11,335
pages

Kanrokoti 1 minute ago
0

## Buchholz sequence

30jl02 Jason Liao 3 hours ago
0

## Hyper X notation

This notation is used to make large numbers like most other notation like hyper E notation(ExE)

X10= 10{10}10 (Tridecal)

Xn= n{n}n

XXn=n{Xn}n

Xn#m=XX.....(m X's)XX.....((m-1)X's)..................................XXX(3 X's)XX(2 X's)X(1 X)n

Xn#m#y=Xn#(Xm#y)

Xn##m=Xn#n#n#....n(m #'s)

1.Xn=10(n=0), Otherwise Xn= n{n}n

2.XXn=20(n=0), Otherwise Xn= n{n}n

3.Every X(s) that you add in the front when n=0, you add 10

4.Xn#m=X10(n,m=0),Otherwise Xn#m=XX.....(m X's)XX.....((m-1)X's)..................................XXX(3 X's)XX(2 X's)X(1 X)n

Xn{1}m=Xn#m

Xn{2}m=Xn#m

Xn{#,1}m=Xn{Xn{n+m}m}m

Xn{#,2}m=Xn{Xn{Xn{n*m}m}m}m

Xn{#,3}m=Xn{Xn{Xn{Xn{n^m}m}m}m}m

Xn{#,#}m=Xn{#,m}=Xn{Xn{Xn{Xn{Xn{Xn.....n{(m-2)}m.....m}m}m}m}m}m

Xn{#,#,#…

Hexirp 1 day ago
3

## (2021-10-23) New Policy

This article was written in Japanese and will be translated into English later.

まず、私は、ポリシーとガイドラインの役割分担を明確にしたいです。今のポリシーは複雑になりすぎています。

Arcioko 2 days ago
1

## unnamed factorial

im bad at extending this to make it sound "formal" so i will just dump the definition

Notated as: (not so good approximation) E25#539

5? = no idea how to calculate

any sugestions to how i shall name this? i have considered interrogatorial (or ?torial) and detonatorial (detonate + factorial, kinda obvious), i also thought of making it the fourth superfactorial, since it beats all of them

edit: i dont think i wanna name it detonatorial, so either i name interrogatorial (or ?torial) or i make a superfactorial

Binary198 3 days ago
1

## Sushi number🍣🐠🐡

One of the world's biggest numbers, dedicated to one of the world's best foods! My attempt at making the largest valid googologism. Leave any corrections in the comments, and I will make sure to fix them because I won't let a tiny mistake get in my way :)

I guess I could keep going, but I'm fine with stopping now. Hopefully the fourth sushi number is the largest valid googologism :)

Clyde1000^1000 6 days ago
0

## Is this correct?

Today, I will do a good meaning towards E# and xE#

E A=10^A

A E B=A*10^B (following rules of equations...)

E A#B=E (E A#(B-1) )

E A#B#C=E A#E (A#B#(C-1))

? will now mean unknown part of txt.

E ?#X#Y = E ?# E ?#X#(Y-1)

E X##Y = E X#(X##(Y-1))

E X##1 = E X

E X #^{Y} Z=E X #...# Z Y #'s { because i hate writing ##########################################}

E X#^{Y}Z=E X#^{Y-1}X#^{Y-1}......X#^{Y-1}X#^{Y-1}X Z X's

GODGAHLAH= E 100#^{100}100

that all

Binary198 7 days ago
0

## Idea for CNF-based large function

We assume the reader is familiar with Cantor normal form. This is inspired by Michael Rathjen's coding function in the Art of Ordinal Analysis. We define a coding function $$\textrm{wCNF}: \omega^\omega \mapsto \mathbb{N}$$ like so:

• $$\textrm{wCNF}(0) = 0$$
• If $$s =_{CNF} \omega^{s_1} + \omega^{s_2} + \cdots + \omega^{s_n}$$, $$\textrm{wCNF}(s) = 2^{s_1 + 1} + 3^{s_2 + 1} + ... + p_n^{s_n + 1}$$, where $$p_n$$ is the $$n$$th prime number.

Due to the use of prime numbers, most outputs will be unique (not all sadly), and some outputs will never appear.

$$\textrm{wCNF}(0) = 0$$

$$\textrm{wCNF}(1) = 2^{0+1} = 2$$

$$\textrm{wCNF}(2) = 2^{0+1} + 3^{0+1} = 5$$

$$\textrm{wCNF}(3) = 2^{0+1} + 3^{0+1} + 5^{0+1} = 10$$

$$\textrm{wCNF}(\omega) = 2^{1+1} = 4$$ …

Binary198 8 days ago
1

## There's PTOs, why not MTOs?

As you all probably know, the proof-theoretic ordinal of a theory T is given by the following formula:

$$||T|| = \textrm{sup}\{\textrm{otyp}(\prec) | \prec \textrm{is} \; \textrm{a} \; \textrm{primitive-recursive} \; \textrm{well-ordering}; T \vdash TI(\prec)\}$$.

Well, I suggest the model-theoretic ordinal, which is given by the following formula:

$$MTO(T) = \textrm{min}\{\alpha: L_\alpha \models T\}$$ and $$MTO_2(T) = \textrm{min}\{\alpha: L_\alpha \cap P(\omega) \models T\}$$

For most theories, I believe that the MTO is greater than the PTO.

I'm not sure about MTO(PA), but I think it is $$\omega$$.

$$MTO(KPi) = MTO_2(\Delta^1_2 - CA + BI) = MTO(KP +$$ there exists a transitive collapse for any well-founded relation$$)$$ is the first recursive…

RightLefts 9 days ago
0

## Test

1

2

GerfloJoroZ 12 days ago
1

Lettuce, Pineapples and Cucumbers

(THESE NUMBERS WERE WRITTEN FROM SMALLEST TO LARGEST, IF THERE IS ANY PROBLEM, PUT IT IN THE COMMENTS)

(SOME OF THIS NUMBERS ARE ILL-DEFINED)

(I WILL UPDATE THIS PAGE LIKE A GOOGOL TIMES, COME HERE AGAIN WHEN YOU WANT TO EAT SALAD)

• 1 Googol and Decker
• 2 Goshomity
• 3 Meameamealokkapoowa
• 5 W H Y ?
• 6 REAL EXTENSIONS
• 6.1 Oblivion Extensions

• Googoltriminex, 10-(101010100)
• Googolduminex, 10-(1010100)
• Googolminex, 10-(10100)
• Tri-Little Googol, 0.08100
• Bi-Little Googol, 0.4100
• Bi-Little Googolplex, 0.40.4100
• Unker, 1
• Bker, 4
• Trker, 7625597484987
• Little Googol, 2100
• Googol, 10100
• Googolduteen, 10100+20
• Googoltriteen, 10100+30
• Googolquinteen, 10100+50
• Googolcentiteen, 10100+100
• Googolty, 10101
• Googolt…

Binary198 14 days ago
1

## Analysis of galaxy function

$$\iota(0) = \textrm{Galaxy}(0, 1) = \\omega$$

$$\iota(1) = \textrm{Galaxy}(\\omega, \textrm{Collapse}_{\\omega}(0))$$

$$\iota(2) = \textrm{Galaxy}(\textrm{Galaxy}(\\omega, \textrm{Collapse}_{\\omega}(0)), \textrm{Collapse}_{\textrm{Galaxy}(\\omega, \textrm{Collapse}_{\\omega}(0))}(0))$$

$$\iota(3) = \textrm{Galaxy}(\textrm{Galaxy}(\textrm{Galaxy}(\\omega, \textrm{Collapse}_{\\omega}(0)), \textrm{Collapse}_{\textrm{Galaxy}(\\omega, \textrm{Collapse}_{\\omega}(0))}(0)), \textrm{Collapse}_{\textrm{Galaxy}(\textrm{Galaxy}(\\omega, \textrm{Collapse}_{\\omega}(0)), \textrm{Collapse}_{\textrm{Galaxy}(\\omega, \textrm{Collapse}_{\\omega}(0))}(0))}(0))$$

$$\kappa(0) = f_{\textrm{Galaxy}(\\omega, \textrm{Collapse}_{\\omega}(0))}(2)$$

\…

Binary198 14 days ago
0

## Galaxy function

This is an ordinal notation I developed, partially inspired by Taranovsky's C. I wanted to play around and test out some different ideas and combine them into one. Put any errors or criticism in the comments.

• 1 Overview
• 2 Notation
• 3 Relation
• 4 Standard form
• 5 Cofinality & Fundamental Sequences
• 6 FGH

I was listening to this really cool song called Galaxy Collapse and I thought it would be cool to name a function after, also Galaxy Collapse and Ordinal Collapsing Function share the word "Collapse" although this isn't an OCF. Anyway:

Here, I will define the character string for the notation:

• $$0 \in T$$
• For all $$(a, b) \in T^2$$, $$a+b \in T, PT$$
• For all $$a \in T$$, $$\textrm{Scream}(a) \in T, PT$$
• For all $$(a, b) \in PT \times (T \backslash \{0\})$$, $$\t… TheGoogologist 15 days ago 1 ## Journey to Infinity: Part 1 Hello everyone! This is Avem231, and welcome to the first part of Journey to Infinity, a resurrection of the Hunt for the Largest Number series that never caught on. Unlike in the previous series, where I simply made an ill-defined salad function, this time, I will actually try to create a clear definition for the googolisms I will be making along the way. Without any further ado, let's begin! In Sbiis Saibian's explanation of the Fast-growing hierarchy, he left off with Γ0 as the largest possible ordinal you can plug into f. Because of this, I will be making my own extension of the FGH that picks up from where he left off, and goes even further beyond with the already-insane potential of the hierarchy. There are a lot of infinitely nested power… MultiSoul 16 days ago 0 ## Make a page https://sites.google.com/view/xtremelistofillionnumbers/home Literally Legendary 17 days ago 1 ## Stacked up-arrow notation This notation is in the form Maybe unfinished GerfloJoroZ 17 days ago 0 ## My SI Prefixes Extensions MADNESS STARTS NOW Value -∞ 1 0.5 0.33333... 0.25 0.2 0.16666666... 1/7 0.125 0.11111111... 1/11 1/12 1/13 1/14 1/15 1/16 1/17 1/18 1/19 1/20 1/21 1/22 1/23 1/24 1/25 1/30 1/50 1/90 Value 10-27 10-30 10-33 10-63 10-93 10-123 10-153 10-183 10-213 10-243 10-273 10-303 10-3003 10-30003 10-300003 10-3000003 10-30000003 10-300000003 10-300000003 10-3000000003 10-(3×1012+3) 10-(3×1015+3) 10-(3×1018+3) 10-(3×1021+3) 10-(3×1024+3) 10-(3×1027+3) 10-(3×1030+3) 10-(3×1033+3) 10-(3×10303+3) 10-(3×103003+3) Literally Legendary 18 days ago 4 ## I need ideas I want to invent a function but I don't know what kind of function to make. Binary198 18 days ago 1 ## One of my array notations: SSAN (superspeedy) I created this. It's quite powerful, though not very accurate because it's based on FGH not hyperoperators (exponentiation, tetration, etc). However, it is very powerful: I might be able to snag SAN's title if I try hard enough. NOTE: I only just realised that there exists an SSAN, hopefully I can change its name somehow. Instead of adding some rules for 2-ary and 3-ary, then adding recursion, I will add unique rules for each array length, and if they get large enough I will switch to recursion, which is probably not as powerful. \(SS\{a, b\} = f_b(a)$$, so $$\{n, 4\}$$ is on par with tetration

$$SS\{a, b, c\} = f_{\varphi(c, b)}(a)$$ using fundamental sequences for the Veblen function, so $$\{n, 0, 1\}$$ is on par with Kirby-Paris hydras

$$SS\… Epicdoge121 19 days ago 1 ## Judy's Number Judy's Number is a HUGE number written in Judy's function that is based off of Knuth's up-arrow notation, and is similar to Graham's function. It's based off of some random person named Judy and her favorite number 7. You start with 7↑↑↑↑↑↑↑7 ( with 7 arrows) that is J1 in Judy function 7{7}7 in BEAF or a Trisept J2 = 7↑↑…↑↑7 (with J1 arrows) or 7{7{7}7}7 in BEAF J(n) = 7↑↑…↑↑7 (with J(n-1 ) arrows) Judy's Number = J7 = 7↑↑…↑↑7 (with J6 arrows) 78 in BEAF Judy's Number is \(f_{\omega+1}(7)$$ in the Fast-growing hierarchy

Little Graham is bigger than it because it starts with 2↑↑↑↑↑↑↑↑↑↑↑↑3

J(n) grows faster than G(n) because J(n) starts with 7↑↑↑↑↑↑↑7 while G(n) start with 3↑↑↑↑3

Clyde1000^1000 19 days ago
1

## f notation

Arcioko 19 days ago
0

## the augmentation function

came up with on a night just doing weird things casually:

So first of all, this function doesnt use numerical, it takes operations itselves:

Basically: , passing it through the augmentation function would increase every hyperoperator, so A_1() = 5^^^^^^5. It also works with operations with more hyperoperators, lets say (8^8)^^^3, A_1() = (8^^8)^^^^3, and there is more, A_2(5^^^^^5) = 5^^^^^^^5, as you notice, A_n(m^m) = m^^^ . . (n number of arrows) . . ^^^m, as i said earlier, It also works with operations with more hyperoperators, A_10(8^7^6) = 8^^^^^^^^^^^7^^^^^^^^^^^6.

Binary198 20 days ago
1

## Si function 四

My second notation thingy (not sure what it classifies as). Hopefully it will be well-defined this time.

Parts of this were copied from Kanrokoti's non-restricted 三 function because I'm sorta lazy, go check out the original here.

• 1 Notation
• 2 Relation
• 3 Cofinality
• 4 Fundamental sequences
• 5 Fast-growing hierarchy
• 6 Fast-growing function
• 7 Normal form

Here, I will define the character string which I will use for defining the notation. I define two sets $$T$$ and $$PT$$ of formal strings consisting of $$0$$, $$\textrm{四}$$, $$+$$, $$($$, $$)$$ and commas in the following recursive way:

• $$0 \in T$$.
• For any $$(a, b) \in T^2$$, $$\textrm{四}_a(b) \in PT \cap T$$.
• For any $$(a_1, a_2, ..., a_m) \in PT^m$$ such that $$2 \leq m < \infty$$, …

Tetramur 21 days ago
0

## USGCS(3)

For USGCS(3), ((0,0,0),(0,0,0),(1,1,1)) should work, I think.

I choose 2

Williy luk 22 days ago
1

## My notation

10^5=[10,5]

10^x=[10,x]

10^10^100=[10[10,100]]

10^10^x=[10[10,x]]

10^^100(10{2}100)=[10,2,100]

10{x}y=[10,x,y]

[10,3[10,2[10,2[10,172]]]]=10^^^10^^10^^10^172

Nayuta Ito 22 days ago
1

## PEGG entered The Eon Of Undefined

Can anyone delete this article? I found that Q is complete before September 19th.

Binary198 22 days ago
0

## LaTeX testing blog: just scroll away

I will just use this blog for testing random LaTeX stuff. This blog will likely grow in length over time xD

$$\textrm{四}$$

GerfloJoroZ 22 days ago
0

## Explaining Numbers That I Named xd

I ONLY GONNA EXPLAIN THE NUMBERS FROM HERE https://googology.wikia.org/wiki/User_blog:GerfloJoroZ/Every_Number_that_i_(tried_to)_name.

WARNING: idk

• 1 Sammy
• 2 Undertale C1
• 3 Undertale C2
• 4 Undertale C3
• 5 Undertale C4
• 6 Undertale C5
• 7 Undertale C6
• 8 Undertale C7
• 9 Undertale C8
• 10 Undertale C9
• 11 Undertale C10
• 12 Grabielplex
• 13 Gaxalogue
• 14 Supalogue
• 15 Versalogue
• 16 Multalogue
• 17 Multataxis
• 18 Borcal
• 19 Megaham
• 20 Megorcal
• 21 Multaham
• 22 Multorcal
• 23 Loud, Bi-Loud and Tri-Loud Multorcal
• 24 Multintar
• 25 Hekto-Multintar
• 26 Multo-Multintar
• 27 Randomgirlwanttohavesexwithme-illion
• 28 Ultrafour
• 29 Ultrafourplex
• 30 Ultrasam
• 31 Ultrasamplex
• 32 Gerflo's Number
• 33 Gerflillion
• 34 Ultragerflo
• 35 Ultragerfloplex

Biggest negative undescribable number.

The number of original Undertale AU's (13 Jun 2021)

The number of Undertale AU's …

Tetramur 23 days ago
0

## Situation around BIGG

For years it was considered as one of the largest numbers ever existed. I even wrote somethere that it was around {L,X,2}200,200 in BEAF. Now that BEAF at this level is possibly well-definable (but it is clearly ill-defined for now) and first rule is to create 200 pairs of brackets by following recursive rule: {L,X,2}[1] = L and {L,X,2}[a+1] = {L,{L,X,2}[a]} (it requires expansion of L into 200-part &-chain, though) but for BIGG even first rule is not clear. So my previous statement that n? is about f_{ψ_{χ_0(0)}(ψ_{χ_1(0)}(0))}(n) in Rathjen's "small psi" function is meaningless.

12AbBa 24 days ago
0

## n(k) calculator

I created a calculator to calculate n(k) here a while back:

It uses the tree building method to calculate n(k), aka calculating all strings of a certain length that work, which I think is the fastest method to calculate n(k). It correctly calculates n(1) and n(2). Recently I optimized the code which cuts the run time quite a lot. However, I was still unable to perform the calculation process for n(3) beyond length 9 due to KA's infinite loop protection, which I was unable to get rid of.

Any ideas on how to improve the program?

Also, if there was no infinite loop protection, will the number of valid sequences for a certain length grow googologically large, or will t…

GerfloJoroZ 24 days ago
0

## WORLD WIDE -ILLIONS

• Micresillion: 10(3×(103×(1)+3)+3
• Shelillion: 10(3×(103×(2)+3)+3
• Samonillion: 10(3×(103×(3)+3)+3
• Marhsisillion, Vanutanillion: 10(3×(103×(4)+3)+3
• Palaunillion: 10(3×(103×(5)+3)+3
• Occiharillion: 10(3×(103×(10)+3)+3
• Soloislillion: 10(3×(103×(20)+3)+3
• Vaticitillion: 10(3×(103×(27)+3)+3
• Sapiminillion: 10(3×(103×(31)+3)+3
• Montserrillion: 10(3×(103×(33)+3)+3
• Macanosillion: 10(3×(103×(63)+3)+3
• Malvisnillion: 10(3×(103×(1367)+3)+3
• Barthillion: 10(3×(103×(1548)+3)+3
• Sacrynievillion: 10(3×(103×(1762)+3)+3
• Neerlandillion: 10(3×(103×(1998)+3)+3
• Butanillion: 10(3×(103×(2559)+3)+3
• Isvirbrillion: 10(3×(103×(2642)+3)+3
• Antibarbillion: 10(3×(103×(2815)+3)+3
• Turcaillion: 10(3×(103×(2824)+3)+3
• Sanvylagrillion: 10(3×(103×(3158)+3)+3
• Dominicillion: 10(3×(103×(3197)…

DeltaRXT 24 days ago
0

## Well... Eightanoards flushing

So Sussylion is so big number bcs is sus :flushing:

One sussyllion ecuals 10^(3*(101001↑↑↑110^1010^101^1010011)+3) bcs from binary code 1010011 1010101 1010011 - SUS

And next is SUSSYBAKALLION

So... Im dumb.... Well Sussybakallion - 10^(10^(3*(101001↑↑↑110^1010^101^1010011)+3)) :flushing:

Next number will be... VERY BIG... Big... Bi... Oh no... Now its your chance to ba a flushing: A...

Gerflojoroznaon - Virusgayniard^Sussybakallion^(Undertale C10*Underfell C8)

And

Gerflojoroznaod - Gerflojoroznaon↑↑↑↑↑↑↑Gerflojoroznaon*Sussybakallion :flushing:

Well... It was my stoopid blogpost

Now i will eat chocolate bcs im not playing in gayshit infart :flushing:

KarmaAnecdotes 25 days ago
1

## This Ordinal thing I made

I made this ordinal collapsing function , where y is a positive integer.

GerfloJoroZ 25 days ago
0

## LOUD

LOUD > FORCE > GRAHAM

• 1 HOWWWWWWWWWWWWWWWWWWWWWWWW
• 1.1 Forcal
• 1.2 Borcal
• 1.3 Decorcal
• 1.4 Megorcal
• 1.5 Multorcal

• Forcal: G1(1000000)
• Force Forcal: G2(1000000)
• Bi-Force Forcal: G3(1000000)
• Tri-Force Forcal: G4(1000000)
• Loud Forcal: G1(1)
• Bi-Loud Forcal: GG1(1)(1)
• Tri-Loud Forcal: GGG1(1)(1)(1)

• Borcal: G2(1000000)
• Force Borcal: G3(1000000)
• Bi-Force Borcal: G4(1000000)
• Tri-Force Borcal: G5(1000000)
• Loud Borcal: G2(2)
• Bi-Loud Borcal: GG2(2)(2)
• Tri-Loud Borcal: GGG2(2)(2)(2)

• Decorcal: G10(1000000)
• Force Decorcal: G11(1000000)
• Bi-Force Decorcal: G12(1000000)
• Tri-Force Decorcal: G13(1000000)
• Loud Decorcal: G10(10)
• Bi-Loud Decorcal: GG10(10)(10)
• Tri-Loud Decorcal: GGG10(10)(10)(10)

• Megorcal: G1000000(1000000)
• Force Megorcal: G1000001(1000000)
• Bi-Force Megorcal: G1000002(1000000)
• Tri-…

GerfloJoroZ 25 days ago
0

## Every Number that i (tried to) name.

I GONNA DIE. (Salad Numbers Almost Everywhere)

• Sammy: S(-1), The biggest finite undefinable negative number
• Undertale C1: 1372
• Undertale C2: 1882384
• Undertale C3: 3543369523456
• Undertale C4: 35433695234562
• Undertale C5: 35433695234564
• Undertale C6: 354336952345616
• Undertale C7: 354336952345665536
• Undertale C8: 3543369523456265536
• Undertale C9: 35433695234562265536
• Undertale C10: 354336952345622265536
• Gabrielplex: 10Gabriel's Number
• Gaxalogue: 10↑↑(103×1033)
• Supalogue: 10↑↑(103×1036)
• Versalogue: 10↑↑(103×1039)
• Multalogue: 10↑↑(103×1042)
• Multataxis: 10↑↑↑(103×1042)
• Baham: G2(64)
• Borcal: G2(1000000)
• Decaham: G10(64)
• Decorcal: G10(1000000)
• Megaham: G1000000(64)
• Megorcal: G1000000(1000000)
• Multaham: G(103×1042)(64)
• Multorcal: G(103×1042)(1000000)
• Loud Multorcal: G(103×1…

Tsskyx 26 days ago
1

## A system utilizing uncountable ordinals better than OCFs?

Hi, first post. So this idea came to me as I was messing around with ordinal notation, being inspired by Stephen Brooks's transfinite number line, which was recently updated to include ordinals up to BHO. The concept is to ditch the multiple arguments of the extended Veblen function and instead use ). Again, if you have some sources for reading, do share them.

Kanrokoti 26 days ago
1

## Weak K-psi function

• 1 Overview
• 2 Weak K-psi function
• 2.1 Notation
• 2.2 Ordering
• 2.3 Cofinality
• 2.4 Search function
• 2.5 Fundamental sequence
• 2.6 FGH
• 2.7 Standard form
• 2.8 Naming

We define Weak K-psi function. This notation extends Ordinal Notation Associated to Extended Buchholz's OCF.

Special thanks: p進大好きbot

Let $$T$$ and $$PT$$ are sets of formal strings consisting of $$0$$, $$+$$, $$\psi$$, $$($$, and $$)$$, which are simultaneously defined in the following recursive way:

1. $$0 \in T$$.
2. For any $$(a,b) \in PT \times (T \setminus \{0\})$$, $$a+b \in T$$.
3. For any $$(a,b) \in T^2$$, $$\psi_a(b) \in PT \cap T$$.

$$0$$ is abbrevi…

GerfloJoroZ 26 days ago
1

## Sam's Number?

Binary198 27 days ago
2

## Formalized and fixed plus notation

Plus notation is a notation with correspondence to ordinals which I developed. In this blog post, I will be formalising and fixing it. Note that here, I only use ellipses in a case such as $$\lbrace a_0, a_1, ..., a_n \rbrace$$ to show a set with $$n+1$$ elements. I say this to avoid ambiguity. So, without further ado, let's jump right into it!

We work in a set $$\Sigma$$ of symbols: $$+$$, $$0$$, $$($$, $$)$$, $$*$$, $$\textrm{^}$$, $$\langle$$, $$\rangle$$ and $$/$$. The set $$\Sigma*$$ is the set of formal strings consisting of symbols from $$\Sigma$$.

We define a set of valid formal strings $$OT$$ (i.e. a subset of $$\Sigma*$$) like so:

For a formal string $$S \in \Sigma*$$, $$S \in OT$$ if it satisfies all these requirements:

• It is non-em…

Literally Legendary 28 days ago
1

## blog post

I define (n) as the largest number which is the output of a Turing machine which has n non-limit non-halting states, except when the machine tries to move to the left of its starting point, it halts.

Binary198 28 days ago
1

## Fast-growing combinatorial function I made because I was bored

This is a fast-growing combinatorial function.

• 1 Rules
• 2 BCF(0)
• 3 BCF(1)
• 4 BCF(2)
• 5 BCF(3)
• 6 BCF(4)
• 7 BCF(5)
• 8 BCF(6)
• 9 List of early values and comparison to $$\Sigma(n)$$
• 9.1 BCF
• 9.1.1 vs.
• 9.2 BB(n)

Start with S0 = 0. Then increase it by the following rules:

Sx+1 = 0(S0S1S2...Sx)

For example, S4 = 0(S0S1S2S3) = 0(00(0)0(00(0))). You calculate BCF(n) like so:

Take Sn and reduce it by the following rules:

0A => A

00A => 0

000A => 0A(0A)

Take the amount of steps that reducing until you reach 0 took, and label it as BCF0(n). Then to compute BCFx+1(n), take a string of BCFx(n) consecutive zeroes and count how many steps it took to reach 0 = 3 * BCFx(n). BCF-(n) = BCFBCF0(n)(n) = BCF0(n) * 3^BCF0(n). BCF(n) = BCF0(n) * 3^(BCF0(n) * 3^(..BCF0(n) * 3^BCF0(n))) with BCF0(n) ste…

Binary198 29 days ago
0

## My early notations

Pretend this blog post doesn't exist. Just keep on scrolling.

Binary198 29 days ago
0

## Creating a pointless fast-growing function via recursion on the TREE and weak tree functions

Pretend this blog post doesn't exist.

Binary198 29 days ago
0

## "Mini OCF - Pt. 2"

So, normal OCFs collapse down uncountables to return countables, and this "mini OCF" collapses countables to return finite numbers. Therefore, I guess it's more of a googological notation then ordinal notation or collapsing function. I have created 3 previous mini OCFs, and this article will be about my newest, most powerful and most extensible. This is the first part of TMOAN. For the full version and any future updates, see here.

Thanks: User:Username5243, User:P進大好きbot, User:Denis Maksudov, User:Kanrokoti

• 1 Pre-Alpha Stage
• 1.1 I: Stage 0
• 1.2 II: Stage 1
• 1.3 III: Linear

Pre-alpha stage TMOAN covers from basic to nested arrays.

Stage 0 is the lowest level of TMOAN. It involves no infinite ordinals in the expression. Basic TMOAN consists of the follo…

KarmaAnecdotes 23 September
1

## Testing Blog

I Just Made This Blog So That I Could Experiment with Latex and make functions with it the stuff, just scroll away.

generalizes w+n, makes 2w and adds it to the list.

Arcioko 23 September
1

## array notation idea thing

so basically i came up with this like yesterday, still have no idea how tf i came up with it (sorry if some things are horribly badly explained)

{a} =

Anything(~^b)a = Anything(~^b-1)Anything(~^b-1)Anything . . . Anything(~^b-1)Anything(~^b-1)Anything

thats all i guess

MultiSoul 23 September
0

## More illions\

Define t$A,B = Bth tier A illion 1. million = t$1,1
2. billion = t$1,2 3. trillion = t$1,3
4. quadrillion = t$1,4 5. quintillion = t$1,5
6. sextillion = t$1,6 7. septillion = t$1,7
8. octillion = t$1,8 9. nonillion = t$1,9
MultiSoul 23 September
0

## The long list of illion numbers

1. Meillion - 10^0.00219726562
2. Dillion - 10^0.00439453125
3. Tillion - 10^0.0087890625
4. Teillion - 10^0.017578125
5. Pillion - 10^0.0234375
6. Hxillion - 10^0.046875
7. Hpillion - 10^0.09375
8. Oillion - 10^0.1875
9. Eillion - 10^0.375
10. Dillion - 10^0.75
11. Hillion - 10^1.5
12. Killion - 10^3
13. Million - 10^6
14. Billion - 10^9
15. Trillion - 10^12
17. Quintillion - 10^18
18. Sextillion - 10^21
19. Septillion - 10^24
20. Octillion - 10^27
21. Nonillion - 10^30
22. Decillion - 10^33
23. Undecillion - 10^36
24. Duodecillion - 10^39
25. Tredecillion - 10^42
26. Quattuordecillion - 10^45
27. Quindecillion - 10^48
28. Sexdecillion - 10^51
29. Septemdecillion - 10^54
30. Octodecillion - 10^57
31. Novemdecillion - 10^60
32. Vigintillion - 10^63
33. Trigintillion - 10^93
35. Quinquagintillion - 10^153
36. Sexagintillion - 10^183
37. Septuagintillion - 10^…

Literally Legendary 22 September
1

## SMB1-plex

One of the previous googolisms I invented to estimate how many one-screen-big levels possible there would be in Super Mario Bros. took almost everything into account... except the sprite limit. This blog post will be doing that.

The original Super Mario Bros. could only have 5 entities loaded at a time. The new expression is (83)^(13*16)*3*5*2^5*(13*16*36)^5 and I define SMB1-plex as (83)^(208)*(7488)^5*480 which is the most simplified form of the expression.

Binary198 22 September
0

## "Mini OCF - Pt. 1"

So, normal OCFs collapse down uncountables to return countables, and this "mini OCF" collapses countables to return finite numbers. Therefore, I guess it's more of a googological notation then ordinal notation or collapsing function. I have created 4 mini OCFs in the past, and this article will be about them.

• 1 OAN (Ordinal Array Notation)
• 2 NOBAN 1.0 (New Ordinal-Based Array Notation)
• 3 NOBAN 2.0
• 4 TMOAN (Third, Modified Array Notation)

OAN is a fairly weak notation, but it's a start. Here is a definition:

• $$[\alpha] = \alpha$$
• $$[\alpha, n] = \alpha n$$
• $$[\alpha, \omega] = \alpha^2 + 1$$
• $$[\alpha, \omega^n] = \alpha^{n+1} + 1$$
• $$[\alpha, \omega^\omega] = \alpha^{2n + 1} + 1$$

You could go past to ordinals like $$\varepsilon_0$$, and I wrote down som…

Binary198 22 September
1

## Super fast-growing hierarchy (powerful recursion-based function)

Created this random family of super fast-growing functions defined via extreme recursion on each other that I made a while ago. I don't know whether they are uncomputable or not, but they seem to have the fastest growth rate out of all my functions, excluding the hyper hierarchy. Don't confuse anything below with transfinite ordinals, OCFs, etc.

• $$\psi_1(n) = n^n$$
• $$\psi_{\alpha+1}(n) = \psi_\alpha^n(n)$$
• $$\phi_m(n) = \psi_{m^n}^{mn}(m+n)$$
• $$\varepsilon_1(n) = \psi_n(\phi_n(n)$$
• $$\varepsilon_{\alpha+1}(n) = \varepsilon_{\alpha}^{\varepsilon_\alpha^{\varepsilon_\alpha((\alpha + 1) \uparrow \uparrow n)}((\alpha+1)^n)}((\alpha+1)n)$$
• $$\Omega_0(n) = \psi_n(\varepsilon_n(n))$$
• \(\Omega_{\alpha+1}(n) = \Omega_\alpha^{\Omega_\alpha(n)}(n \uparrow \…