Recent posts

I just started getting into googology again and I have created 2 numbers since. The first one is called Jonah's Number (My name is Jonah Nichols) and it is equal to 1.919...x10^10^200.

My next number is much larger and is called Ultra Behemoth. It is equal to 10^^^10^^10^1500. Comment what you think of them or maybe even make pages on them!

Updatd: I made a third number called goggle which is equal to {10,10^100,100}

This will (hopefully) be my most refined (recursive) notation

\(\#_k\) is the labeled rest of the notation

\(\{ \text{and} \}\) is any amount of ( ) brackets

\(a\) will always denote the number before the \(:\)

\(\#_i\rightarrow\#_j\) means \(\#_i\) turns into \(\#_j\)

\(f_0(c)= (c+1)^{c+1}\) and \(f_k(c)= f_{k-1}^c(c)\) where \(f^0_k(c)=c\) and \(f^{d}_k(c)=f_k(f_k^{d-1}(c))\)

This is the 1st part of ? parts and it is the most simple

\(a:=a\)

\(0:\#=0\)

\(1:\#=1\)

\(a:\#_1b\rightarrow f_a^a(b):\#_1\)

Start looking from right to left until you find a number inside a pair of bracket.Call that number \(b\).Then:

1.If \(b=0\) then \((b)\rightarrow a\)

2.If \(b>0\) then \((b)\rightarrow(b-1)(b-1)...(b-1)\) with (in total) \(a\) \((b-1)\)'s

The limit of \(n:…

• one base pair of DNA has length ~ 0.33 nanometers, mass ~ 1 zeptogram, and has 4 "states" (A-T, G-C, T-A, C-G), which can encode 2 bits.
• diploid human genome has ~ 6 billions base pairs, length ~ 2 meters, mass ~ 6 picograms, and contains ~ 1.5 gigabytes of data.
  • diploid nuclear human genome consists of 46 DNA molecules.
    • largest of them are in chromosomes 1 (~ 249 millions base pairs, ~ 8.5 centimeters, ~ 250 femtograms, ~ 62 megabytes).
    • smallest of them are in chromosomes 21 (~ 47 millions base pairs, ~ 1.6 centimeters, ~ 50 femtograms, ~ 12 megabytes).
    • human mitochondrial DNA is much less than any human nuclear DNA (16569 base pairs, ~ 5.4 micrometers, ~ 20 attograms, ~ 4 kilobytes).
  • female diploid human genome is 3 centimeters longer than of …

This is a program that factors prime numbers using Turing machines. This may seem like a silly project because it took me ~6 seconds to factor 21 (3*7) and it printed the number in unary. This project is studying the amount of time that it takes to factor each semiprime that is:

1. Greater than or equal to 21

2. The larger factor is less than 1.6 times the square root.

3. The semiprime is at least 3 integers away from the last prime being tested.

Because we are not testing the ability to factor, we are testing the number of steps required to factor and dividing it by the number. This function, so far, has not grown that fast. But, because of the way that busy beavers behave, I believe there is a possibility that it is an uncomputable one.

21: 2…

There was an OCF which collapsed an ordinal;

I don't know why it collapsed an ordinal - on purpose it'll be ill-defined!

There was an OCF which collapsed a cardinal;

That nested and expanded forever inside it!

It collapsed the cardinal to make the ordinal;

I don't know why it collapsed an ordinal - on purpose it'll be ill-defined!

There was an OCF which collapsed an inaccessible cardinal;

It's a sigh to collapse an I.

It collapsed the inaccessible cardinal to make the cardinal;

That nested and expanded forever inside it!

It collapsed the cardinal to make the ordinal;

I don't know why it collapsed an ordinal - on purpose it'll be ill-defined!

There was an OCF which collapsed a Mahlo cardinal;

That's not a gem, but it's an M.

It collapsed the Mahlo cardin…

Three months ago I thought: we can fill (-∞; +∞) with rational numbers, and it may seem that it is the most "dense" set of numbers within (-∞; +∞), because there are no "gaps" between rational numbers, that is there are rational numbers between any two different rational numbers. But it is not true: there are real numbers, and some of them are not rational. Set of rational numbers is countable, and set of real numbers has cardinality of continuum. And I asked myself: is set of real numbers the most "dense" in (-∞; +∞)? Or maybe there are numbers in (-∞; +∞), which are not real, and cardinality of set of them is larger than cardinality of continuum? (Maybe, some of these numbers are located "between" reals, the same way as some of reals are…

So if this definition doesn't work in any way then please tell me what is wrong with it.

\(f\) is the FGH

\(g\) is the SGH

when i say \(f_\alpha(n)\approx g_\alpha(n)\) i am saying that for that \(\alpha\) there exists an\(n\) for which holds \(g_\alpha(n+1)\geqslant f_\alpha(n))\)

\(\gamma(\beta)\) is the ordinal needed that \(g_\alpha(n)\approx f_\alpha(n)\) is the case for the \(\beta\)'th time

the way you get that for succesors is easy but if you come across a limit then :

\(\eta\) is the limit ordinal which you come acros

so when \(\alpha=\eta+\delta\) then \(\gamma(\alpha)=\gamma(\eta[n]+\delta)\)

\(f_\alpha(n)\approx g_\alpha(n)\) means \(\exists(n|f_\alpha(n)\leqslant g_\alpha(n+1))\)

for \(\alpha=\delta+1\):

\(\gamma(\alpha)[n]=\beta|\beta …

This question is not just related to googology but more so maths in general. Is 343 the only non-trivial cube in the form 1 + k + k^2?

A(1) = 1

A(2) = 8

A(3) = 81

A(4) = 1024

A(5) = 15625

A(6) = 279936

A(7) = 5764801

A(8) = 134217728

A(9) = 3486784401

A(10) = 100000000000

https://lllllllllwith10ls.github.io/san-solver/

The top secret number is coming from a special way in the wiki: Super Numbers! 6 special numbers that is so much and high:

1. Supergoogoltillion: Number is super big. 1E+10Duotrigintillion

2. Alottillion: Inferno number that goes to infinity. 1E+1Uncentillion

3. Luckyseventillion: Lucky seven number with scientific notation of 7s. 7.7777777E+77Septenvigintillion

4. Infinitillion: A infinity number. 1E+Infinity

5. Mastertillion: A powerful number with multi number notation. 1E+1234567890 Decillion

6. Ultratillion: Number that doubles the notation value and soo big. 2E+2Quindecillion

Those numbers are strong, but they are the 6 secret numbers that was never discovered, but the numbers have unlimited zeros. And the one of our secret numbers is calle…

Just curious, what is your sexual orientation

Hello,here i am again.Sorry for the not so frequent post,but i still have a lot of w.i.p. stuff(In case you want to see it you can look at my user page).But i don't want to talk too much so here it is.

\(G_k\) is tree nr \(k\) and it has at most \(n^k+\text{the number of conections between nodes in the previous tree}\) in a set of trees (\(\{G_1,G_2,G_3...G_i\}\))

I will define \(i\) as the number of trees in a given set.

I will define \(j\in\mathbb{N}

Link to Google doc, as Formatting doesn’t transfer well to Wikia.

[1]

E(n) is the number of times Edwin Shade vandalizes somebody’s userpage in n days.

Here is the T array function, which now includes Tetrational arrays. This should reach the BHO.

Attempt 2.

A string is any expression in the outer brackets that is a number, comma, dimensional separator, or bracket.

An entry is a integer greater than or equal to 0.

@ is any non-empty string.

$ is any number of opening brackets (can be 0 opening brackets)

& is everything inside the outer brackets.

&& is everything inside the outer brackets, but a is decreased by 1.

Entry a is at Layer x, where x is the amount of curly braces an entry is enclosed by. The base layer is x=0.

Rules: [APPLY IF POSSIBLE BEFORE STARTING SCANNING PROCESS]

1. Base Rule: T_n[0]=n+1

2. Recursion Rule: T_n[$a#]=T_(T_(...(T_(T_n[$a-1#])[$a-1#]…

The Dog is the first in a number sequence or astronomically large numbers.

The Dog is equal to 15^15^15.

This is annex to Fixed program blog with lists of ordinals, generated using Ordinal Explorer v3.1.

March 24, 2019 I published 9 lists of ordinals, generated using v3.0 of my program.

Bubby3 (thank you Bubby3!) wrote in a comment that I skip many ordinals like [[[M][M]][M]][[M][M]] = psi(W_2+psi_1(W_2)) = psi(e_(W+1)*2) and [[[[[M]M]]M][[M]M]] = psi(psi_I(0)*2). In that blog I wrote that limit of these lists is "Rathjen ordinal" (#1201 in Scorcher007's list of 2000 ordinals), but it meant that this limit is lesser.

I think that Bubby3 meant [[[[M][M]][M]][M][M]], not [[[M][M]][M]][[M][M]]. I think that

[[[M][M]][M]][[M][M]] = [[Ω₂]Ω][Ω₂]

is non-standard form of

[[M][M]][[M][M]] = [Ω₂][Ω₂]

At least my program generates almost the same fundamental s…

I fixed v3.0 of my program and created its new version Ordinal Explorer v3.1.

Note: the alphabet is the same as in v3.0.

The notation uses 3 symbols:

[

]

M

Note: strings are the same as in v3.0.

There are 3 kinds of strings:

'

M

[X]α, where X and α are strings

(Some examples of correspondence "string - ordinal": empty string is 0; "M" is least weakly Mahlo cardinal; "[]" is 1; "[[]]" is ω (least infinite ordinal)).

For string [X]α base is α:

base([X]α) = α

For empty string and M base is empty string:

base() =

base(M) =

For string [X]α booster is X:

booster([X]α) = X

For empty string and M booster is empty string:

booster() =

booster(M) =

Countable limit of this notation ('Rathjen ordinal') is designated as "L".

Note: this comparison algorythm is the same as in v…

Here is my analysis of SAN past DDAN with BM4. A lot of you will object and say "It doesn't work" and will not analyze it. However, there is no function that is as strong as behaves the same. So, I am going to analze second generation SAN anaway, despite what people say.

With any non-working notation, there are two strength, it's working and hyphoteical. Iyts working strength is how the notations which don't produce a loop and terminate are. Hypothecial strength represents how strong would it be if it were fixed, and it didn't loop and always halt. Almost always, the hypotheical strength is much stronger, but there can be excpetions, namely where a problem causes the notation to be stronger than expected, then causes it to loop after that.

…

Alright, i think we finally figured out where we're moving to.

Maybe part 1?

RULES:

1.)  φ1(n) = adding all numbers before n, so n+(n-1)+(n-2).........+7+6+5+4+3+2+1+0

2.)  φ2(n) = Multiplying all numbers before n, or just n!

3.)  φ3(n) = Exponentiating all numbers before n, or n^(n-1)^(n-2)...........^5^4^3^2^1^0

4.)  φx(n) = Repeated use of the (x-1)th hyper-operation, until you reach 0.

5.)  φφ1(n) = φn(n)

6.)  φφ2(n) = φ(φ(φ(φ.............φ(φ(φ(φ(φn(n)(n)(n))).....)))))) with n layers, from the inner-most out.

7.)  φφ3(n) = φφ2(φφ2(φφ2(φφ2............φφ2(φφ2(φφ2(φφ2(n))(n))))............. with n layers, from the inner-most out.

8.)  φφx(n) = φφ(x-1)(φφ(x-1)..............φφ(x-1)(φφ(x-1)(φφ(x-1)(n)))(n))))..................)))) with n layers, from the inner-most out.

That's it for now :D

The bee movie but every time they say bee it's every star wars movie but every 2 seconds it's all of the simpsons but every 7 seconds it's every video on YouTube but every minute all of gumball plays at %25 speed but every 3 seconds it's all of seinfeld but every 4 seconds is all of mlp:fim but every 2 seconds all of dr who plays but every 9 seconds all of emoji movie plays at %0.1 speed but every 7 seconds all of rick and morty plays but every 2 seconds it's all of rugrats but every 3 seconds it's all of tmnt but every 6 seconds it's all of spongebob at %50 speed but every 6 seconds all of adventure time plays at %25 speed but every 4 seconds all of pokemon plays but every 9 seconds it's all of gravity falls at %5 speed but every 5 second…

Happy April Fools everyone!

Sorry, it's a day late, and I didn't really plan anything, hence the silence, but, nonetheless, someone else apparently did...

On the Garfield Wiki, you will readily observe accounts have been vandalizing and editing the place which are the same as my old accounts here! Believe me, I was surprised too, because they aren't me.

Now, here’s a now-deleted blog post where I publicly announced the password to all accounts made prior a certain date is “bowers314” in the title. This post is also visible in the deletion history of the googology wiki, so yes, someone could find it if they were searching for it.

From this, it stands to reason that anyone possessing both this list and knowledge of this omni-password could use my old…

As you know,

\(\epsilon_{\alpha} = \varphi(1,\alpha)\)

\(\zeta_{\alpha} = \varphi(2,\alpha)\)

\(\eta_{\alpha} = \varphi(3,\alpha)\)

and what comes after eta? It's theta! Therefore:

\(\theta_{\alpha} = \varphi(4,\alpha)\)

And we can keep going on:

\(\iota_{\alpha} = \varphi(5,\alpha)\)

\(\kappa_{\alpha} = \varphi(6,\alpha)\)

\(\lambda_{\alpha} = \varphi(7,\alpha)\)

\(\mu_{\alpha} = \varphi(8,\alpha)\)

\(\nu_{\alpha} = \varphi(9,\alpha)\)

\(\xi_{\alpha} = \varphi(10,\alpha)\)

\(\omicron_{\alpha} = \varphi(11,\alpha)\)

\(\pi_{\alpha} = \varphi(12,\alpha)\)

\(\rho_{\alpha} = \varphi(13,\alpha)\)

\(\sigma_{\alpha} = \varphi(14,\alpha)\)

\(\tau_{\alpha} = \varphi(15,\alpha)\)

\(\upsilon_{\alpha} = \varphi(16,\alpha)\)

\(\phi_{\alpha} = \varphi(17,\alpha)\)

\(\chi_{…

https://sites.google.com/site/pointlesslargenumberstuff/home/3_3/introductionsequences

Feedback much appreciated! It isn't big, just my attempt at overviewing that subject.

So uh it is like called OmegaNum.js. Why? Because it holds numbers up to 10{1000}10, or approximately \(f_\omega(1001)\).

You can find it here.

How it works:

The number is in the form of the sign(1 or -1) and an array representing absolute value, in the form:

\([n_0,n_1,n_2,n_3,\cdots]=\cdots(10\uparrow^3)^{n_3}(10\uparrow^1)^{n_2}(10\uparrow)^{n_1}n_0\).

Please inform me about any bugs, or missing features, I obviously had missed something.

\(E\) function takes place on a plane with starting point \((0|0)\) at the bottom left corner and the thick ness of the line is \(n^{-n}\) and the lenght is \(n^{-1}\).And the size of the units doesn't matter

\(\alpha_n\) is the angle that line nr.\(n\) deviates from the x plane

\(G_1\) is the starting graph.And \(G_i\) is the graph that appears if you repeat the rules for the transformation of \(G_1\) for the \(i\)-th time

\(\star\) denotes homeomorphical embeding lets say i.e. \(G_i\star G_j\) means that \(G_i\) is homeorphically embedable into \(G_j\)

\(\omega=\text{sup}(0,1,2,3...)\)

\(\text{int}(n)=\) n rounded too the nearest integer.(you start rounding up from n.5)

\(L_i^{G_n}[x,y,\alpha]\) means that the coordinates of \(L_i^{G_n}\) are …

SORRY,please ignore

n - add n

[ - start section

] - end section

rn - repeat preceding section 

(#) - write string # (if there are parentheses inside other parenthese you write a subscript next to that parentheses to indicate layer (_0)_0/() is first layer (surface), (_1)_1 is second, and so on.)

starting value is 0. sections are evaluated left to right

Some strings may not evaluate to an actual number

For example, [3[([)r3(3)[(]r3)]r3 = [[[3]r3]r3]r3

Okay, so let me tell you what happened. Back in October, some user under the alias DrCocktorr vandalized my userpage on beyond universe wiki. He wrote stuff about how i poop and fart. So I blocked him for an infinite amount of time. A couple months later, my userpage got vandalized again probably by the same person under a different alias. This time, he uploaded a piece of shit on my userpage in addition to saying how I poop and fart. When I went to block this user it said this username does not exist. My userpage on beyond universe wiki has never been vandalized since then.

I use short hand notation for recursion of nested functions. They are a further development of notation I use that is not in common use.

Here is a summary review of my notation:

Decremented Function \(C\) is short-hand notation. For any arbitrary function:

\(M^{c + 1}(n)\) then \(C = M^{c}(n)\)

e.g.

\(M^{c + 1}(n) = M(C) = M(M^c(n))\)

Parameter Subscript Brackets is shorthand for functions with multiple parameters:

\(M(x,0_{[2]}) = M(x,0,0)\)

\(M(x,y_{[2]}) = M(x,y_1,y_2)\)

\(M(x,0_{[2]},y_{[3]},z) = M(x,0,0,y_1,y_2,y_3,z)\)

Leading Zeros Assumption applies to any function I define which accepts a variable number of input parameters. All leading zeroes can then be ignored:

\(M(0_{[x]},0_{[2]},y_{[3]},1) = M(0_{[x + 2]},y_{[3]},1) = t_0(y_{[3]},1) = M(y_…

This is annex to New program (ordinals up to Rathjen ordinal) blog with lists of ordinals, generated using Ordinal Explorer v3.0. I posted them in the separate blog, since they are too large.

These lists are expansions of Rathjen ordinal, designated here as "L" (see #1201 in Scorcher007's list of 2000 ordinals).

Also I inserted inside each pair of ordinals (α, β) 8 elements of fundamental sequence of β larger than α (if β is not a successor).

Sizes of "clean" lists (that is without additional fs elements):

Single expansion: 3 ordinals

Double expansion: 12 ordinals

Triple expansion: 73 ordinals

Quadruple expansion: 592 ordinals

Pentuple expansion: 6233 ordinals

Hexuple expansion (not published here): 86326 ordinals

Sizes of lists with additional fs e…

I modified my program I published in my previous blog post and created its new version Ordinal Explorer v3.0. I think that its countable limit is ordinal #1201 of Scorcher007's list of ordinals ('Rathjen ordinal'). (Countable limit in v2.0 was #691 or 'Small Rathjen ordinal').

The notation uses 3 symbols:

[

]

M

(In v2.0 was "I" instead of "M").

There are 3 kinds of strings:

'

M

[X]α, where X and α are strings

(Some examples of correspondence "string - ordinal": empty string is 0; "M" is least weakly Mahlo cardinal; "[]" is 1; "[[]]" is ω (least infinite ordinal), see Fundamental sequence algorythm section).

For string [X]α base is α:

base([X]α) = α

For empty string and M base is empty string:

base() =

base(M) =

For string [X]α booster is X:

booster([X]α) =…

I made myself a new website with a few new ideas and googolisms

check it out pls

woah, big numbers!

This is an English translation of my Japanese blog post submitted for a Japanese googological event last year.

I denote by \(\textrm{Lim}\) the class of non-zero limit ordinals.

Throughout this blog post, a function means a map \(\mathbb{N} \to \mathbb{N}\), and I denote by \(\textrm{Func}\) the set of functions. For functions \(g\) and \(h\), I define the relation \(g < h\) by the strict order of their growth rates. Namely, the relation \(g < h\) holds if and only if there is an \(N \in \mathbb{N}\) such that for any \(n \in \mathbb{N}\), \(n > N\) implies \(g(n) < h(n)\).

A family \((g_{\beta})_{\beta \in \alpha} \in \textrm{Func}^{\alpha}\) of functions indexed by an ordinal \(\alpha\) is said to be a linear hierarchy if for any \((\beta,\ga…

A(n) is the number of times DrCocktor vandalizes Cloudy176’s userpage in n days.

RULES:

1.)  c(0) = w

2.)  c(1) = e0

3.)  c(x) = the xth symbol.

4.)  c_0(x) = c(x)

5.)  c_1(x) = c(c(c(c...........c(c(c(c(c(c(c(c(c(x)))).....))))))) with c(x) c's.

6.)  c_2(x) =  c(c(c(c...........c(c(c(c(c(c(c(c(c(x)))).....))))))) with  c(c(c(c...........c(c(c(c(c(c(c(c(c(x)))).....))))))) with  c(c(c(c...........c(c(c(c(c(c(c(c(c(x)))).....))))))) with  c(c(c(c...........c(c(c(c(c(c(c(c(c(x)))).....))))))) with...............  c(c(c(c...........c(c(c(c(c(c(c(c(c(x)))).....))))))) with  c(c(c(c...........c(c(c(c(c(c(c(c(c(x)))).....))))))) with  c(c(c(c...........c(c(c(c(c(c(c(c(c(x)))).....))))))) with c(x) c's. )c's.)c's.)................................c's)c's.  The amount of times it says "c's" is c(x).

7.)  c_w(x) =   c(c(c(c...........c…

Hey Googology Wikia, Edwin Shade here.

For reasons primarily relating to the posting of the “Chasing Shade” blog post - in addition to the prior consumption of coffee and using the computer past my proper bedtime - I was unable to sleep last night, and spent my time from the late night to the early morning hours in mental turmoil, entertaining thoughts that are unspeakable as I look back on them over the divide of just a half-day.

This is how it was at least, until six-o’clock a.m., when - unable to stop the dark thoughts that were piling in my mind - I turned to prayer as a last resort. I am a bit ashamed now to say “last resort”, but my thoughts were racing so much I did not think of it sooner.

Anyways, I prayed multiple times, and when I w…

Hello Googology Wiki! As I’ve watched Edwin Shade’s thousand-sockpuppet attack on this wiki, I’ve long wondered: just who is this guy? I’m sure many of you have wondered that very same thing, so I set out to “connect the dots” and see what kind of public information I could find on him. Let’s examine the clues, shall we?

1. Last October, someone uploaded Edwin has uploaded here in the past.
2. The title of the aforementioned video contains the name Kaleb Cook. Could this be Edwin’s real name?
3. Equally interesting is the second part of the title - “The Wizard of Village Park.” I assume it’s a (tongue in cheek?) reference to Thomas Edison’s Menlo Park, so perhaps this “Village Park” is where Edwin lives?
4. Then, another clue: this IP address was blocked…

Rules:

[n] = n

[n, 0] = n+1

[n, 1] = n+2

[n, x] = n+(x+1)

[n, 1, 2] = [n, n]

[n, 2, 2] = [n, n+1]

[n, x, 2] = [n, (n+(x-1))] = [n, [n, x-2]]

[n, 1, 3] = [n, n, 2] = [n, (2n-1)] = n+2n = 3n

That's it for now :D

It’s now time for the second part of Hyper Tower Notation. I’ll be defining how multi-symbol separators will decompose. Last time, we left off at n#_nn, and found a growth limit of w^w. The behaviour of the expressions won’t be fundamentally changed by having multiple symbols in a separator, so I’ll just get into each case:

1. #_k+1#_m#_p...→#_k#_m#_p...

2. ##...##→#_n#_n…#_n

3. ##...##_k+1#_m#_p...→#_n#_n…#_n#_k#_m#_p...

Let’s see how those work in practice:

3##3=3#₃3#₃3^(3##2)=3#₃3#₃3^(3#₃3#₃3^(3#₃3#₃3))

3#₂#3=3##3##3^(3#₂#2)=3##3##3^(3##3##3^(3##3##3))

3##₂#3=3#₃##3#₃##3^(3#₃##3#₃##3^(3#₃##3#₃##3))

I’ll now compare the notation to the FGH:

n##n~w^w

n##n#n~w^w+1

n##n#₂n~w^w+w

n##n#₃n~w^w+w^2

n##n##n~w^w*2

n##n##n##n~w^w*3

n#₂#n~w^(w+1)

n#₃#n~w^(w+2)

n##₂n~w^(w*2)

n##₃n~w^(w*3)

n###n~…

Ok. I'm bad at this. Please a9kqw{#=Mz[x-v,3/

0:1. \(\lt\) is partial well-ordering relation of two expressions.

0:2. For any expressions \(a\), \(b\), and \(c\), \(a\lt b\land b\lt c\implies a\lt c\).

0:3. For any expressions \(a\) and \(b\), \(a=b\) means that it is equivalent, and it can be interchanged. Or, \(a=b\implies\forall P[P(a)\iff P(b)]\)

0:4. For any expression \(a\), \(a=a\).

0:5. For any expressions \(a\) and \(b\), \(a\lt b\oplus b\lt a\oplus a=b\).

P1

1:1. The expression consists of \(t\), which then must be followed by \((\), which must then be followed by a \()\), or another expression of this, then \()\). For example, \(t(t())\).

1:2. \(r\), if \(r\ne t()\) then \(t()\lt r\).

1:3. \(r\lt t(r)\), where \(r\) is an expression.

The …

Hyper Tower Notation is an exponentiation-based notation. The expressions of this notation will decompose into power-towers.

1. ...n#1=...n

2. ...n#m+1=...n^(...n#m)

3. ...n#_k+1m+1=...n#_kn#_k...n n’s...n#_kn^(...n#_k+1m); #₁=#

4. ...a#_mn#_k+11=...a#_mn#_kn#_k...n n’s...n#_kn^(...a#_mn); n#_k+11=n#_kn#_k...n n’s...n#_kn

Let’s look at some examples:

4#3=4^(4#2)=4^4^4

5#3#3=5#3^(5#3#2)=5#3^(5#3^(5#3))=5#3^(5#(3^5^5^5))=5#3^5^5^...3^5^5^5 5’s...5^5

4#4#4#3=4#4#4^(4#4#4#2)=4#4#4^(4#4#4^(4#4#4))

n#m is basically n^^m, and each new sign and term added roughly moves the expression up a hyper-operator level. n#n#n…#n#n roughly corresponds to recursion level omega(w) in the FGH.

4#₂3=4#4#4#4^(4#₂2)=4#4#4#4^(4#4#4#4^(4#4#4#4))

4#₂4#4=4#24^(4#₂4#3)=4#₂4^(4#₂4^(4#₂4^(4#₂4)))

4#₂4#₂=4#₂4#4…

Addition is the weakest of the hyper-operators. In the FGH, it compares to f1(n). It’s a weak notation, but it’s a great starting point for googological functions. So, let’s take 3+3+3. We have three-plus-three-plus-three. An average person can easily tell you that this is equal to nine. But, what happens when I do this: 3++3++3? Well, I’ll show you how I’d solve this:

3++3++3=3+3+3++2++3=9++2++3=9+9+9+9+9+9+9+9+9++1++3=81++1++3=81+81...81 81’s…+81++3=6561++3=43046721++2~1.853*10^15++1~3.433*10^30.

Yeah, that actually defeats 3^^3, amazingly enough. But it’s nothing compared to 3+++3:

3+++3=3++3++3+++2~3.433*10^30+++2. That’s big. So, what’s the point of this? Well, these expressions are part of my new notation, Cascade Notation. This notatio…

Let we have an ordinal notation (that is correspondence "string - ordinal"). An let we have a fundamental sequence algorythm in this notation (that is correspondence "string, corresponding to ordinal - string, corresponding to an element of fundamental sequence of this ordinal").

If we have this notation and this algorythm, is a system of fundamental sequences defined?

Maybe not. Because there may be different strings in the notation, corresponding to the same ordinal. And the fundamental sequence algorythm, applied to these strings, may produce different fundamental sequences, so we can get different fundamental sequences of the same ordinal. So, system of fundamental sequences may be not well-defined.

There may be different ways to solve th…

This is part 1: Beginning of One Entry Arrays

RULES:

(a) = a

(a)(1) = a+1 = f_0(a)

(a)(2) = a+1+1+1........+1+1+1 with a nests, or 2a = f_1(a)

(a)(3) = 2*2*2*2*2..............*2*2*2*2*2*a or (2^a)a = f_2(a)

(a)(x) = f_x(a) (I think).

(a)(1)(2) = (a)(a)

(a)(2)(2) = (a)(((((...(((((a)((((...(((((a)(a))(a)))......a)))))))........)))))) with a nests (Right to left)

That's it for now :D

PrEpArE fOR ToP TeN AnImE PlOt TwiSts WheN wE rEaCh ThE LImIt Of NormAl OnE EntRy ArrAys XdDddd

This idea came from wikipedia where I read something about commutative hyperopperations, but they didn't even reach tetration so I made this.

The goal is to define a commutative hyper operation function, \( A(a,b)=A(b,a). \)

This is the definition:

1. \( A(a,b)=A(b,a) \), from now on the first argument is assumed to be larger or equal, without loss of generality.
2. \( A(a,0)=a+1 \)
3. \( A(a+1,b+1)=A(A(a,b+1),b) \)

For \( a\ge b \) we get:

• \( A(a,0)=a+1 \)
• \( A(a,1)=a+2 \)
• \( A(a,2)=2a+2\)
• \( A(a,3)>2^{a+1} \)
• \( A(a,4)>2↑↑a\)
• in general \( A(n,n) \) hase the same growth rate as \( f_\omega \)

\(\eta\) is any function it could even mean that nothing happens.But \(\eta\) does not size down the input.

\(\Theta(0)[n]=n\) and \(\Theta_0=\Theta\) here \(\alpha[n]\) is the n-th term in the fundamental sequence of \(\alpha\)

\(\Theta(\eta\alpha)[n]=\Theta(\eta\alpha[n])\) if the next case does not apply.And if the \(\Omega\) case does not apply

\(\Theta(\eta\alpha+1)[n]=\Theta(\eta\alpha)\uparrow\uparrow n\) n times

\(\Theta(\eta\Omega)=sup(\Theta(\eta\omega),\Theta(\eta\omega+\Theta(\eta\omega)),\Theta(\eta\omega+\Theta(\eta\omega+\Theta(\eta\omega)))),...)\)

\(\eta\Omega_0=\eta\Omega\) and \(\eta\Omega_{\alpha+1}[n]=\eta\Omega_\alpha^{\dots^{\eta\Omega_\alpha}}​\) or \(\eta\Omega_\alpha\uparrow\uparrow n\)

Thats it for part 1?

Oliwier's Moskalewicz Growth Notation (OMGN) is a notation to define large numbers created by Olwier Moskalewicz in 2019.

Rules aren't similar to BEAF or BAN.

• for \({a}\) we have just a.
• for \({a,b}\) we've got \(a -> a -> b times