) 14:20, May 23, 2013 (UTC)

Congratulations with your birthday (and age \(2 \downarrow\downarrow 3\), this time for real :P) Wythagoras (talk) 18:54, May 23, 2014 (UTC)

- That also happens to be equal to \(^3\!2\), which is equal to \(2\uparrow^23\). 80.98.179.160 16:18, December 24, 2017 (UTC)

## Please, stop undo edits

Please, stop undo Ikosarakts edits. It is easy to fix. Wythagoras (talk) 09:13, August 8, 2013 (UTC)

Sorry, didn't know you edit them too. LittlePeng9 (**talk**) 09:18, August 8, 2013 (UTC)

Temporarily I got a laptop just now, so I can edit normally. Ikosarakt1 (talk ^ contribs) 11:45, August 8, 2013 (UTC)

Hey, so you're an admin now, considering that I won't be around much and Ikosarakt could use some support. FB100Z • talk • contribs 20:42, September 22, 2013 (UTC)

## name

i kinda...found out your real name while stumbling upon a publicly available source. i won't disclose anything publicly to avoid drawing attention to it, but next time on chat i'll PM you where i found it so you can cover up. sorry :/ you're.so.pretty! 08:54, April 12, 2014 (UTC)

- You don't have to be sorry, I expected you to find it eventually :P I wouldn't touch that topic if I wasn't aware of possible consequences. I don't consider my name to be any sort of top secret, if you asked me I'd tell you. But still I'm quite curious about where you have found it. LittlePeng9 (
**talk**) 11:18, April 12, 2014 (UTC)

## prime counting function in FOA

so i have all the pieces in place for defining \(\pi(n)\), except one that seems potentially problematic, namely a function that counts the number of 1's in the binary expansion of *n*. any ideas? you're.so.pretty! 22:15, April 14, 2014 (UTC)

- okay so OEIS tells me that this function is also equivalent to the largest integer
*a*such that \(2^a | \binom{2n}{n}\), or \(2^a | \frac{(2n)!}{n!^2}\), but i don't see how to do factorials. every step i take seems to do some sort of goalpost moving you're.so.pretty! 00:51, April 15, 2014 (UTC)

- I think I have found everything we need: here (pages 294-298) are preliminary definitions for defining exponentiation, but I'm pretty sure this can be used to define sequence of primes, and thus also \(\pi(n)\). Idea is to have a number which codes all numbers up to a given point, and from this it is possible to add next number to this sequence. This will work, but won't be easy to formalize. LittlePeng9 (
**talk**) 10:51, April 18, 2014 (UTC)

## check it

bam you're.so.pretty! 22:57, May 23, 2014 (UTC)

bam LittlePeng9 (**talk**) 04:20, May 24, 2014 (UTC)

## surreal fundseqs

i realize now that if we allow non-monotonic fundseqs, we can do something like w/2 = lim(0, w, 1, w-1, 2, w-2, ...). however i'm not aware of any definitions of the limit for surreal numbers, so maybe i'm rejoicing too soon you're.so.pretty! 20:59, June 10, 2014 (UTC)

- I don't think there is any useful notion of limit in surreal numbers world, because, for example, between two "walls" of this limit there still is w/2+1, w/2-1, not to mention the whole wilderness of sums of infinitesimals. I, however, don't rule out existence of limits in surreals, though I can't even see when we can call sequence convergent. LittlePeng9 (
**talk**) 21:22, June 10, 2014 (UTC)- w/2+1 and w/2-1 seem like they would fall on the other side of a classification of surreals analogous to limit vs. successor ordinals. Although I've unfortunately misplaced my copy of ONAG (it'll turn up sooner or later), I wouldn't be surprised if Conway has already discovered and formalized that distinction. Maybe it's something like having a limit ordinal birthday, I'm not sure. you're.so.pretty! 23:11, June 10, 2014 (UTC)

- I think of the following definition for non-monotonic sequence a_1,a_2,... It's a number a with the least birthday value, and, for every natural number n, there are k,l>n such that a_k<a<a_l. I don't know how this'd work, and it doesn't define unidirectional limits. (what is ONAG, btw?) LittlePeng9 (
**talk**) 05:28, June 11, 2014 (UTC)*On Numbers And Games*by John Conway, which is about surreal numbers and their applications in combinatorial game theory.- Here's another guess. Define lim
_{2}(*S*) for a__countably infinite__set*S*of surreals as follows. Partition*S*into two disjoint sets*L*and*R*so that 1)*L*is well-ordered by <, having order type at most w, 2)*R*is well-ordered by >, having order type at most w 3) all elements in*L*are strictly less than all those in*R*. Then lim_{2}(*S*) = {*L*|*R*}. lim_{2}is not total (at least not over the power set of On_{2}) but I think it is single-valued — that is, it seems that the*L*/*R*partition is unique. you're.so.pretty! 07:32, June 11, 2014 (UTC)

- I think of the following definition for non-monotonic sequence a_1,a_2,... It's a number a with the least birthday value, and, for every natural number n, there are k,l>n such that a_k<a<a_l. I don't know how this'd work, and it doesn't define unidirectional limits. (what is ONAG, btw?) LittlePeng9 (

Proof of uniqueness: Let *L*_{1}/*R*_{1} and *L*_{2}/*R*_{2} be two such partitions of *S*. We wish to show that *L*_{1} = *L*_{2}.

If *L*_{1} and *L*_{2} are both empty, then we are done. Otherwise, let *x* be the least element of *L*_{1}, which exists because *L*_{1} is well-ordered by <. *x* must therefore be the least element of *S* (any smaller elements must belong to *R*_{1}, which contradicts condition 3). Now *x* belongs to either *L*_{2} or *R*_{2}. Suppose *x* belongs to *R*_{2}: *L*_{2} is empty and *x* is the least element of *R*_{2}. But since *S* is infinite, so is *R*_{2}, and therefore its order type with respect to < is at least w. Since *R*_{2} has a least element, its order type cannot be w, which contradicts condition 2. Therefore, *x* is in *L*_{2}.

Since *x* is in both *L*_{1} and *L*_{2}, we consider *S* \ {*x*} as partitioned by (*L*_{1} \ {*x*})/*R*_{1} and (*L*_{2} \ {*x*})/*R*_{2}. We can make the same argument by induction.

you're.so.pretty! 17:57, June 11, 2014 (UTC)

- Note: induction will work here because, by assumption,
*L*_{1}and*L*_{2 }are both well-ordered. LittlePeng9 (**talk**) 18:10, June 11, 2014 (UTC)

## Continued fractions: Copeland-Erdos vs. Champernowne

Here's why the Champernowne constant has such a spiky continued fraction and Copeland-Erdos does not. Far into the digits of the former, you get very close to periodic behavior:

- ...184732861184732862184732863184732864184732865184732866...

which is a few digits off a perfectly repeating decimal. This means that the Champernowne constant will be extremely close to certain rational numbers. We have to take a bit of leap of faith to assume that some of these will be its convergents, but if this link is legitimate, it explains the spikes very well. An unusually close convergent needs to be compensated for with a large term in the continued fraction.

The lack of spikes in the Copeland-Erdos constant is a consequence of prime gaps getting wider and wider as we approach infinity, so there's less room for the periodicity as seen in Champernowne. However, spikes DO happen. They're just rarer than that of the Champernowne constant since the primes grow faster.

I would like to try a simulation to test the theory that concatenation of slower-growing integer sequences results in spikier continued fractions. you're.so.pretty! 21:20, July 30, 2014 (UTC)

- Thanks for that explanation. I actually had an idea on why this is so. I wonder if Champernowne's constant is Liouville. Such enormous terms in continued fraction strongly suggest so, but I wasn't able to find any mention of that fact. LittlePeng9 (
**talk**) 21:30, July 30, 2014 (UTC)

## TM specialist

Yes, I think I would agree with that. Wythagoras (talk) 12:02, August 3, 2014 (UTC)

- I'm a TM specialist too:

0 * 1 r 0

- King2218 (talk) 13:12, August 3, 2014 (UTC)
- <sarcasm>Neat! Yes, you are certainly a TM specialist. Maybe you'll get a Fields Medal for this discovery!</sarcasm> Wythagoras (talk) 14:08, August 3, 2014 (UTC)
- dude this is a new chapter in TM research. i cant believe it man, thats brilliant you're.so.pretty! 19:56, August 3, 2014 (UTC)
- You wouldn't believe it guys, but this machine can solve the halting problem! :) King2218 (talk) 14:19, August 4, 2014 (UTC)
- Oh my god, yes, you are 100% right - I wouldn't believe it :) LittlePeng9 (
**talk**) 14:54, August 4, 2014 (UTC)- I hope I get this right. Is is that if it runs longer than this machine it doesn't halt and if it runs less steps it halts or so? :P Wythagoras (talk) 16:33, August 4, 2014 (UTC)

- Oh my god, yes, you are 100% right - I wouldn't believe it :) LittlePeng9 (

- You wouldn't believe it guys, but this machine can solve the halting problem! :) King2218 (talk) 14:19, August 4, 2014 (UTC)

k guys ive improved King's work and have proven the following:

0 * 1 r

you're.so.pretty! 07:39, August 29, 2014 (UTC)

- This reminds me of a bug on old Windows versions which makes the upper-right buttons of windows appear as "0 1 r", among other graphical glitches. So that was a theorem all along?!?!? -- ☁ I want more clouds! ⛅ 12:16, August 29, 2014 (UTC)

## Bachmann OCFs

It took a while, but I finally located the original paper by Heinz Bachmann describing one of the first ever ordinal collapsing functions. There's one problem: it's in German.

Some help deciphering it would be appreciated. you're.so.pretty! 00:23, August 30, 2014 (UTC)

- I can help you, if you want. Go to the wiki chat (not IRC) and we'll start. I'll join when I see you are in. Also, it looks nice. Wythagoras (talk) 16:48, August 30, 2014 (UTC)
- Oh man, that'd be great! If you could find some way to decipher the psi/phi function in the paper and get a complete definition on ordinal collapsing function, I would probably love you forever you're.so.pretty! 19:57, August 30, 2014 (UTC)
- Sorry, I've been super absent-minded and have been forgetting to open Wikia Chat. Even worse, our time zones are nine hours apart, so I've probably been asleep for most of the times you've been on and vice versa :c you're.so.pretty! 09:45, August 31, 2014 (UTC)
- Well, you now went to sleep, and for me and Wyth it's around noon right now. LittlePeng9 (
**talk**) 09:51, August 31, 2014 (UTC)

## Friedman

Did you contact him? What did he say about other problem? Wythagoras (talk) 16:48, August 30, 2014 (UTC)

- I had to restate the other question, because the way I stated it was a bit too vague. I'm waiting for a reply right now. LittlePeng9 (
**talk**) 17:00, August 30, 2014 (UTC)- Hasn't he replied yet? Wythagoras (talk) 05:57, September 6, 2014 (UTC)
- Nope ;-; LittlePeng9 (
**talk**) 06:26, September 6, 2014 (UTC)

## stackoverflow

I'm not surprised that the stackoverflow folks are picky about your question. (They always are.) Here's my advice: since you are talking about code, *post actual code*. This is probably the main reason it was suspended. Also, make sure to post the minimal amount of code necessary for a complete stranger to understand it. it's vel time 07:10, September 17, 2014 (UTC)

## Editing blogs.

DONT. EDIT. http://googology.wikia.com/wiki/User_blog:Alejandro_Magno/My_ordinal_is_smaller_Remake EVER. AGAIN -- A Large Number Googologist -- 21:13, October 14, 2014 (UTC)

## dude

i'm going to have to ask you to stop leaving comments on Alejandro's posts, or in general interacting directly with him. you're just stirring up trouble. it's vel time 21:36, October 14, 2014 (UTC)

- Okay, I understand. Sorry for causing trouble. LittlePeng9 (
**talk**) 04:23, October 15, 2014 (UTC)

## w_1^ck = w_1

recall that the church-kleene ordinal is defined as the least ordinal that is not the order type of a computable well-ordering of a subset of the natural numbers, and that the first uncountable ordinal is defined as the least ordinal that is not the order type of any well-ordering of a subset of the natural numbers. is it possible for these to be equal in a "reasonable theory"? it's vel time 03:21, November 9, 2014 (UTC)

- I would believe this to be true. I think it might be possible in Kripke-Platek set theory. In the model \(L_{\omega_1^\text{CK}}\) neither if these ordinals exists, so I wouldn't be surprised if we could find different model in which both ordinals are the same. LittlePeng9 (
**talk**) 09:34, November 9, 2014 (UTC)- Is there such as KPC (Kripke-Platek set theory+Axiom of Choice)??? It's more possible there than in KP. 80.98.179.160 16:23, December 24, 2017 (UTC)

## IRC

goto irc —Preceding unsigned comment added by Vilius2001 (talk • contribs)

- Don't tell me what to do. LittlePeng9 (
**talk**) 09:29, December 18, 2014 (UTC)

have been waiting 5 hours on the irc ._.

## Problem 1

I have a solution, come to chat if you want to see it... Wythagoras (talk) 14:08, February 20, 2015 (UTC)

"Linking page to itself" > "Undoing it" is the best way for getting the WIKI EXPERT badge!!!!!!!

Just kidding, sorry :(

Antares 3^^^3 10:05, March 7, 2015 (UTC)

- Making a random edit and reverting it is never a problem, but I'd rather think of something more creative for a daily edit, e.g. replying to you on my talk page. LittlePeng9 (
**talk**) 12:26, March 7, 2015 (UTC)

- How many days do you have until you get the "Wiki Hero" badge? \(\ Antares.H \) 07:36, March 8, 2015 (UTC)
- He edited last not at January 17, 2015, so you could calculate that (why did I bother to look this up?) Wythagoras (talk) 15:40, March 19, 2015 (UTC)
- wut LittlePeng9 (
**talk**) 15:55, March 19, 2015 (UTC)- Seems like you are lucky... Wythagoras (talk) 16:22, March 19, 2015 (UTC)
- I'm not exactly sure what happened, but my guess is that on that day I made an edit really late, and because of time zone difference between me and the server it got recorded in my contributions page as an edit on the next day, but I dunno. Maybe I was just lucky. LittlePeng9 (
**talk**) 17:31, March 19, 2015 (UTC)

- I'm not exactly sure what happened, but my guess is that on that day I made an edit really late, and because of time zone difference between me and the server it got recorded in my contributions page as an edit on the next day, but I dunno. Maybe I was just lucky. LittlePeng9 (

- Seems like you are lucky... Wythagoras (talk) 16:22, March 19, 2015 (UTC)

- wut LittlePeng9 (

- He edited last not at January 17, 2015, so you could calculate that (why did I bother to look this up?) Wythagoras (talk) 15:40, March 19, 2015 (UTC)

## BIG HUGE REMINDER READ NOW SUPER IMPORTANT

work on the article on second-order arithmetic sometime later (he asked me to remind him on the irc) Cookiefonster (talk) 21:05, March 26, 2015 (UTC)

- Thanks dude, but you should've made this thing on Saturday. I'm not looking on my talk page unless I have someone edit there. LittlePeng9 (
**talk**) 21:06, March 26, 2015 (UTC)

## Labelled graph minor

Still I don't understand how your definition of graph minor on labelled graphs works.

For example,

for (1) to (5), is graph A a minor of graph B? And for (1) to (3), is graph B a minor of graph A? {hyp/^,cos} (talk) 05:56, May 2, 2015 (UTC)

- Are we using an ordering of labels in which labels form an empty order (i.e. no two labels are in relation) or the usual order on natural numbers? It is important here because the answer depends on this. LittlePeng9 (
**talk**) 07:26, May 2, 2015 (UTC) - What about "no two different labels are in relation"? If so, it seems that 5A isn't a minor of 5B, but I don't know the others. {hyp/^,cos} (talk) 08:37, May 2, 2015 (UTC)

- I believe that 4A is a minor of 4B, but the answer to all your other questions is no. For 4A/4B, the vertices labelled 2 and 3 can be combined into one vertex labelled with both 2 and 3, and the vertex labelled 3 in 4A can be mapped to it. For the others, observe that if graph A has a label that isn't <= any label from graph B, or graph A has more vertices with labels from set S than graph B has vertices with labels greater than or equal to labels from set S, graph A can't be a minor of graph B. Deedlit11 (talk) 09:33, May 2, 2015 (UTC)
- Precisely as Deedlit says. LittlePeng9 (
**talk**) 10:18, May 2, 2015 (UTC)

- Precisely as Deedlit says. LittlePeng9 (

- I believe that 4A is a minor of 4B, but the answer to all your other questions is no. For 4A/4B, the vertices labelled 2 and 3 can be combined into one vertex labelled with both 2 and 3, and the vertex labelled 3 in 4A can be mapped to it. For the others, observe that if graph A has a label that isn't <= any label from graph B, or graph A has more vertices with labels from set S than graph B has vertices with labels greater than or equal to labels from set S, graph A can't be a minor of graph B. Deedlit11 (talk) 09:33, May 2, 2015 (UTC)

- Okay. Is that definition (no two different labels are in relation) equivalent to this one?
*Graph A is a minor of graph B iff A can be obtained from B by contracting some edges, deleting some edges, and deleting some isolated vertices. Where "contracting an edge" merges two vertices (labelled a and b) into one with label a or b.*- And is this a well-quasi-order? {hyp/^,cos} (talk) 11:36, May 2, 2015 (UTC)
- As long as we have finitely many labels with empty relation betwen them, then that's it, and this indeed is a well-quasi-order. LittlePeng9 (
**talk**) 13:28, May 2, 2015 (UTC)

- As long as we have finitely many labels with empty relation betwen them, then that's it, and this indeed is a well-quasi-order. LittlePeng9 (

## Birthday

Congrats with your birthday and age 17. Wythagoras (talk) 08:13, May 23, 2015 (UTC)

Hey, thanks a lot :D LittlePeng9 (**talk**) 10:30, May 23, 2015 (UTC)

happy birthday Cookiefonster (talk) 11:37, May 23, 2015 (UTC)

Thanks. I've just noticed that today is \(2^8\)-th consecutive day of me editing the wiki. LittlePeng9 (**talk**) 13:19, May 23, 2015 (UTC)

- Also, the previous one was my 3333th edit on the wiki. LittlePeng9 (
**talk**) 13:22, May 23, 2015 (UTC)

Happy ~~brithday~~ birthday to you -- ☁ I want more clouds! ⛅ 14:32, May 23, 2015 (UTC)

htanks a lot. LittlePeng9 (**talk**) 15:02, May 23, 2015 (UTC)

happy 17th !! ! !! -- vel! 17:16, May 23, 2015 (UTC)

- Thanks dude ! !!! LittlePeng9 (
**talk**) 17:54, May 23, 2015 (UTC)

## Wiki Hero

If I'm correct you have only approximately 5 days to go to get the badge! Wythagoras (talk) 16:32, September 3, 2015 (UTC)

- After I post this, I'll be at 359/365, so if basic arithmetic didn't fail me, on Wednesday I'll get the badge! LittlePeng9 (
**talk**) 17:30, September 3, 2015 (UTC)

## Couldn't you solve this?

I would like to state that ]n {1} 2 {1} 2[ *is* defined. I clearly stated that you take the last two members of the string, and therefore, ]n {1} 2 {1} 2[ = ]n {1} ))2(([ = )))...)))n(((...(((, with ))2(( nested functions. Even though I do understand why you deleted it (The source needs to be external)... If you think it wasn't well-defined, you could simply edit it, or ask my to do so.

KthulhuHimself (talk) 06:17, October 14, 2015 (UTC)

- i think i can speak for littlepeng9 here — please see my comment on your post about TaN. -- vel! 06:34, October 14, 2015 (UTC)

- It wasn't even me who said your notation is ill-defined, it was User:Fluoroantimonic Acid. LittlePeng9 (
**talk**) 10:51, October 14, 2015 (UTC)

I can see that now. Hope he sees this.

KthulhuHimself (talk) 11:35, October 14, 2015 (UTC)

The rules you added in the article were different of the current rules in the blog post and they were ill-defined. The actual rules from the blog post are fine Fluoroantimonic Acid (talk) 15:31, October 14, 2015 (UTC)

Good to hear, I'll keep that in mind.

KthulhuHimself (talk) 16:05, October 14, 2015 (UTC)

## Milestone

4000 EDITS!

Be proud of yourself!

Boboris02 (talk) 17:21, October 10, 2016 (UTC)Boboris02Boboris02 (talk) 17:21, October 10, 2016 (UTC)

## Vandalism

Please remove http://googology.wikia.com/wiki/BESTEST_TRUE Mush9 (talk) 15:21, December 20, 2016 (UTC)

- Apparently this is written by a sockpuppet of that guy who made that KKK page. He should be permabanned. Hit (talk) 15:31, December 20, 2016 (UTC)

- Sorry, I was having a lecture. Couldn't act sooner. LittlePeng9 (
**talk**) 15:44, December 20, 2016 (UTC)

Sinister http://googology.wikia.com/wiki/Talk:Baker%27s_Dozenplex?diff=140223&oldid=140220 :P Anyway, I think it violates the rule about not publishing original work on the mainspace. Mush9 (talk) 10:57, December 31, 2016 (UTC)

- As for the edit on the talk page, its shouldn't have been cleared, so I have reverted that edit. As for the page itself - indeed, it violates the rule about not publishing one's own work on the wiki. However, I don't think it makes much sense right now to delete the page, if we were to do that only for someone else to recreate it in a day or two. LittlePeng9 (
**talk**) 11:13, December 31, 2016 (UTC)

Very true. Mush9 (talk) 11:53, December 31, 2016 (UTC)

More vandalism! Can you protect the title such as ddd??? ARsygo (talk) 10:52, March 30, 2017 (UTC)

## A few questions about uncomputable functions

Hey,Peng!

I don't want to be annoying so I will be quick.Since I tried to dive into the strange world of \(|\mathbb{U}|\) (uncomputable in MBOT.),I have realised there are many holes in my knowledge.So I wanted to know the answer for a few questions and desided that you're the best person to ask.

1.Do all ordinals \(\alpha\) above the order \(\omega^\text{CK}_1\) the same streanght in all ordinal hierarchies?

That is,\(f_{\alpha}(n) = H_{\alpha}(n) = g_{\alpha}(n) = N_{\alpha}(n) = ......\).

2.How can you prove that one uncomputable function eventually overgrows all computable ones?

If we take for example the busy beaver function (\(\Sigma(n)\)),how could one prove it does eventually dominate all computable functions?

3.I still don't get how you get from \(\text{Ord}_j\) to \(\text{Ord}_{j+1}\).More specifically what's the fundamental sequence of \(\text{Ord}_{j+1}\)?

4.If we add an extra truth predicate \(T\) to a set language and have it increase the possible statements that could be defined in that language,then is there any limit to how many truth predicates can be defined?

Thanks.Boboris02 (talk) 20:40, January 11, 2017 (UTC)

- I'm in bed now, so I'll be pretty brief. I might elaborate on these tomorrow if you wish.
- 1. First and foremost, \(f_\alpha\) and all others depend on the definition of the fundamental sequences for all limit ordinals up to \(\alpha\). Whether \(f_\alpha\) grows much faster (by whatever measure of "much faster") than \(g_\alpha\) might very well depend on the choice of FSes.
- Second, I doubt we have an exact equality between any two functions in the hierarchy. If by = you mean some sort of "the same growth rate", first define what this means, then ask again.
- Third, even if such an equality held for \(\alpha\), it most likely won't for \(\alpha+1\).
- 2. Please see the relevant Wikipedia page. Most of the outgrows-all-computable-functions proofs are based on the same principles.
- 3. This is somewhat technical. Basically \(\text{Ord}_{j+1}\) is the next ordinal after \(\text{Ord}_j\) such that \(V\) and \(V_{\text{Ord}_{j+1}}\) satisfy the same formulas in FOST + symbols for all \(\text{Ord}_\alpha,\alpha\leq j\). The FS is not very relevant, I just use it to justify existence of \(\text{Ord}_{j+1}\), but if you insist on seeing how it's defined, remind me tomorrow.
- 4. Once we have \(T\), we can consider the language of FOST augmented by \(T\) and consider the truth predicate of this language. We can repeat this any finite number of times, we can even do this transfinitely many times. This is, basically, what Emlightened does with her Little Bigeddon. LittlePeng9 (
**talk**) 21:44, January 11, 2017 (UTC)

## Anti-Nazi policy

Should all pages with references to the Wikipedia articles Strafgesetzbuch section 86a and Swastika be checked? --84.61.133.52 14:48, February 3, 2017 (UTC)

I found two: 1072 and 6976. --84.61.133.52 14:55, February 3, 2017 (UTC)

- Thank you very much. I have missed the two when doing a search, but a Google search now confirms there aren't any others. For anyone interested, the reason I remove the mentions of the swastika is that, first of all, these claims aren't mathematical, let alone googological, and second, the claim that the symbol itself is banned is wrong anyways. LittlePeng9 (
**talk**) 15:07, February 3, 2017 (UTC)

Since 101 is already protected from creation, and 102 = 14 + 88 = 18 + 84 = 28 + 74, I think that 102 needs also to be protected from creation. --84.61.133.52 14:29, February 5, 2017 (UTC)

Was 86#In German criminal law a good idea? --84.61.154.157 09:11, July 27, 2017 (UTC)

## BIG FOOT source

There seems to be an error with the page http://snappizz.com/foot, the primary source for BIG FOOT. Have any mirrors usable as a reference? ~εmli 15:27, June 1, 2017 (UTC)

## Turing machines

I created a 4-color turing machine that converts a solid string (that is, a member of the set G where ""∈G and if A∈G, A concatenated with "1" ∈ G.) of n ones into a solid string of 2n ones. Is it possible to use a 3-color turing machine to turn a solid string of n ones into 2n ones (not necessarily in a solid string)? A 2-color machine? 73.220.135.34 19:12, June 30, 2017 (UTC)

- Two colors, three states. LittlePeng9 (
**talk**) 19:42, June 30, 2017 (UTC)

LittlePeng9 i would like to ask you some questions about Turing machines. So a Turing machine computing a function is like a person making a function. Is this correct? A Turing machine's infinite tape is like a person with an infinite supply of paper and pens. The read/write head of a Turing machine is like a persons body, since it can move. The states of a Turing machine is like the mind of a person. Is all of this correct? Also check the Talk:Busy beaver function page, because i posted questions on the dates of June 23rd, June 28th, and July 18th of this year 20172607:FB90:4713:CE4D:D475:D783:C3B3:A5EB 21:48, August 5, 2017 (UTC)

## IRC

When will you be active on the IRC channel again? Boboris02 (talk) 17:20, January 9, 2018 (UTC)

- Some time ago I have abandoned the IRC channel, moving to Simple Art's Discord server. It's not googology-centered, but if you would be interested to join, message me on Discord at Wojowu#0414. LittlePeng9 (
**talk**) 17:23, January 9, 2018 (UTC) - And why have you abandoned the IRC channel? Boboris02 (talk) 18:13, January 9, 2018 (UTC)
- Because Discord is better. LittlePeng9 (
**talk**) 19:08, January 9, 2018 (UTC)

- Because Discord is better. LittlePeng9 (
- Yet you laughed at me when I said that a year ago....Boboris02 (talk) 20:31, January 9, 2018 (UTC)
- haha you LittlePeng9 (
**talk**) 20:36, January 9, 2018 (UTC) - On a more serious note, it wasn't Discord itself which I've disliked, but rather the Googology Server. SA's server is more varied in the discussion topics and I like it more. LittlePeng9 (
**talk**) 20:46, January 9, 2018 (UTC) - I got that. Boboris02 (talk) 21:48, January 9, 2018 (UTC)

- haha you LittlePeng9 (
- I sent you a friend request. I am the one with Albert Einstein saying "Git Gud" as a profile picture.Boboris02 (talk) 07:27, January 12, 2018 (UTC)
- I'm afraid I might have deleted the request along with some spam requests... I've tried to recover it, but I didn't take note of your ID. I promise I will pay more attention if you try again :P LittlePeng9 (
**talk**) 22:43, January 12, 2018 (UTC) - I sent you another one just now. Also if you don't want people to send you spam requests,don't show your IP on the internet. I mean,I know most spambots don't go to Googology Wiki but it's still the internet after all. (pssst it would be very helpful if FANDOM had private messaging.)Boboris02 (talk) 18:30, January 15, 2018 (UTC)

- I'm afraid I might have deleted the request along with some spam requests... I've tried to recover it, but I didn't take note of your ID. I promise I will pay more attention if you try again :P LittlePeng9 (

## Relation of admissiblity and stablity

Could you explain relation of admissiblity and stablity as easily as possible.

The hierarchy that I understand:

ω_{1}^{CK }- 1st 1-admissible ordinal

ω_{ω}^{CK} is П^{1}_{1}-comprehension. 1st noncomputable ordinal is not admissible. The collapse of this ordinal gives PTO of П^{1}_{1}-CA_{0}

recursively inaccessible - equivalent for cardinal is weakly inaccessible cardinal

Mahlo inaccessible - equivalent for cardinal is weakly Mahlo cardinal

After according to Feferman:

П^{1}_{1}-indescribable cardinal = 2-regular cardinal = weakly copmact cardinal

Equivalent for countable ordinals is П_{3}-reflection ordinal = 1st 2-admissible ordinal

In the more general case:

П^{1}_{n}-indescribable cardinal = n+1-regular cardinal

Equivalent for countable ordinals is П_{n+2}-reflection ordinal = n+1-admissible ordinal

П^{1}_{ω}-indescribable cardinal = ω-regular cardinal = П^{2}_{0}-indescribable cardinal is sup of П^{1}_{n}-indescribable cardinal

Equivalent for countable ordinals is П_{ω}-reflection ordinal = ω-admissible ordinal = П^{1}_{0}-reflection ordinal is sup of П_{n}-reflection cardinal

ω-admissible ordinal = П^{1}_{0}-reflection ordinal also is (+1)-stable

How to relate admissiblity and stablity after?

Are my guess correct?

ω+1-admissible ordinal = (+2)-stable ordinal

ω×2-admissible ordinal = (+ω)-stable ordinal

ω_{1}^{CK}-admissible ordinal = (+ω_{1}^{CK})-stable ordinal = (^{+})-stable ordinal = П^{1}_{1}-reflection cardinal

ω_{2}^{CK}-admissible ordinal = (+ω_{2}^{CK})-stable ordinal = (^{++})-stable ordinal

I-admissible ordinal = inaccessibly-stable ordinal

M-admissible ordinal = Mahlo-stable ordinal

(+1)-stable-admissible ordinal = doubly (+1)-stable ordinal

after:

(+1)-stable-stable-admissible ordinal

ω_{1}^{CK}-limit of ...-stable-stable-admissible ordinal

(^{+})-stable-limit of ...-stable-stable-admissible ordinal

e t.c. up to

(unbounded in any stable ordinal)-admissible ordinal = nonprojectible ordinal is П^{1}_{2}-comprehension. The collapse of this ordinal gives PTO of П^{1}_{2}-CA_{0}

And can we somehow relate a-indescribability and stability? Scorcher007 (talk) 16:45, January 17, 2018 (UTC)

- I'm afraid my understanding of stability is basically void, so I can't help you too much. All I know is the definition of stability, namely that we requre certain level of the constructible hierarchy to satisfy the same sentences as the full constructible universe, but I imagine you know that already. LittlePeng9 (
**talk**) 15:55, January 17, 2018 (UTC)

- Thanks for the answer. Unfortunately, it is difficult to find detailed explanations for this question.
- Here are a few more facts:
- According to cantorsattic.info and D. Taranovsky
- Π
_{ω*a}-reflecting means a-stable - so I made a mistake, and:
- ω*2-admissible ordinal = (+2)-stable ordinal
- ω
^{ω}-admissible ordinal = (+ω)-stable ordinal - Then ε
_{0}-admissible ordinal = (+ε_{0})-stable ordinal

- According J.S. Stegert:
- 0-indescribable cardinal is П
^{1}_{n}-indescribable cardinal - m-indescribable cardinal is П
^{m+1}_{n}-indescribable cardinal - 1-indescribable cardinal equivalent (+1)-stable ordinal
- n-indescribable cardinal equivalent (+n)-stable ordinal
- Then (some level of stability)-indescribable cardinal equivalent (some level of stability)-admissible ordinal.

- And more. According to D. Taranovsky: ω-ly (+1)-stable ordinal already is nonprojectible ordinal or П
^{1}_{2}-comprehension. - Scorcher007 (talk) 16:45, January 17, 2018 (UTC)
- Not every nonprojectible ordinal gives a collapse for \(\Pi^1_2-\text{CA}_0\). In fact,\(\Pi^1_2-\text{CA}_0\) is equiconsistent with KP + nonprojectible universe,so it is rather the limit to nonprojectible ordinals and is not bounded by any nonprojectible ordinal.Boboris02 (talk) 14:41, February 16, 2018 (UTC)
- Thank you for the clarification. I corrected the table, now everything converges.Scorcher007 (talk) 08:23, February 17, 2018 (UTC)

- And more. According to D. Taranovsky: ω-ly (+1)-stable ordinal already is nonprojectible ordinal or П

PENGUINS!!!

BlauesWasser (talk) 00:06, February 25, 2018 (UTC)

- (")> LittlePeng9 (
**talk**) 09:42, February 25, 2018 (UTC)

## a completely random question

LittlePeng9 what's the difference between functions, programs, and algorithms? Is multiplication a function or an algorithm?2607:FB90:44A7:25EB:60D0:D993:FD2B:A549 02:38, March 9, 2018 (UTC)

- In short: program and algorithm are essentially the same thing, meaning a set of instructions which lets us compute functions. Multiplication is a function, but a specific way to compute products (like long multiplication) is an algorithm. LittlePeng9 (
**talk**) 18:32, March 9, 2018 (UTC)

A program is a description of an algorithm and an algorithm is the actual movement to produce the output of the function from the input of the function. A program is a description of an algorithm. Is this correct? How does the Halting problem go beyond recursion?2607:FB90:4704:CAA:6B20:263:A35B:1633 01:11, March 10, 2018 (UTC)

## Judge the rate of growth

Would you be willing to have a brief glance on the function I design, posted on my blog. I do not know how to estimate its rate or growth.Boris Huller (talk) 23:09, May 12, 2018 (UTC)

## Your lower bound for tree (3)

You found a lower bound of 2^17+8 for tree(3) in the version of the weak tree function which you call utree. Although I'm aware of newer bounds, I'm still interested how that bound was derived. So can you please help me to find it? George Albert Lee (talk) 00:04, 9 November 2020 (UTC)

## Doubts

Hey LittlePeng 9I have a doubt regarding your J letter notation read the below this is a big number library used for an increment game,which uses your letter notation:

J

Using arrows, large numbers such as 10{16}(10{12})^51 (10^^^)^6 eeee104 can be easily written, and numbers not talked about here are relatively close to other numbers, thus is discarded. However, we hit a limit; the notation is strongest if we increase the n in 10{n}10 as if it were a variable. No worries, we can just put numbers with hyperoperators as the number of arrows! Well, we will get some problems. We will get something like 10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10^^^(10^^)^520 ee500}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10. It is very hard on eyes, and very large in comparison, defeating the purpose of notation numbers in the first place. It also takes a lot of space as text, and it takes so long to write it by hand. I only put this because I could make the computer repeat pieces of text set amount of times. Worst of all, to compare which number are larger, you need to count each individual layers, and that is impractical. There must be a better way to handle such large numbers, right?Luckily, there is! Function f(x)=10{x}10 is written as Jx. This is defined as a part of PsiCubed2's letter notation. In addition, we can treat *this* as another operation, this time unary, but we always treated 10{n}x as unary operation "10{n}", so no worries. So, we can now see the above number clearly, written J^45 10^^^(10^^)^520 ee500.

Ok I have a doubt in the above description,what exactly is J^45, can you explain in detail I mean how did he J^45 from the number above,also how can the above number 10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10{10^^^(10^^)^520 ee500}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10}10 be written in Beaf form of expansion that is {a,b,1,2}, also I were to write something like J^9000000000000000 10{9000000000000000}10^10^10^10^10^10^10^100000000000000 ,what would J^90000000000000000 mean here and how can this number be expressed in Beaf expansion form.can you please answer?