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Recently I was trying to extend set X{·}a (see Definitions update blog) beyond Ω^{ΩΩΩ...}. (Now I like to represent arrays of ordinals as "larger" ordinals).
I did not want to give up X{·}a, since it was used in short and independent of fundamental sequence systems definitions of ordinal array functions [X]a and generalized Veblen function φ(X). Also, fundamental sequences was not used in these definitions.
When I failed to extend X{·}a, I started to formulate equal values and comparison of generalized Veblen function in terms of "larger" ordinals instead of arrays of ordinals. I noticed that apparently it does work beyond Ω^{ΩΩΩ...}.
Then I suddenly realized, that sets used there can be used instead of X{·}a. (Later I found out that I already used…
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Hi there. I like to keep openings simple you see ? Oh wait, now this is not simple anymo ... Moving on.
So I discovered this wiki a few months ago, and ever since then this math has been a ton of fun, mainly because there's a lot of room for creating new stuff (because the numbers are big, get it ? Get it ? Sorry.). Ever since then, I wanted to make a notation of my own. But it didn't turn out well... at all. Let's not bother talking about my previous attempts at making notations.
So that's why I'm looking for other ideas. I need some brain food, some ideas to extend on, to define, to enhance, and hopefully, to present to you guys.
I had already a few ideas by myself, but I still wanted your input on them :
 Putting notations out of context (e.g…

curious if anyone has seen any surreal analysis or combinatorial game theory applied to tree(3) or any other googology.
since surreal numbers stem from the analysis of go, it seems to be a good fit for tree(3). i'm not entirely sure at this point on how to go about it tho..
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Since this wiki is falling apart, I'm thinking of creating an alternative googology website. One that is open only for verified registered members.
Would anybody here be interested in such a thing?
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considering setting up a stackexchange for googology on https://area51.stackexchange.com/i did the thing : https://area51.stackexchange.com/proposals/118201/googology
need community support! here's part of the procress from the faq :
 Interested parties propose and discuss sample questions to define what the site is — and is not — about.
 Users are asked to commit to participate in the site to assure that the site will have enough participation — we don't want to create ghost towns.
 The site is launched for a beta period to seed it with questions, develop the FAQ, appoint temporary moderators, and refine its design.
 If a site reaches critical mass, it becomes a full member of the Stack Exchange Network.
any thoughts?
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Here, I'm going to write only what we've agreed on.
κ is any ordinal, including stage one.
 S(1)=Ω
 S(κ+1) is the next cardinal after S(κ)
 S(κ+T) does NOT the first fixed point of α→S(κ+α).
 The nth fixed point of α→S(κ+α) is denoted by ψ_(S(κ+T))(n),
 S(κ+T) is the first fixed point of α→ψ_(S(κ+T))(α).
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Have we already agreed on larger cardinals (Specifically, S(T^2) and larger)?Some people say S(T2)=M_2, instead of I_2. It's a hell! 
Hi, just hi. Just because a lot of works in school and on my other hobbies, I have been inactive for few months. I seldom checked this wiki for the past few months but I will check it more now. I would like to say that I started to play the mobile version of Googology City and Googology FGH Battles about a month ago.
Firstly, I would like to talk about my notation. Now it is notation level 28. I got a lot of useful upgrades but I got no ordinals or hydras. I find this math very fun.
So, I would like to ask if anyone does this math and what are the beautiful ordinals or functionss I should use. Dont mind the math difficulty, just consider I have infinite amount of number. I just want some ideas for the math to be prettier.
Secondly, I am talki…
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Hi everyone, I’m Pi.jayk. I have created an extension to the googomump family of numbers using numerical prefixes. Here is the list below.
Gogoomump = 10^12000
Dogoomump = 10^24000
Trogoomump = 10^36000
Tetrogoomump = 10^48000
Pentogoomump = 10^60000
Hexogoomump = 10^72000
Heptogoomump = 10^84000
Octogoomump = 10^96000
Ennogoomump = 10^108000
Decogoomump = 10^120000
Centogoomump = 10^1200000
Millogoomump = 10^12 million
Microgoomump = 10^12 billion
Nanogoomump = 10^12 trillion
Picogoomump = 10^12 quadrillion
Femtogoomump = 10^12 quintillion
Attogoomump = 10^12 sextillion
Zeptogoomump = 10^12 septillion
Yoctogoomump = 10^12 octillion
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re: nnn6nnn's recent constitution post, i'd like to formally nomite the following members for a steering committee for gWiki: emlightened, hyp cos, littlepeng9, cloudy176, deedlit11, psicubed2 & nnn6nnn
cast nominations & votes in the comments!
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1 and a half years ago I wanted to make my own super huge number so I went crazy with recursion and ended up with this...
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I have no idea how these compare to the other numbers in this wiki. 
Tomaszewski’s number is so large it cannot be described using mathematics. It makes Sam’s number look like a grain of sand. It would fill a megafugagargantugoogolplex multiverses & a megafugagargantugoogolplex heavens. It is so complicated it would give God a headache. It is undescribable.
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Red guys number is defined as 07494795709  the number on the redtardis booth in the video DON'T HUG ME I'M SCARED 6 on youtube. It is a little under 7.5 billion and almsot equalt o the world's population. very big!!!
here's the video of same title that is an easter egg:
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EDIT : Okay, so I just realized that this notation was still pretty weak because I didn't grasp the full power of dimensional arrays in other notations. I will try harder to come up with a strong notation (hopefully a more original one, too), and I'll come back then. In the meantime, I guess this is another failed attempt. But hey, you know what they say, third time's the charm.
tl;dr this is still shit, ignore it for now, but I will return
Okay, so a while ago I tried to make my own array notation, and it didn't go too well. Having learned quite a bunch of things, I present you my own, better array notation.
Basic notation (S[]) :
Rule #1 :
Rule #2 :
Rule #3 :
Rule #4 :
Rule #5 :
represents the (possibly empty) rest of the array.
The limit function…
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Various people have analyzed BMS in the past; however, these analyses aren't detailed enough, which can cause contradictions between people. Also, some analyses might be wrong, and their creators might've not realized that yet. So we, the people at the googology Discord server, created a BMS analysis sheet as an attempt to create a master analysis for BMS. (Note: It contains the analyses of various people, so when there are contradictions between those analyses, their creators can make more detailed analyses in the contradicted areas to doublecheck)
Here's the link:
https://docs.google.com/spreadsheets/d/1ZbtPFzImztKn3VV5gtSsGuQqqp8U_ECwQhXaxdXhQ
(note: you need to give Aarex your email so he can give you permissions to edit the sheet)
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My Generalized Factorial Function!
Here is the definition:
Take any twoparameter function F.
F(n, !) = F(n, F(n  1, !))
F(1, !) = 1
For example
4 + ! = 4 + (3 + !) = 4 + 6 = 10
3 + ! = 3 + (2 + !) = 3 + 3 = 6
2 + ! = 2 + (1 + !) = 2 + 1 = 3
1 + ! = 1
Therefore, 4 + ! = 10
Another example (Ackermann Function):
A(3, !) = A(3, A(2, !)) = A(3, 5) = 253
A(2, !) = A(2, A(1, !)) = A(2, 1) = 5
A(1, !) = 1
Therefore, A(3, !) = 253
And you could even do it for the original factorial (multiplication):
4 * ! (or 4!) = 4 * (3 * !) = 4 * 6 = 24
3! = 3(2!) = 3 * 2 = 6
2! = 2(1!) = 2 * 1 = 2
1! = 1
My questions are:How can I extend it to more arguments/parameters?
What is the growth rate of this function?
Has this been discovered before?
Is this even a valid function (Because…
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I was told to try making a notation, so here it is
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When I made my aray ordinals I noticed linear BEAF grows faster than it's array ordinal (array without the prime and base) in the fgh. This is beacause BEAF always iterates in the most important value, in the limit case to. This is why I define:
 \( A_0(x)=x+1 \)
 \( A_{\alpha+1}(x+1)=A_\alpha(A_{\alpha+1}(x)) \)
 \( A_{\alpha+1}(0)=A_\alpha(1) \)
 \( A_\alpha(x+1)=A_{\alpha[A_\alpha(x)]}(x+1) \)
 \( A_\alpha(0)=A_{\alpha[0]}(1) \)
For natural numbers \( A_\alpha(x)=Ack(\alpha,x) \).
For ordinals this iterates in the collapsing argument so \( A_\omega(x)\approx f_{\omega+1}(x) \)
Succesor ordinals iterate the normal argument again so \( A_{\omega+n}(x)\approx f_{\omega+n+1}(x) \)
We can conclude that \( A_\alpha\approx f_{\alpha+1} \) so it's not tha…
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It is obvious that Edwin is still active on this wiki. It is equally obvious that the wiki staff are not going to allow the admins here to have access to the tools needed to deal with the situaiton.
Also, the architecture of the FANDOM wikia sites pretty much prevents fixing the other problems we have here. I do care for the community here, but I feel like there's nothing more that I can do. Worse: it is interfering with my actual life, which is complicated enough at this moment in time.
I'll be frank: As much as I care about the productive people here (both veterans and newbies), I don't believe this place can be saved. The combination of having:
(1) an impractical site architecture
(2) an uncooperative staff
(3) a cunning malicious troll who …
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This is the 2 argument Veblen function:
 \( \varphi_0(\beta)= \omega^\beta \)
 \( \varphi_{\alpha+1}(\beta)=\text{the $\beta^{th}$ ordinal in } \{ \gamma\mid\gamma=\varphi_{\alpha}(\gamma)\} \)
 \( \varphi_{\alpha}(\beta)[n]=\varphi_{\alpha[n]}(\beta) \)
It is usually extend to mutliple arguments, then to transfinitly many arguements and dropped to be replaced by unintuitive OCFs.
But I defined:
 \( \varphi_0(\beta)= \omega^\beta \)
 \( \varphi_{\alpha+1}(\beta)=\text{the $\beta+1^{th}$ ordinal in } \{ \gamma\mid\forall\delta

Since I have a lot of ideas related to googology. I've decided to make a blog post to organise my other blogposts (even tough I don't have a lot of posts yet) this will make it easier to link to my previous notations.
normal text = todo
These are articles inwhich I introduce the ideas on which I'll base further extensions
 Functions as fundamental sequences
 Array Ordinals
 Weak fixed points
 Uncountable indexed veblen function
 Hydra OCF
 Super Ackermann Hierachy
These are more powerfull (and complex) versions of ideas I introduced
 Ordinal array notations II
 Large cardinally indexed Veblen funtion

Here is the proposed Constitution for the Googology Wiki. Ratification requirements can be found in Article V Section II. Please vote via a comment
We, googologists, in order to form a more perfect field of mathematics and to protect the sociology on this wiki, do ordain and establish this as the Constitution for the Googology Wiki.
This name of this organization shall be The Googology Wiki, hereby refered to as GWiki
The purpose of this organization shall be to create large numbers, functions, and definitions, as well as to further expand the interest of googology to those outside of the field.
Article II: Members and Policy Voting Procedure
The general body of the organization shall be occupied by active googologists: those who have contribut…
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Are there any article I can read to familiar with how to calculate the rate of growth? I am new here so I do not much about it yet. Several days before I post a function designed by myself, and some kind people @Alemagno12 tolerate my horrible handwriting and give me an answer. I do not mean that I don't trust him, but I am not sure whether my words can clarify my meaning or not. So I think it is better to calculate by myself. I will be very happy if anyone else can take a brief glance of the function I defined.
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In the fastgrowing hierarchy, the googolo is \(f_googol(googol)\), where a googol is defined as 10 with n f's.
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The Omega Function Ω is a function that signifies exponential factorialization.
Definition:
Ω(n) = n^{n1....1}
f(n)Ω(n) = n^{f(n1)...f(1)}
Where the ellipses represent exponentiation between n1 and 2.
Instantaneous Rate of Growth:
The instantaneous rate of growth of the omega function can be found by taking the derivative of Ω(n)
We know that Ω(n) = n^{n1....1}.
Using the power rule of derivatives, Ω(n)' = Ω(n1)[n^{(Ω(n))1}].
Creator:
The Omega Function was coined by Googology wikia user IIEnDeRwITHeRII in 2018, in the early days of May.
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lvl(x, y, z) = lvl(x1, y, lvl(x1, y, lvl(...lvl(x  1, y, y))...)) with z  1 number of lvl()s
lvl(1, y, z) = y + z
For example:
lvl(1, y, z) = y + z
lvl(2, y, z) = y * z
lvl(3, y, z) = y ^ z
lvl(4, y, z) = y tetrated to the z
lvl(5, y, z) = y pentrated to the z
AND SO ON...
I'm trying to make an extended version of it, but it's currently not welldefined.
Can anybody tell me the growth rate of the level function and any suggestions on it (Or just any comments)?
Thank you for reading.
Created by MilkyWay90
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The Node Function n(m) is defined as the minimum time elapsed for m numbers ( for any m has 100000000 possible values.
This function is uncomputable because of its highly variable nature due to different computer's different processing speeds. As such, it is impossible to find a definite value for any m in n(m). It is only possible for upper limits to be calculated for any m in n(m).
Googology Wikia user IIEnDeRwITHeRII first coined this function in 2018, on May 11th.
He has also found some upper values of n(m) for n = 1 and n = 2.
For n(1), the result is 0, because comparing something against itself is completed instantly. n(1) is also the only node value that can be exactly calculated.
n(2) has an upper limit of 341334.
For n(3), the function …
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Hey this is my own new array notation which can show very large numbers. First, you have to start out with one of these brackets  ( ) on each end. The first number (left) is b, or the base. It can be any number. The last number (right) is p, or the power. Right now our array looks like this. (bp) This is not the final product. There is r, the repeat variable. You have to put this in another set of brackets, like this. (b(r)p) This is why it is called the BRP array. Here is how it works. Let's do an example, and decode it. Here it is (3(2)1) First, we will find out 3 to the 1 power (b to the p power) This is 3. But we don't stop here, as the repeater variable says two. So we take our product, and find out our product to the power of itself…
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It's Nathan. Yes, I'm still alive.
Let me start by saying thank you. Even was I was immature and completely left because of one fight. I was not in a good time when that "thing" occurred and I didn't want to risk staying with him still on here. I forgive him for he said, and I don't hold anything against him, but he is honestly not the only reason I left.
I love googology, don't get me wrong. In fact, I have still been working ruthlessly on my textbook since Novenmber, 2016. However, the type of googology here, where it's mostly people coming and going and phases occuring isn't as exciting for me. I like the kind of stuff on Sbiis' site. The nature of number and how ambiguity (with number's value) is kinda ok when it gets to large numbers. I…
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So.. yesterday I created an article on a number "hyperbuttermite", and it was deleted. It had a source, and it had quite a bit of information.
It was deleted, it doesn't say why it was deleted, and I would like to ask; why was it deleted?
This is the source of where I got the number from, and please, answer me, why was the page deleted?
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For Ordinal array functions see User blog:Rgetar/Definitions update.
Here are my Ordinal array functions: [X]a, where a is ordinal, X is array of ordinals, and [X]a is ordinal.
Then I started to designate arrays of ordinals as "larger" ordinals, for example, array of finite and countable ordinals
1, ω + 1, 10, ε_{0}, 5
as uncountable ordinal
Ω^{4} + Ω^{3}(ω + 1) + Ω^{2}10 + Ωε_{0} + 5
etc.
Here are new function cp(X) and new definitions of functions leo(X), lest(X; α), X^{0}
cp means "cardinality part"
leo means "last element of"
lest means "last element set to"
("Element", since I named leo and lest when I considered X as array, now I consider X as "larger" ordinal, but I didn't change names of these functions).
Ordinal X of cardinality card(X) can be represented as sum
X = cp(X…
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Let's start with the Big Number Game. So you start with a certain set of numbers, each having an infinite amount of ones. Using these numbers you can plug it in to a modified version of BEAF where {a,b,1} now means addition, and {a,b,2} means multiplication and so on. Now once you form a number, all the numbers you previously used are kept, but you keep your new number. Each time you form a new number is a turn. Now lets move on to Uber's Function. Basically, this takes a number of turns, and a set of numbers which are not one. When these are plugged in, Uber's Function creates the largest number in n turns in the Big Number Game with the set of numbers specified.
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There probably isn't, but if there somehow is it would be a dream come true. Despite my efforts in trying to put a hard ban on all those ridiculous categories, it's been near impossible to get people to pay attention to all this (though how much of it is just that one obsessive IP is unknown to me). A few of us already discussed in a prior thread what categories should and shouldn't be on the wiki.
I'd also love it if there was a way to purge all occurrences of a certain category for a wiki (like numbers with 25 to 42 digits or whatever the hell). Unfortunately, I doubt there is. God, Wikia is such a terrible website.
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LUCO = Large Unrecursive Countable Ordinal
Typically appears when trying to find models \(L_\alpha\) of theories. Become increasingly important for stronger theories. In this blog post I will be using various LUCO notions, such as \(\Pi_n\)reflection and stability, so some background understanding will be required to extract the essence of my calculations.
For this blog post I will use constants \[a=C(\Omega_22,0)\] \[\pi_+=C(\Omega_2,\pi)\] \[\kappa=\text{some ordinal }
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We all know \(\alpha\) is a fixed point of \(f\) if \( \alpha=f(\alpha) \) . This is the basis of the Veblen hierachy, nameing the ordinals you can't name yet (using previous functions).
But not all strickly increasing functions have fixed points. Example: the succesor function doesn't have any fixed points (by definition). So I would like to define \( \alpha \) is a weak fixed point of \( f \) if \( \beta < \alpha \Rightarrow f(\beta)
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Did you know that BMS is relatively easy to extend to transfinite entries. In this notation, we can chop off. Here is a litany of examples of how this function will work
(0)(1[1]1) is the limit of (0)(1), (0)(1,1), (0)(1,1,1), (0)(1,1,1,1), etc.
(0)(1[1]1)(1) is the limit of (0)(1[1]1), (0)(1[1]1)(0)(1[1]1), (0)(1[1]1)(0)(1[1]1)(0)(1[1]1), etc.
(0)(1[1]1)(1,1) is the limit of (0)(1[1]1), (0)(1[1]1)(1)(2[1]1), (0)(1[1]1)(1)(2[1]1)(2)(3[1]1), etc.
(0)(1[1]1)(1,1,1) is the limit of (0)(1[1]1), (0)(1[1]1)(1,1)(2,2[1]1), (0)(1[1]1)(1,1)(2,2[1]1)(2,2)(3,3[1]1), etc.
(0)(1[1]1)(1[1]1) is the limit of (0)(1[1]1)(1), (0)(1[1]1)(1,1), (0)(1[1]1)(1,1,1), etc.
(0)(1[1]1)(2) is the limit of (0)(1[1]1), (0)(1[1]1)(1[1]1), (0)(1[1]1)(1[1]1)(1[1]1), etc.
(0)(1…
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This number is from a Japanese googology competition held (technically) a month ago. I am just translating other person's number, so my translation might be wrong. According to the original author, it's bigger than BIG FOOT.
Number:
The alphabets of a formal language FOL is defined as followings:
in A. Combine P and 16x((P)48(x=t)) by AND operator in FOL and call it Q.
Whether 80(Q) is provable with A or not depends on m, A, and P. If you take m big enough, 80(Q) is always not provable with A no matter what P is. There exists the mininum of such number m, so take it and add 1. That's the f(n).
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"fight for not being able to write the smallest proof" number is defined as f(10^100). 
I'm the fourth person to earn the "Wiki Hero!" badge! Contributing to the wiki every day for 365 days!
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See "illdefined"
 The most doubtful and strong system ever.
 What computers can't do.
 Did you mean: Fish number 1
 Did you mean: Fish number 2
 Did you mean: Fish number 3
 Did you mean: Fish number 4
 Did you mean: Fish number 5
 Did you mean: Fish number 6
 Did you mean: Fish number 7
 The smallest googologism.
 The biggest googologism.
 The biggest number ever.
 Just a small number.
See "badly defined"
 What googologists hate the most.
 What googologists love the most.
 A magic which can get any value of any notation.
 Another way of saying ε_0.
 Another way of saying zerillion in short scale.
 Another way of saying halfillion in long scale.
 The only programming language in googology.
 The most powerful calculator ever existed.
 A number bounded by 10^10^6.
 Another way of saying …
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A triggonogogol is equal to {10,10 (248238473) googol}
Blog posts ain’t need no sources!
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As I was reading about array notations, I realised that most could be thought of as a combination of:
 A base: for refilling numbers
 A prime: for iteration
 An "ordinal array': this names the functions and diagonises by "collapsing" the array with the value of the array with the prime decreased by 1.
So you could define:
 \( \{a\cdots\}[n]=\{a1\cdots\} \)
 \( \{ 0 \cdots 0,p \cdots \} [n]=\{0 \cdots n,p1 \cdots \} \)
And to formalise:
 \(\{\cdots\}=sup\{\{\cdots\}[0], \{\cdots\}[1], \{\cdots\}[2], \cdots \}\)
Then:
 \(\{a_1,a_2,\cdots,a_b\}=ω^n ⋅ a_n + ω^{n1} ⋅ a_{n1} + \cdots + ω⋅a_2 + a_1\)
And a noation like
 \(A\{0,\cdots\}=2 \) (1 doesn't generate enough nesting)
 \(A\{a,0\cdots\}=a+2 \) (+2 to be consistent with th…
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Most of us know that finite numbres are cardinals, and infinite numbers are ordinals.
We consider that zero is a cardinal, but... I really don't agree! Why? There isn't any operational way that you can use zeros to make a number less, or larger than zero. (0+0 = 0, 00 = 0, 0*0 = 0, 0/0 = 0.. Wait, does 0/0 equal 1 or 0? XD)
Zeros are just like ordinals if you compare them to cardinals (excluding zero of course), there is no way to turn a cardinal into an ordinal without using some infinite function, or notation just like omega. Same with zero! You can't turn it into another number!
I would like to name zero it's special number group, or theory, but I fear it already has a name, please tell me, does it? Or have I just come up with something?
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Have been doing a lot of research on games lately & just found this tweet issued by Vi Hart almost 6 years ago:
Challenge: figure out how (and if) Surcomplex PseudoNumbers work. They are of the form a + bi, where a and b are games (illegal surreals).
"Surcomplex PseudoNumbers" returns exactly 1 result in Google: Vi's tweet. Clearly not something that has gained much traction since 2012..
Pseudosurreal numbers are 'nonnumeric' games (surreal numbers are 'numberic' games : they result in a single number vs a range or 'gap') which do not require the left term of the game to be less than the right term, for instance { 0  0 } or { 1  1 }.
Since we know that surcomplex numbers are of the form { a + bi  x + yi } where a + bi < x + yi, we als…
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I'd like to hear opinions and suggestions regarding the notability criteria for:
1. Notations.
2. Specific numbers.
In the wiki mainspace.
We've already decided (the vote was something like 8 vs 2) that there will be notability criteria of some sort, but we haven't decided on the specifics. Also, I'd like to find a middle ground that will satisfy those who opposed my original proposition. Their main objection was that having notability criteria will discourage beginners to post their work, and I agree that this could be a potential problem if we are not careful.
So keep that in mind as well, when you're making your suggestions.
Let the discussion begin!
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My new HyperRex function is compared here to recursive functions such as Veblen and my previous Hyper function. The HyperRex function is a set of two functions \(H()\) and \(r()\) and has a growth rate well beyond \(f_{\vartheta(\Omega\uparrow\uparrow\omega)}(n)\)
The notation I use here is not in general use, but I find helpful. They are parameter subscript brackets, leading zeros assumption, recursion parameter subscript \(*\), and the decremented function \(C\).
Parameter Subscript Brackets, where:
\(M(a,0_{[2]}) = M(a,0,0)\)
\(M(a,0_{[2]},b_{[3]},1) = M(a,0,0,b_1,b_2,b_3,1)\)
Leading Zeros Assumption, where:
\(M(0_{[x]},0_{[2]},b_{[3]},1) = = M(0_{[x + 2]},b_1,b_2,b_3,1) = M(b_1,b_2,b_3,1)\)
Recursion Parameter Subscript \(*\), where:
\(M^2(a)…
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In a few of my googological programs I've used functions to represent the fundamental sequence of ordinals. Since I thought it's an intersting concept and I coudn't find anything about it online.
The basic idea is to let an functionordinal α be a function so
 α(0)=α[0]
 α(1)=α[1]
 α(2)=α[2]
 …
So in general α is a function for which α(n)=the nth element of α's fundamental sequence (there are multiple fundamental sequences, so there are multiple functions for a given ordinal) 0 isn't a function (it doens't have fundamental sequence)
1 is the function that returns 0 for all input λn.0, this satisfies 1[n]=0
2 is the function that returns 1 for all input λn.1 = λn.(λm.0)), this satisfies 2[n][m]=1[m]=0
…
Ordinal Function fundamental sequence
0 0 isn't a f…
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This blog will compare recursive functions such as Veblen to my new Hyper function. The Hyper function is a set of two functions \(H()\) and \(p()\) and has a growth rate \(\approx f_{\vartheta(\Omega\uparrow\uparrow\omega)}(n)\)
The notation I use here is not in general use, but I find helpful. They are parameter subscript brackets, leading zeros assumption, recursion parameter subscript \(*\), and the decremented function \(C\).
Parameter Subscript Brackets, where:
\(M(a,0_{[2]}) = M(a,0,0)\)
\(M(a,0_{[2]},b_{[3]},1) = M(a,0,0,b_1,b_2,b_3,1)\)
Leading Zeros Assumption, where:
\(M(0_{[x]},0_{[2]},b_{[3]},1) = M(b_1,b_2,b_3,1)\)
Recursion Parameter Subscript \(*\), where:
\(M^2(a) = M^2(a_*) = M(M(a))\) and \(M(a,b_*) = M(a,b)\)
\(M^2(a,b_*) = M(a,…
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