User talk:Hyp cos

Can i ask who are you? JiawhienIsBackNoEvadeBlockOnceAgainIComeBack (talk) 09:10, October 1, 2013 (UTC)


 * Hyp cos. FB100Z &bull; talk &bull; contribs 19:02, October 1, 2013 (UTC)


 * The average of ex and its reciprocal. -- ☁ I want more clouds! ⛅ 03:18, October 3, 2013 (UTC)

SGH
Please look here. I gave reason why SGH is weaker than you think Wythagoras (talk) 19:00, October 16, 2013 (UTC)

SGH grows from outside --- \(g_{\theta(\Omega^\omega,1)}(n)\) stands above \(g_{p(SVO)}(n)\) for any function p which has lower level than SVO. So it's easy for SGH to grow from outside and hard to grow from inside. For example, \(g_{\theta(\Omega^{\omega+1})}(n)\) expands to \(g_{\theta(\Omega^\omega\theta(\Omega^\omega...\theta(\Omega^\omega)))}(n)\), \(g_{\theta(\Omega^\omega+1)}(n)\) expands to \(g_{\theta(\Omega^\omega,\theta(\Omega^\omega,...\theta(\Omega^\omega)))}(n)\), and even \(g_{\theta(\Omega^\omega,1)}(n)\) can expand to \(g_{\theta(\Omega^n,\theta(\Omega^\omega)+1)}(n)\).

However, many notations grow from inside, such as BEAF, BAN, HH, etc. .For example, \(H_{\omega^{\omega^\omega}+1}(n)=H_{\omega^{\omega^\omega}}(n+1)\), and {3,3,1,2}={3,3,{3,3,3}}. It's hard to compare them with SGH.

FGH can grows from outside or inside. \(f_{\alpha+1}(n)=f_\alpha^n(n)\). And there're 2 very different ways to get \(f_\alpha^2(n)\) from \(f_\alpha(n)\). One way is \(f_\alpha(n+1)\), \(f_\alpha(2n)\), ..., \(f_\alpha(p(n))\), ..., \(f_\alpha(f_\alpha(n))\), and thet other way is \(f_\alpha(n)+1\), \(2f_\alpha(n)\), ..., \(p(f_\alpha(n))\), ..., \(f_\alpha(f_\alpha(n))\). Thus, FGH becomes a general way to compare other notations. Notations growing from inside or outside suit. For comparison between FGH and SGH, we should grow FGH from outside and not inside. &#123;hyp&#60;hyp··cos&#62;cos&#125; (talk) 01:36, October 17, 2013 (UTC)
 * \(f_\alpha^2(n)\) just means \(f_\alpha(f_\alpha(n))\) and generally \(f_\alpha^n(n)\) is \(f_\alpha(f_\alpha(\cdots(f_\alpha(n))\cdots))\) with n f's. Ikosarakt1 (talk ^ contribs) 07:36, October 17, 2013 (UTC)

google site
DO you have your own google site about the large numbers? JiawhienIsBackNoEvadeBlockOnceAgainIComeBack (talk) 07:05, November 17, 2013 (UTC)

No. Actually I'm green to googology. hyp$hyp?cos&#38;cos (talk) 10:31, November 17, 2013 (UTC)

For Aarex
Make an extension of latest extension of Extended Up-arrow Notation. AarexTiao 01:50, December 1, 2013 (UTC)

Number navigator
Can you explain this a little more? FB100Z &bull; talk &bull; contribs 01:46, December 5, 2013 (UTC)

Number navigators in numbers such as Hyper-E numbers, Bowers' gongulus series, and lower hyperfactorial array numbers can be deleted because we can use the templetes as our new and powerful "number navigator". We can see other similar numbers in templetes. But, in nucleaxul we cannot delete it because there's no templete for nucleaxul numbers. If we delete it, we can't find numbers similar to nucleaxul on this page. hyp$hyp?cos&#38;cos (talk) 01:53, December 5, 2013 (UTC)
 * So why not just add a template? FB100Z &bull; talk &bull; contribs 02:44, December 5, 2013 (UTC)
 * Done! But there's still something wrong. In Extremexul template, the "Extremexul" in the first row under the title is bold, and it doesn't link to anywhere. However, in gigantixul template I newly add, the "Gigantixul" in the first row under the title links to page Gigantixul. But I don't know how to fix it. Okey now. Nothing serious in the new templates. hyp$hyp?cos&#38;cos (talk) 11:59, December 5, 2013 (UTC)

Extension of Nested Array Notation
Make an extension of Nested Array Notation
 * I found \(nR\{0\{1\text{^}\}*\} = nR\{0\{0\underbrace{**...**}_n\}\), \(nR\{0\{1\text{^}\}**\} = nR\{0\{0\underbrace{**...**}_n\{1\text{^}\}*\}*\}\), etc.


 * And \(nR\{0\{2\text{^}\}*\} = nR\{0\underbrace{\{1\text{^}\}...\{1\text{^}\}}_n*\}\), \(nR\{0\{3\text{^}\}*\} = nR\{0\underbrace{\{2\text{^}\}...\{2\text{^}\}}_n*\}\), etc.


 * \(\{0\{0\{0*\text{^}\}1\text{^}\}*\} = \{0\{0\{0\{...\text{^}\}1\text{^}\}1\text{^}\}*\}\)


 * The extension of NAN name is HNAN (Hyper-Nested Array Notation.) AarexTiao 14:47, December 29, 2013 (UTC)

Extra: Growth rate of PNAN = psi(X(e(0))) AarexTiao 02:08, December 27, 2013 (UTC)
 * I don't think so. First, we have different ideas from {0{0,1*}1} on. We know the nR{0{0,1*}1} and the nR{0{{0{0,1*}1}*}1} both have growth rate \(\psi(\psi_{I(1,0,0)}(0))=\psi(\psi_{\chi(M)}(0))\), but what's the equvalence? We know {0{1*}1} is equvalent to \(\psi_{I(\omega,0)}(0)\) (not \(\psi(\psi_{I(\omega,0)}(0))\) or \(I(\omega,0)\)), and {0{1*}2} is equvalent to \(\psi_{I(\omega,0)}(1)\), the {0{1*}0,1} is equvalent to \(I(\omega,0)\).
 * More, the {0{{0{1*}1}*}1} is equvalent to \(\psi_{I(\psi_{I(\omega,0)}(0),0)}(0)\),
 * the {0{{0{1*}1}*}0,1} is equvalent to \(I(\psi_{I(\omega,0)}(0),0)\),
 * the {0{{0{1*}0,1}*}1} is equvalent to \(\psi_{I(I(\omega,0),0)}(0)\),
 * and the {0{{0{1*}0,1}*}0,1} is equvalent to \(I(I(\omega,0),0)\).
 * The {0{{0{{0{1*}1}*}1}*}1} is equvalent to \(\psi_{I(\psi_{I(\psi_{I(\omega,0)}(0),0)}(0),0)}(0)\), and the {0{{0{{0{1*}0,1}*}0,1}*}0,1} is equvalent to\(I(I(I(\omega,0),0),0)\).
 * Now think of this: {0{0,1*}1} is the diagonalizer of {0{ ____ *}1}, so {0{{0{0,1*}1}*}1} is equvalent to \(\psi_{I(1,0,0)}(0)\), the {0{{0{{0{0,1*}1}*}1}*}1{{0{0,1*}1}*}1} is equvalent to \(\psi_{I(\psi_{I(1,0,0)}(0),1)}(0)\), the {0{{0{0,1*}1}*}2} is equvalent to \(\psi_{I(1,0,0)}(1)\), and the {0{{0{0,1*}1}*}0,1} is equvalent to \(I(1,0,0)\).
 * And the {0{0,1*}1} is equvalent to M, then.
 * So nR{0,1{0,1*}1} has growth rate \(\psi(\psi_{\chi(\varepsilon_{M+1})}(0))\),
 * nR{0,0,1{0,1*}1} has growth rate \(\psi(\psi_{\chi(\psi_{I_{M+1}}(0))}(0))\),
 * nR{0,0,0,1{0,1*}1} has growth rate \(\psi(\psi_{\chi(\psi_{I(1,M+1)}(0))}(0))\),
 * nR{0{1*}1{0,1*}1} has growth rate \(\psi(\psi_{\chi(\psi_{I(\omega,M+1)}(0))}(0))\),
 * nR{0{{0{{0{0,1*}1}*}1}*}1{0,1*}1} has growth rate \(\psi(\psi_{\chi(\psi_{I(\psi_{I(1,0,0)}(0),M+1)}(0))}(0))\),
 * nR{0{{0{0,1*}1}*}1{0,1*}1} has growth rate \(\psi(\psi_{\chi(\psi_{I(M,1)}(0))}(0))\),
 * nR{0{0,1*}2} has growth rate \(\psi(\psi_{\chi(M_2)}(0))=\psi(\psi_{\chi(\psi_{I(1,0,M+1)}(0))}(0))\),
 * nR{0{0,1*}3} has growth rate \(\psi(\psi_{\chi(M_3)}(0))\),
 * nR{0{0,1*}{0{0,1*}1}} has growth rate \(\psi(\psi_{\chi(M_M)}(0))\),
 * So next to a comma means inaccessible, next to a {0,1*} means mahlo, and then next to a {0,2*} means compact. we can use {0{0,A*}1} for stage function of A, and the diagonalizer, {0{0,0,1*}1}, is the stage cardinal. Next to a {0,0,1*} means stage, so {0{0,0,1*}0,1} is inaccessible stage, {0{0,0,1*}0{0,1*}1} is mahlo stage, {0{0,0,1*}0{0,2*}1} is compact stage, {0{0,0,1*}0{0,0,1*}1} is stage stage. etc. hyp$hyp?cos&#38;cos (talk) 02:36, December 27, 2013 (UTC)
 * Well? {0{1,0,1*}1} is w-ex-stage, {0{{0,1},0,1*}1} is W-ex-stage, {0{{0,0,1},0,1*}1} is I-ex-stage, {0{{0{1*}1},0,1*}1} is I(w,0)-ex-stage, {0{{0{{0{0,1}1}*}1},0,1*}1} is chi(M)-ex-stage, {0{{0{0,1*}1},0,1*}1} is M-ex-stage, {0{{0{0,2*}1},0,1*}1} is compact-ex-stage, {0{{0{0,0,1*}1},0,1*}1}, and {0{0,1,1*}1} is stage-ex-...-ex-stage AarexTiao 02:50, December 27, 2013 (UTC)

In my system, stage-ex-stage-ex-...-ex-stage is just the first ordinal in the second row of n-dimensions, or psi(X(ww+12)) Wythagoras (talk) 07:42, December 27, 2013 (UTC)
 * I stronger than you! Check it out! AarexTiao 01:11, December 29, 2013 (UTC)
 * Well, if you have some strong ordinal notations, can you compare it to nested array notation? hyp$hyp?cos&#38;cos (talk) 02:27, December 29, 2013 (UTC)

HNAN II
Can you add Hyper-Nested Array Notation II on R function series? AarexTiao 15:17, February 1, 2014 (UTC)


 * click here AarexTiao 02:41, February 9, 2014 (UTC)

Thanks, Hyp cos. I see you have more experience with the categorization than I do.

Questions
Hello Hypcos. I would like to ask a few questions about your work with large numbers. I would like to ask a couple of questions about your website called WordPress Steps Toward Infinity. How come parts 8-12 don't work in your Strong Array Notation? Do parts 1-7 work in your Strong Array Notation? Does your R function work? Two people on this website told me that your R function has an infinite loop.
 * Some expressions in NDAN cause infinite loop, such as s(3,3{1(2(1)○△2)△3}2). That makes NDAN and beyond don't work. But I've just know what's the problem, and I'll fix it soon.
 * Parts 1-7 work, and they're as strong as R function.
 * But I don't know what's wrong with R function now. &#123;hyp/^,cos&#125; (talk) 00:42, July 1, 2016 (UTC)
 * iirc it was littlepeng9 who found the error. -- vel! 17:21, July 1, 2016 (UTC)
 * No, it wasn't me. For what I know it was Fluoroantimonic Acid. I have happened to meet him on IRC recently and I've asked him about this. I don't have access to logs right now, but he said that the flaw regarded an old version of the notation and that error isn't there anymore. I'm pretty sure Deedlit was there at the time of the conversation, so he might be able to provide logs. LittlePeng9 (talk) 19:51, July 1, 2016 (UTC)


 * I don't think the error was ever there. Here is the passage that FA thought meant there was a problem with the notation:


 * To solve s(a,b{1{2}1,2}2), we start the process, then meet case B2, then meet case B1. Before case B1 applies, the array is s(a,b{1{2}1,2}2{1{2}1,2}1); after case B1 applies, the array becomes s(a,a{1{2}b,1}2{1{2}1,2}1). Then rule 2a and 2b applies, so s(a,b{1{2}1,2}2) = s(a,a{1{2}b}2). What if we use the case B1 in Linear array notation? s(a,b{1{2}1,2}2) will be s(a,a{a{a}b}2). When a ≥ 3 and b ≥ 2, it’ll reduces to something containing {1{2}1,2} again – it can never be solved. So a change on case B1 is necessary.


 * But what Hyp Cos was saying was that if we used the version of B1 in "Linear array notation" in his "Extended array notation", then we would get an infinite loop, which is why he changed that particular rule for "Extended array notation". Deedlit11 (talk) 00:18, July 2, 2016 (UTC)


 * That cause only LAN will work and since NDAN is fixed, why DEN will have 2 types includes weak and strong? GoogleAarex2001 17:18, July 7, 2016 (UTC)

My feeling is that as a notation becomes more complex, the more necessary it becomes for someone to prove that it terminates. -- vel! 04:49, July 7, 2016 (UTC)

Question from Googleaarex
Is wDEN parts finally completed? GoogleAarex2001 23:32, July 11, 2016 (UTC)
 * It depends on how you define "wDEN parts". My plan for the next part is converting superscripts on droppers into a very special separator - that's a bit similar to "from mDEN to pDAN". &#123;hyp/^,cos&#125; (talk) 13:12, July 12, 2016 (UTC)
 * Since wDmEN is completed, what is your next part plan? AarexWikia04 (talk) 01:35, July 18, 2016 (UTC)
 * The case B3 in wDmEN makes it hard to extend. So I may make a new version of wDmEN before extend it. &#123;hyp/^,cos&#125; (talk) 01:39, July 18, 2016 (UTC)
 * Can you do the new case B3 changing instead? AarexWikia04 (talk) 15:47, July 21, 2016 (UTC)

Will you ever make number names past two-row arrays? Username5243 (talk) 01:43, July 18, 2016 (UTC)
 * Yes, but I'll finish it much later. &#123;hyp/^,cos&#125; (talk) 01:45, July 18, 2016 (UTC)
 * But I have the ideas! AarexWikia04 (talk) 01:46, July 18, 2016 (UTC)

I will beat your biggest number on SAN.
I am nearby passing Pentenlinel = s(3,2,2{2}3,3,3,3,3)!

I am currently on Googolis-exchgain = s(10,100{2}1,1,1,2). AarexWikia04 (talk) 15:44, July 21, 2016 (UTC)

Question about PDAN
Hi Hyp Cos, I had a question about PDAN, in particular the evaluation of {a,b {1 {1,,1,,2} 2} 2}.

I get that A_1 = {1 { 1,,1,,2} 2}, A_2 = {1,,1,,2}. Then B_3 = ",,", B_2 = {1,,1,,2,,1} = {1,,1,,2}, and B_1 = {1 {1,,1,,2} 2 {1,,1,,2} 1} = {1 {1,,1,,2} 2}. (Actually, do we apply rule 2 at this point, or not?) Then X = "{1,,1" and Y = "}", so we compare lvl(A_1) to lvl(X ,, n-1 Y) = lvl({1,,1,,1}) = lvl({1}), and lvl (A_1) is greater.

So we go to case B2.5: P = "{1", Q = "2}", so S_1 = ",", S_2 = {1,2}, S_3 = {1{1,2}2}, S_4 = {1{1{1,2}2}2}, and so on. Is this correct? If so, that seems problematic, as that should be how {1,,2} reduces and not {1,,1,,2}. The problem seems to be that we want {1,,1,,2} to drop down to something like {1,,X}, but in your comparison you compare to {1,,1,,1} = {1}. Deedlit11 (talk) 04:14, August 2, 2016 (UTC)
 * You're right, and s(a,b {1 {1,,1,,2} 2} 2) = s(a,b {1 {1,,2} 2} 2). That seems problematic because it's a "wrong way" of pDAN. Another example for "wrong way" is that (in \(\psi\) function up to \(\psi(\Omega_\omega)\)) \(\psi(\psi_1(\Omega_\omega))=\psi(\Omega_2)<\psi(\Omega_\omega)\), and making \(\alpha\) in \(\psi(\psi_1(\alpha))\) larger is a "wrong way". Since we have many ways to extend the expressions, we can't avoid the appearence of "wrong ways" for notations complicated enough.
 * In pDAN, the main reason for some expressions running into "wrong way" is that the Case B2.6 doesn't work correctly on them. Then, how to find expressions in the "right way" of pDAN? We use the most simple expressions, e.g. s(a,b {1,,1,,2} 2), s(a,b {1,,1,,3} 2), s(a,b {1,,1,,1,,2} 2) and so on. They meet Case B2.6 for the first reduce, and, after that, they'll continue the "right way". &#123;hyp/^,cos&#125; (talk) 05:40, August 2, 2016 (UTC)

Thanks! I'm curious, have you made comparisons with your notation and ordinal collapsing functions beyond \(\psi(\psi_I(0))\)? For the R function, you've said that LAN reaches \(\psi(\psi_{I(\omega,0)}(0))\), and for PNAN you've compared {0{0,1*}1} to M and {0{0,2*}1} to K; so, if my interpretation to the new array notation is correct, that should mean that PDAN reaches \(\psi(\psi_{I(\omega,0)}(0))\) (I'm inclined to agree with this) and that {1 {1,,2,,,2}2} is analogous to M and {1{1,,3,,,2}2} is analogous to K (I'm less sure about this). Do you agree with this, or do you get something different? Deedlit11 (talk) 10:58, August 2, 2016 (UTC)
 * No, PNAN reaches over C(e(W2+1)). AarexWikia04 - 11:00, August 2, 2016 (UTC)


 * I didn't talk about the limit of PNAN in my post. Are you referring to Taranovsky's notation? What makes you say PNAN surpasses C(e(W2+1))? Deedlit11 (talk) 11:11, August 2, 2016 (UTC)


 * My guess (originating from some comparisions between the R function and strong array notation) is that \(\psi(\psi_{I(\omega,0)}(0))\) is just the growth rate of s(a,b {1,,1 ... ,,1,,2} 2), and that {1{1,,2,,}2} is analogous to M and {1{1,,3,,}2} is analogous to K. -- ☁ I want more clouds! ⛅ 11:39, August 2, 2016 (UTC)
 * Oh wait, {1{1,,2,,}2} and {1{1,,3,,}2} is equivalent to {1{1,,2,,,2}2} and {1{1,,3,,,2}2}, respectively. -- ☁ I want more clouds! ⛅ 11:42, August 2, 2016 (UTC)
 * pDAN is as strong as PNAN in R function and Taranovsky's "Degrees of Reflection" notation, but they're much weaker than \(C(C(\varepsilon_{\Omega_2+1},0),0)\). &#123;hyp/^,cos&#125; (talk) 12:24, August 2, 2016 (UTC)
 * s(n,n{1{2,,,2}2}2) reaches \(\psi(\psi_{I(\omega,0)}(0))\), {1{1,,2,,,2}2} is analogous to M, but {1{1,,3,,,2}2} is just analogous to \(\Xi[1]\) (the least 1-weakly Mahlo), and {1{1,,1,,2,,,2}2} is analogous to K. &#123;hyp/^,cos&#125; (talk) 12:24, August 2, 2016 (UTC)
 * But this is sDAN. AarexWikia04 - 12:30, August 2, 2016 (UTC)
 * {whatever,,,2} is equivalent to {whatever,,}, so this could still be considered a part of pDAN. -- ☁ I want more clouds! ⛅ 16:09, August 2, 2016 (UTC)

Thanks again... according to your PNAN calculations, you would have

{1{1,,2,,,2}3} is the second Mahlo cardinal {1{1,,2,,,2}a} is the ath Mahlo cardinal {1{1,,2,,,2}1,,2} is the first inaccessibly Mahlo cardinal {1{1,,2,,,2}2,,2} is the second inaccessibly Mahlo cardinal {1{1,,2,,,2}1,,3} is the first 2-inaccessibly Mahlo cardinal {1{1,,2,,,2}1,,1,,2} is the first hyper-inaccessibly Mahlo cardinal {1{1,,2,,,2}1{1,,2,,,2}2} is the first 2-Mahlo cardinal

but apparently this is incorrect... where does it go wrong?

Also, would it be correct to say that {1{1,,1,,2,,,2}a} is analogous to the ath weakly compact cardinal? Deedlit11 (talk) 16:28, August 2, 2016 (UTC)
 * I might make analysis soon but starting with pDAN. AarexWikia04 - 16:37, August 2, 2016 (UTC)
 * I still haven't fully understand ordinal collapsing notation on K, so the comparisons are done between my array notation and Taranovsky's notation. He claimed that \(C(\Omega_2+C(\Omega_2,C(\Omega_22,0)),0)\) is the least recursive inaccessible, \(C(\Omega_2+\omega^{C(\Omega_2,C(\Omega_22,0))2},0)\) is the least recursive Mahlo, and \(C(\Omega_2+\omega^{\omega^{C(\Omega_2,C(\Omega_22,0))2}},0)\) is the least recursive \(\Pi_3\)-reflecting, and I get the results from here. As a result, my PNAN calculations are wrong beyond M. &#123;hyp/^,cos&#125; (talk) 00:56, August 3, 2016 (UTC)

Ah, I see. I believe I more or less understand the ordinal collapsing function with K, but unfortunately I don't quite grasp your notation or Taranovsky's notation, so we are at an impasse. How far have you compared Taranovsky's notation with your own? Do you know if Taranovsky's main notation is equal to, say, Dropping Array Notation, or if one is stronger than the other? Deedlit11 (talk) 04:50, August 3, 2016 (UTC)
 * Limit of DAN is \(C(C(\Omega_22+C(\Omega_2+C(\Omega_2+1,C(\Omega_22,0)),0),0),0)\) (also limit of R function), limit of NDAN is \(C(C(\Omega_22+C(\Omega_2+C(\Omega_22,C(\Omega_22,0)),0),0),0)\), and limit of WDmEN is \(C(C(\Omega_22+C(\Omega_2+C(\Omega_22+1,0),0),0),0)\) (but I'm not very sure about the last one). So Taranovsky's notation is much stronger. &#123;hyp/^,cos&#125; (talk) 08:43, August 3, 2016 (UTC)
 * He made new part, so my limit ordinal of pDDN is C(C(W_2*2+C(W_2*2,0),0)). AarexWikia04 - 17:25, August 31, 2016 (UTC)

Woah, really?! The notations for Taranovsky's system barely budge even all the way up your notation! I guess Taranovsky's notation is really that strong. Deedlit11 (talk) 17:55, August 3, 2016 (UTC)


 * Hyp cos is wrong. I check and C(1,1) is stronger, which is 2nd fixed point of x -> C(0,x). AarexWikia04 - 11:33, August 3, 2016 (UTC)
 * NVM AarexWikia04 - 15:10, August 3, 2016 (UTC)

After WDmEN?
How about soWDmEN (Second order weak dropper-expanding notation?) AarexWikia04 - 14:13, August 2, 2016 (UTC)
 * According to pDDN after WDEN, will you make like sDDN, DDN, and nDDN? AarexWikia04 - 15:43, August 11, 2016 (UTC)
 * I pause making parts now, and go to naming numbers. &#123;hyp/^,cos&#125; (talk) 03:50, August 12, 2016 (UTC)
 * When will you make parts again? AarexWikia04 - 00:28, August 13, 2016 (UTC)
 * He made the new part, pDDN. Will you make sDDN after or before EAN numbers? AarexWikia04 - 15:07, August 23, 2016 (UTC)
 * Ummm, hyp cos will analyze entire SAN first; then he will make EAN numbers or extensions included:

* Secondary Dropper-Dropping Notation * Dropper-Dropping Notation * Nested Dropper-Dropping Notation * Second-Order Weak Dropper-Expanding Notation * Higher-Order Weak Dropper-Expanding Notation * Strong Dropper-Expanding Notation
 * Googleaarex (talk) 02:25, February 20, 2017 (UTC)

Hello, I'm back in here again, and I have revamped idea for sDDN.
 * Imagine A without grave accents in the end is first mark of the definition of A.
 * Define A with adding 1 grave accent in the end is the next mark of the definition of B, where B is like A but removing grave accents in the end.
 * Define A with adding comma in the end is dropper of the definition of A.
 * ,, is a simplest dropper and (n+1)-tuple comma is a simplest n-dropper, which is dropper of (n-1)-dropper.
 * Define A with adding colon in the end is weak dropper-expanding of the definition of A.
 * Define A with adding colon then n grave accents in the end is weak (n+1)th mark of dropper-expanding of the definition of A.
 * Let dropper-(nth mark of A) separators are same as (dropper-A)-(nth mark of A).
 * Finally, we could define 1-dropper-2-dropper separators, which is a strongest separator of sDDN.

We can easily extend this:
 * DDN defines separators up to the limit of dropper-dropper separators
 * nDDN defines separators up to the limit of the fixed point A, which is dropper-A separators

Good luck for defining 3 extensions using my idea! Googleaarex (talk) 22:23, August 5, 2017 (UTC)

Can you define ntepAAN?
First, read the incomplete definition right here:. AarexWikia04 - 00:45, August 4, 2016 (UTC)

Is your site resumed for more analysis?
Please answer me... AarexWikia04 - 00:42, November 19, 2016 (UTC)
 * It continued up to sDAN analysis. Googleaarex (talk) 02:23, February 20, 2017 (UTC)

Adding strong array notation approximation
Can I add approximations using strong array notation on non-SAN numbers? For example, number pages that coined and defined using other notations especially Saibian's ExE (e.g. grangol, greagol, etc.) or BEAF. ARsygo (talk) 16:03, June 29, 2017 (UTC)
 * Yes. &#123;hyp/^,cos&#125; (talk) 00:10, June 30, 2017 (UTC)

My ideas up to SDEN
Nishada 00:33, August 6, 2017 (UTC) EDIT: I fixed a mistake. Nishada 00:34, August 6, 2017 (UTC) EDIT 2: I fixed another mistake. Nishada 00:35, August 6, 2017 (UTC)
 * Grave accent is 1-separator.
 * ,, is 2-separator.
 * ,,, is 2-separator over 2-separator = 3-separator.
 * ,,,, is 2-separator over 2-separator over 2-separator = 4-separator.
 * : is 2-separator over x-separators = 1-separator-separator.
 * + is 2-separator-separator.
 * Now, here is the ideas beyond pDDN:
 * ++ is 2-separator-separator over 2-separator-separator = 3-separator-separator.
 * +++ is 2-separator-separator over 2-separator-separator over 2-separator-sepator = 4-separator-separator.
 * [1]□□◊ is 2-separator-separator over x-separator-separators = 1-separator-separator-separator.
 * [2]□□◊ is 2-separator-separator-separator.
 * [3]□□◊ is 2-separator-separator-separator over 2-separator-separator-separator = 3-separator-separator-separator.
 * [1]□□□◊ is 2-separator-separator-separator over x-separator-separator-separators = 1-separator-separator-separator-separator.
 * [x]□y◊ is x-separatory, where ab = aaa... b times (concatenation of strings, not multiplication)
 * sDDN (Secondary Dropper-Dropping Notation) goes up to ++
 * DDN (Dropper-Dropping Notation) goes up to [1,2]□◊
 * NDDN (Nested Dropper-Dropping Notation) goes up to {1[1]□□◊2}
 * SNDN (Strong Nested Dropping Notation) goes up to [1]□□□...◊
 * You are wrong, 2-separator-separator is stronger than you thought. Googleaarex (talk) 00:36, August 6, 2017 (UTC)
 * Well, my ideas for future parts seems like the logical extension to the current parts.
 * Forgot the signature Nishada 00:38, August 6, 2017 (UTC)
 * Also, your 3-separator-separator corresponds to my 2-dropper-(1-dropper) separators. That's why my sDDN is stronger than yours. Googleaarex (talk) 00:57, August 6, 2017 (UTC)

OCF vs DAN
From http://googology.wikia.com/wiki/User:Hyp_cos/OCF_vs_Array_Notation_p2

...fitting Taranovsky's correspondences (x,x{1{1{2,,}2,,}2}2) - the growth rate limit in KP + $$\Pi_n$$ -reflection

...not fitting Taranovsky's correspondences (x,x{1{1`,,2,,}2}2) the growth rate limit in KP + $$\Pi_n$$ -reflection

KP + $$\Pi_n$$ -reflection equivalent KP + $$\Pi_\omega$$ ?

Then:

What the growth rate has $$\Pi_{OCF(\Pi_\omega)}$$ in DAN?

What the growth rate has $$\Pi_{OCF(\Pi_{OCF(\Pi_\omega)})}$$ in DAN?

etc. up to $$\alpha \mapsto \Pi_{OCF(\alpha)}$$ ?

Or what stable ordinals will be equivalent? If I correctly understand the stable ordinals are equivalent to transfinite $$\Pi_n$$ -reflection?


 * Here is the answer, and it's not that easy:
 * \(\Pi_{OCF(\Pi_\omega)}\) = \(\Pi_{\psi_{\Omega}(\Pi_{\psi_{\Omega}(\Pi_{\omega}-\text{reflecting})}-\text{reflecting})}-\text{reflecting}\) approx. corresponds to \(\{1,,1\{1',,1\{1\{1,,1,2\}2\}2\}2\}\)
 * \(\Pi_{OCF(\Pi_{OCF(\Pi_\omega)})}\) = \(\Pi_{\psi_{\Omega}(\Pi_{\omega}-\text{reflecting})}-\text{reflecting}\) approx. corresponds to \(\{1,,1\{1',,1\{1\{1,,1\{1',,1\{1\{1,,1,2\}2\}2\}2\}2\}2\}2\}\)
 * \(\alpha \mapsto \Pi_{OCF(\alpha)}\) = \(\psi_{\Omega}(\Pi_{\Omega}-\text{reflecting})\) approx. corresponds to \(\{1,,1\{1',,1\{1,,2\}2\}2\}\)
 * Googleaarex (talk) 09:06, August 6, 2017 (UTC)