10,220 Pages

• ## Classes of ordinals ｛-｝a, ｛*｝a, ｛+｝a

May 24, 2018 by Rgetar

Recently I was trying to extend set X{·}a (see Definitions update blog) beyond ΩΩΩΩ.... (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 ΩΩΩΩ....

Then I suddenly realized, that sets used there can be used instead of X{·}a. (Later I found out that I already used…

• ## Looking for some ideas

May 23, 2018 by Syst3ms

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…
• ## surreal analysis of tree(3)

May 23, 2018 by Meowzz

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..

• ## A new googology community site - anyone interested?

May 23, 2018 by PsiCubed2

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?

• ## googology.stackexchange

May 23, 2018 by Meowzz

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 :

1. Interested parties propose and discuss sample questions to define what the site is — and is not — about.
2. 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.
3. The site is launched for a beta period to seed it with questions, develop the FAQ, appoint temporary moderators, and refine its design.
4. If a site reaches critical mass, it becomes a full member of the Stack Exchange Network.

any thoughts?

• ## What We've Agreed on Stage Cardinal

May 21, 2018 by Nayuta Ito

Here, I'm going to write only what we've agreed on.

κ is any ordinal, including stage one.

1. S(1)=Ω
2. S(κ+1) is the next cardinal after S(κ)
3. S(κ+T) does NOT the first fixed point of α→S(κ+α).
1. The n-th fixed point of α→S(κ+α) is denoted by ψ_(S(κ+T))(n),
2. S(κ+T) is the first fixed point of α→ψ_(S(κ+T))(α).

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!

• ## It's Judgement Day.

May 21, 2018 by That's The Way The Cookie Crumbles II

Don't you make fun of me or I'll sic Hal 9000 on me and make you vanish like Bowie.

• ## Hi everybody

May 19, 2018 by BestBMCplayerr

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…

• ## Gogoomump and related numbers

May 17, 2018 by Pi.jayk

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

Femtogoomump = 10^12 quintillion

Attogoomump = 10^12 sextillion

Zeptogoomump = 10^12 septillion

Yoctogoomump = 10^12 octillion

• ## steering committee

May 17, 2018 by Meowzz

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

• ## Some big numbers

May 16, 2018 by DWither

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...

I have no idea how these compare to the other numbers in this wiki.

• ## Tomaszewski’s number

May 16, 2018 by Pi.jayk

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.

• ## Red guys number

May 14, 2018 by Maccur Jaccor

Red guys number is defined as 07494795709 -- the number on the red-tardis 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:

• ## Syst3ms's Array Notation

May 14, 2018 by Syst3ms

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…

• ## BMS analysis sheet!

May 14, 2018 by Alemagno12

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 double-check)

(note: you need to give Aarex your email so he can give you permissions to edit the sheet)

• ## Generalized Factorial Function

May 13, 2018 by MilkyWay90

My Generalized Factorial Function!

Here is the definition:

Take any two-parameter 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…

• ## HBN: Hierarchical Bracket Notation

May 13, 2018 by Planetn9ne

I was told to try making a notation, so here it is

• ## SAH: Super Ackermann Hierachy

May 13, 2018 by Fejfo

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:

1. $$A_0(x)=x+1$$
2. $$A_{\alpha+1}(x+1)=A_\alpha(A_{\alpha+1}(x))$$
3. $$A_{\alpha+1}(0)=A_\alpha(1)$$
4. $$A_\alpha(x+1)=A_{\alpha[A_\alpha(x)]}(x+1)$$
5. $$A_\alpha(0)=A_{\alpha[0]}(1)$$

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…

• ## Well... It was nice while it lasted.

May 12, 2018 by PsiCubed2

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 …

• ## Uncountable indexed veblen function

May 12, 2018 by Fejfo

This is the 2 argument Veblen function:

1. $$\varphi_0(\beta)= \omega^\beta$$
2. $$\varphi_{\alpha+1}(\beta)=\text{the \beta^{th} ordinal in } \{ \gamma\mid\gamma=\varphi_{\alpha}(\gamma)\}$$
3. $$\varphi_{\alpha}(\beta)[n]=\varphi_{\alpha[n]}(\beta)$$

It is usually extend to mutliple arguments, then to transfinitly many arguements and dropped to be replaced by unintuitive OCFs.

But I defined:

1. $$\varphi_0(\beta)= \omega^\beta$$

• ## Omega Function

May 12, 2018 by IIEnDeRwITHeRII

The Omega Function Ω is a function that signifies exponential factorialization.

Definition:

Ω(n) = nn-1....1

f(n)Ω(n) = nf(n-1)...f(1)

Where the ellipses represent exponentiation between n-1 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) = nn-1....1.

Using the power rule of derivatives, Ω(n)' = Ω(n-1)[n(Ω(n))-1].

Creator:

The Omega Function was coined by Googology wikia user IIEnDeRwITHeRII in 2018, in the early days of May.

• ## My Level Function

May 11, 2018 by MilkyWay90

lvl(x, y, z) = lvl(x-1, y, lvl(x-1, 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 well-defined.

Can anybody tell me the growth rate of the level function and any suggestions on it (Or just any comments)?

Created by MilkyWay90

• ## Node Function

May 11, 2018 by IIEnDeRwITHeRII

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 …

• ## Down arrow notation

May 11, 2018 by Boris Huller
• ## BRP Array Notation

May 9, 2018 by Hellotheretrespasser

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…

• ## Well... This is gonna be awkward

May 8, 2018 by Nnn6nnn

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…

• ## Why was my page deleted?

May 7, 2018 by BlauesWasser

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?

• ## Generalizations of Ordinal array functions and FGH

May 7, 2018 by Rgetar

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, ε0, 5

as uncountable ordinal

Ω4 + Ω3(ω + 1) + Ω210 + Ωε0 + 5

etc.

Here are new function cp(X) and new definitions of functions leo(X), lest(X; α), X0

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…

• ## Big Number Game and Uber's Function

May 7, 2018 by Ubersketch

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.

• ## Is there any way to ban certain categories from being added to articles?

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.

• ## Algorithm for Generating LUCOs From TON Expressions + Intuitive Analysis

May 6, 2018 by Boboris02

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 }

• ## Weak fixed points

May 5, 2018 by Fejfo

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) Read more > • ## Pure Maltiplication OCF May 5, 2018 by Nayuta Ito I have no idea what I should do. Definition: Read more > • ## Proof that the order type of fusible numbers is exactly e0 May 5, 2018 by Alemagno12 coming soon hint: think of TON, but without Ωxs Read more > • ## Transfinite BMS. May 4, 2018 by Bubby3 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… Read more > • ## Is this bigger than Rayo's number? May 4, 2018 by Nayuta Ito 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). "fight for not being able to write the smallest proof" number is defined as f(10^100). Read more > • ## Wiki Hero! May 4, 2018 by Daniellewiki I'm the fourth person to earn the "Wiki Hero!" badge! Contributing to the wiki every day for 365 days! Read more > • ## The Devil's Dictionary in Googology May 4, 2018 by Nayuta Ito See "ill-defined" 1. The most doubtful and strong system ever. 1. What computers can't do. 1. Did you mean: Fish number 1 2. Did you mean: Fish number 2 3. Did you mean: Fish number 3 4. Did you mean: Fish number 4 5. Did you mean: Fish number 5 6. Did you mean: Fish number 6 7. Did you mean: Fish number 7 1. The smallest googologism. 2. The biggest googologism. 1. The biggest number ever. 2. Just a small number. See "badly defined" 1. What googologists hate the most. 2. What googologists love the most. 1. A magic which can get any value of any notation. 1. Another way of saying ε_0. 1. Another way of saying zerillion in short scale. 2. Another way of saying halfillion in long scale. 1. The only programming language in googology. 2. The most powerful calculator ever existed. 1. A number bounded by 10^10^6. 1. Another way of saying … Read more > • ## Triggonogogol May 4, 2018 by FaxulFaze A triggonogogol is equal to {10,10 (248238473) googol} Blog posts ain’t need no sources! Read more > • ## Array notation arrays as ordinals May 3, 2018 by Fejfo 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]=\{a-1\cdots\}$$
• $$\{ 0 \cdots 0,p \cdots \} [n]=\{0 \cdots n,p-1 \cdots \}$$

And to formalise:

• $$\{\cdots\}=sup\{\{\cdots\}[0], \{\cdots\}[1], \{\cdots\}[2], \cdots \}$$

Then:

• $$\{a_1,a_2,\cdots,a_b\}=ω^n ⋅ a_n + ω^{n-1} ⋅ a_{n-1} + \cdots + ω⋅a_2 + a_1$$

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…

• ## Why Zero Shouldn't be considered a cardinal

May 1, 2018 by BlauesWasser

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, 0-0 = 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?

• ## *games!

April 30, 2018 by Meowzz

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 Pseudo-Numbers work. They are of the form a + bi, where a and b are games (illegal surreals).

"Surcomplex Pseudo-Numbers" returns exactly 1 result in Google: Vi's tweet. Clearly not something that has gained much traction since 2012..

Pseudo-surreal numbers are 'non-numeric' 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…

• ## Notability Guidelines for the mainspace: looking for your input

April 29, 2018 by PsiCubed2

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!

• ## The HyperRex Function

April 29, 2018 by B1mb0w

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:

$$M(a,0_{[2]}) = M(a,0,0)$$

$$M(a,0_{[2]},b_{[3]},1) = M(a,0,0,b_1,b_2,b_3,1)$$

$$M(0_{[x]},0_{[2]},b_{[3]},1) = = M(0_{[x + 2]},b_1,b_2,b_3,1) = M(b_1,b_2,b_3,1)$$

Recursion Parameter Subscript $$*$$, where:

$$M^2(a)… Read more > • ## Using functions as fundamental sequences for the fgh April 28, 2018 by Fejfo 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 function-ordinal 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 n-th 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… Read more > • ## The Hyper Function April 28, 2018 by B1mb0w 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:

$$M(a,0_{[2]}) = M(a,0,0)$$

$$M(a,0_{[2]},b_{[3]},1) = M(a,0,0,b_1,b_2,b_3,1)$$

$$M(0_{[x]},0_{[2]},b_{[3]},1) = M(b_1,b_2,b_3,1)$$
Recursion Parameter Subscript $$*$$, where:
$$M^2(a) = M^2(a_*) = M(M(a))$$ and $$M(a,b_*) = M(a,b)$$