one more post
I'm assume you've already seen 'No Subject'
Just one more question, How big is Quadrit(3) to Quadrit(5)?
Comparing 3-Row Bashicu Matrix System with Taranovsky's Ordinal Notation
This shows values for Taranovsky's Ordinal Notation compared with 3-Row Bashicu Matrix System. It's still a work in progress. It'll be done soon.
Extended Arrow Notation
This is a fixed version of my new notation and there will be analysis on it.
The notation, as I called, "extended arrow notation" is a bit silly considering that there are no arrows in the notation...
Lets define it!
a, b and c are natural numbers.
@ represents inactive "blocks" or nothing.
A1 and A2 stands for different arrays or nuffin.
As stands for smallest array that is able to satisfy the rule application conditions, and inversely AL stands for largest array tiatstrac.
stands for an array with only ones.
Any other letter stands for a integer number larger than 1.
NOTE: THIS ONLY WILL COVER ARRAY NOTATION.
a@1=a
a@[A1,]b=a@[A1]b
a[1]b=a^b
a@[1](b+1)=a@a@[1]b
a@[c+1,A1]b=a@[c,A1][c,A1][c,A1]... b times... [c,A1][c,A1][c,A1]b
a@[As,1,c+1,AL]b=a@[As,a@…
No Subject
Base-2 Function: (Binary Function)
Bit(0) = 0
Bit(1) = 1
Bit(2) = 10
Bit(3) = 11
Bit(4) = 100
Bit(5) = 101
Bit(6) = 110
Bit(7) = 111
Bit(8) = 1,000
Bit(9) = 1,001
Bit(10) = 1,010
Trinary Function: Not to be confused with Trinary Base-3
Trit(n) = Bit(Bit(Bit(...Bit(Bit(n))))... (n's Bit)
Trit(0) = 0
Trit(1) = 1
Trit(2) = 1,010
Trit(3) = 1,111,110,011
Trit(4) = 1,010,110,101,111,001,011,010,001,101,011,011,001,100,110,110,011,001,100,101,110,100,100
Trit(5) = 1.001100101... × 10^219
Trit(6) = 1.011010110... × 10^727
Trit(7) = 1.000010100... × 10^2,415
Trit(8) = 1.101101000... × 10^12,049
Trit(9) = 1.101010000... × 10^40,023
Trit(10) = 1.010001010... × 10^132,951
Quadrinary Function:
Quadrit(n) = Trit(Trit(Trit(...Trit(Trit(n))))... (n's Trit)
Quadrit(0) = 0
Quadrit(1) …
Googolnovemicosiplex
The googolnovemicosiplex (also known as googolennaisokaplex or googolplexplexplexplexplexplexplexplexplexplexplexplexplexplexplexplexplexplexplexplexplexplexplexplexplexplexplexplexplex) is equal to E100#30 = EEEEEEEEEEEEEEEEEEEEEEEEEEEEEE100 in Hyper-E notation.
It is also equal to , or a 1 followed by a Googoloctoicosiplex
zeroes. It is 10^^30^100+1 ( exact ) digits long.
Strength of TON considering non-Gandy ordinals
In this blog I want to present a hypothesis about the strength of TON. Taranovsky's main hypothesis assumed that the strength of the TON was Z2+PD. However, new analysis of Hypcos shows that the 2nd system of the main notation is weaker than previously thought because the rich structure of non-Gandy ordinals was underestimated. However, perhaps this should not affect the full strength of the TON, only weakening the systems by n-1.
According to Hypcos analysis:
- Ω0 = C(Ω1,0) - ε0 = PTO of PA
- C(εΩ1+1,0) = C(C(Ω2,Ω1),0) = BHO = PTO of KPω or Δ11-CA+BI
- ... ≤ C(C(C(...Ωn×2...,0),0),0) n times - ordinals with order type of a well-ordering Δ10 subset (computable) of ω or recursive ordinals
- Ω1 = C(Ω2,0) = ω1CK = 1st ordinal with order type of a well-ord…
A Semi-Unoriginal Tier-4 -illion Number Extension
One day I was just finding ideas for random -illions when i had the idea to turn the hierarchy chain from All Dimensions Wiki into -illion numbers.
So what did i do? Self explanatory ig.
Anyway here’s the list:
Kalillion - 10^(3x10^(3x10^3000)+3)
Mejillion - 10^(3x10^(3x10^3000000)+3)
Gijillion - 10^(3x10^(3x10^3000000000)+3)
Astillion - 10^(3x10^(3x10^(3x10^12))+3)
Lunillion - 10^(3x10^(3x10^(3x10^15))+3)
Fermillion - 10^(3x10^(3x10^(3x10^18))+3)
Jovillion - 10^(3x10^(3x10^(3x10^21))+3)
Solillion - 10^(3x10^(3x10^(3x10^24))+3)
Betillion - 10^(3x10^(3x10^(3x10^27))+3)
Glocillion - 10^(3x10^(3x10^(3x10^30))+3)
Gaxillion - 10^(3x10^(3x10^(3x10^33))+3)
Supillion - 10^(3x10^(3x10^(3x10^36))+3)
Versillion - 10^(3x10^(3x10^(3x10^39))+3)
Multillion - 10^(3x10^(3…
Holer's Number
Holer's Number is a huge number built on many fast-growing functions defined by Holersh.
There are two Holer's Number: the Small Holer's Number (𝔥) and the Large Holer's Number (ℌ).
The Basic Holer Function is defined as folows:
(WIP)
Isra miraj and the time of 50
This article was copied from a powerpoint created by Fahmi Basya, one of the lecturers at the Tarbiyah Faculty of Islamic State University Syarif Hidayatullah, everything in this article was not created by me, only translated into English, I got the presentation from the slideplayer website
This is not officially sourced from a real religion, just people who want to play with religion, but the principles here are not found in religious literature
There are several series of presentations in this article called flying books, I marked them in the form of headings in my blog post, Later I will provide the most complete sources for all presentations
I DID NOT MAKE THIS PRESENTATION
For those who feel offended by matters of religion and belief, plea…
X ordinal
1 separator "ω^2"
X=ω
Χ/1=ω+1
Χ/2=ω+2
X/X=ω*2
X/X/1=ω*2+1
X/X/X=ω*3
…
2 dividers "ε0"
X//1=X/X/…=ω^2
(X//1)/1=ω^2+1
(X//1)/2=ω^2+2
(X//1)/X=ω^2+ω
(X//1)/(X//1)=ω^2*2
(X//1)/(X//1)/(X//1)=ω^2*3
X//2=ω^3
(X//2)/(X//2)=ω^3*2
X//3=ω^4
X//X=ω^ω
X//X/1=ω^ω+1
X//X/X=ω^ω*2
X//X//1=ω^ω^2
X//X//X=ω^ω^ω
X//X//X//X=ω^ω^ω^ω
…
3 dividers "ζ0"
X///1=ε0
(X///1)/1=ε0+1
(X///1)/(X///1)=ε0*2
(X///1)/(X///1)/(X///1)=ε0*3
(X///1)//1=ε0*ω
((X///1)//1)/(X///1)=ε0*ω+1
((X///1)//1)/((X///1)//1)=ε0*ω*2
(X///1)//2=ε0*ω^2
(X///1)//3=ε0*ω^3
(X///1)//X=ε0*ω^ω
(X///1)//(X//X)=ε0*ω^ω^ω
(X///1)//(X//X//X)=ε0*ω^ω^ω^ω
(X///1)//(X///1)=ε0^2
(X///1)//(X///1)/1=ε0^2*ω
(X///1)//(X///1)/2=ε0^2*ω^2
(X///1)//(X///1)/(X///1)=ε0^3
(X///1)//(X///1)/(X///1)/(X///1)=ε0^4
(X///1)//(X///1)//1=ε0^ω
(X///1)//(X///1)//2=ε0^ω^2
(X///1)…
φ(ω,0) in ES5
EDIT: THIS IS ABSOLUTELY NOT ES5, it just has more ES5 than ES6
lprss([0,n],n) is about \(f_{\varphi_\omega(0)}(n)\) in the fast growing hierarchy.
493 characters, entirely compatible in the ES5 Enviroment, meaning you are always able to run it on any javascript console!
\(()[n]=n+1\)
\((a_1,a_2,a_3,...a_{k-2},a_{k-1},0)[n]=(a_1,a_2,a_3,...a_{k-2},a_{k-1})^n[n]\)
\((a_1,a_2,a_3,...a_{k-2},a_{k-1},a_{k})[n]=\text{expand}(a_1,a_2,a_3,...a_{k-2},a_{k-1},a_{k})[n]\)
\(m=a[\text{len}(a)], r = \min\{l\in a|l< m\}, \delta=m-r-1\)
\(G=(a_1,a_2,a_3,\ldots,a_{r-2},a_{r-1}),B = (a_r,a_{r+1},a_{r+2},\ldots,a_{k-2},a_{k-1})\)
\(\text{expand}(a,p)=G\cup \bigcup_{t=0,t< p}B+t\delta\) where \(A+b\) means all of \(A\)'s entries added by \(b\).
This is an implementat…
Creating my YouTube Channel
Hey guys,I have created my YouTube Channel,Im still going to upload videos soon,But im already preparing it,Anyways,If you want to find it,Here is the link:
https://www.youtube.com/@ValentinoScientist
The whole "k" massif in 1 post
part l
k{1}=10
k{2}=10^10
k{3}=10^10^10
k{1,2}=k{10}
k{1,,2,2}=k{1(2*)2,,1,2}
k{1,,1,3}=k{1,,10,2}
k{1,,1,1,2}=k{1,,1,10}
k{1,,1(2)2}=k{1,,1,1,…,1,2}
k{1,,1,,2}=k{1,,1(2*)2}
k{2,,1,,2}=k{1,,1(2**)2}
k{1,,2,,2}=k{1(2*)2,,1,,2}
k{1,,1,,3}=k{1,,1(2*)2,,2}
k{1,,1,,1,,2}=k{1,,1,,10}
…
part Vl (α version)
k{1(2,,)2}=k{1,,1,,…10}
k{1(2,,)3}=k{1,,1,,…,,1,,2(2,,)2}
k{1(3,,)2}=k{1(2,,)1(2,,)…10}
k{1(4,,)2}=k{1(3,,)1(3,,)…10}
k{1(1,2,,)2}
k{1(1(1,2,,)2,,)2}
k{1(2*,,)2}=k{1(1(…(1(1(1,2,,)2,,)2,,)…)2,,)2}
k{1(2**,,)2}=k{1(1(…(1(1(1,2*,,)2*,,)2*,,)…)2*,,)2}
k{1(2***,,)2}=k{1(1(…(1(1(1,2**,,)2**,,)2**,,)…)2**,,)2}
k{1,,,2}=k{1(2*,,)2}
k{2,,,2}=k{1(2**,,)2}
k{1,,,3}=k{1(2*,,)2,,,2}
k{1,,,1,,,2}=k{1,,,1(2*,,)2}
k{1(2,,,)2}=k{1,,,1,,,…10}
k{1(1,2,,,)2}
k{1(1(1,2,,,)2,,,)2}
k{1,,,,2}=k{1(…
HNS extension 1
(I analyzed it in my head and the limit ordinal is only \(\varepsilon_1\))
A Hyper N-Ary Double-Sequence is in the form \(\langle A_{1}\rangle\langle A_{2}\rangle\langle A_{3}\rangle\ldots\langle A_{k-2}\rangle\langle A_{k-1}\rangle\langle A_{k}\rangle[n]\)
If the sequence is \(\langle 0 \rangle\), then return n+1.
Otherwise, if it ends with \(\langle 0 \rangle\), then \(\langle A_{1}\rangle\langle A_{2}\rangle\langle A_{3}\rangle\ldots\langle A_{k-2}\rangle\langle A_{k-1}\rangle\langle A_{k}\rangle[n] = (\langle A_{1}\rangle\langle A_{2}\rangle\langle A_{3}\rangle\ldots\langle A_{k-2}\rangle\langle A_{k-1}\rangle)^n[n]\), where \(f^a(b)\) represents function iteration.
Otherwise, if the last sequence ends with \(\langle\ldots0\rangle\), then…
Solving the functional equation a◦f = g◦a for f and for g
Abstract
The functional equation \(a \circ f = g \circ a\) with \(f: X \to X\) and \(g: Y \to Y\) is an Abel equation when \(f = s\), the successor function. That is generalization of the iterate of \(g\).
We provide an expression for \(g\) in terms of \(a\) and \(f\). That is an "unitetate" of \(g\) when \(f = s\).
- 1 Main text
- 1.1 Lemma 1.1.
- 1.2 Definition 1.2.
- 1.3 Lemmma 1.3.
- 1.4 Lemma 1.4.
- 1.5 Proposition 1.5.
- 1.6 Corollary 1.6.
- 1.7 Example 1.7.
PDF: https://drive.google.com/file/d/1D1QFIPzsM0I8kiK8m9SLL4cFzhPlQQV6/view?usp=sharing
Old PDF: https://drive.google.com/file/d/1dgi1s4isEMUAmoQQcA22oszKpPGmc-F4/view?usp=sharing
\[ \newcommand{\U}[1]{\mathcal{U}_{#1}} \]
From here, let \(\sim_f\) denotes the equivalence kernel of a function \(f\); \(a \sim_f b :\if…
N-ary System Extension Using Broken Integers
Broken integers are numbers such that the previous number minus it is less than -1.
for example, the broken integer in [0,1,2,3,4,2,3,5,6,7] is 5, since 3-5 < -1.
Revised expansion definition:
If the last number of the sequence is 0, stop this process, otherwise:
Let p be the last number of a sequence
If there exists p-1 in the sequence, "select" everything from the last number that is p-1, delete p from the sequence, and duplicate the sequence p times.
If there doesn't exist p-1 in the sequence, and p is less than or equal to the number before it, duplicate the sequence with each duplicate's number being increased by p-1. E.g. [0,2,2] = [0,2,1,3,2,4,3,5,...]
If there doesn't exist p-1 is the sequence, and p is greater than the number before it (m…
Inferniation
- a|b=[a][a][a][a][a][a][a][a]... with b copies of [a]-s.
- a||b=a|a|a|a|a|a|a... with b copies of a-s.
- a|||b=a||a||a||a||a||a... with b copies of a-s.
- a||b=a|2b
- a|||||...b with c copies of |-s.
- So a|cb=a|c-1a|c-1a|c-1a|c-1a... with b copies of a-s.
- a||cb=a|da|da|da... with b copies of a-s, d=a|cb.
- a|||cb=a||da||da||da||da... with b copies of a-s, d=a||cb.
- a||||cb=a|||da|||da|||da|||da... with b copies of a-s, d=a|||cb.
- a||||cb=a||||da||||da||||da||||da... with b copies of a-s, d=a||||cb.
- a|ecb=a|e-1da|e-1da|e-1da... with b copies of a-s, d=a|e-1cb.
- So: a|2cb=a||cb, a|3cb=a|||cb, a||||cb=a|4cb
- a||ecb=a|eca|eca|eca|eca|eca|eca|eca... with b copies of a-s.
- a|||ecb=a||eca||eca||eca||eca||eca||eca||eca... with b copies of a-s.
- a||||ecb=a|||eca|||eca…
strong extension of the Veblen function
before ψ(Ω2)
φ(1#ω)=ψ(Ω^ω)
φ(1#φ(1#1))=ψ(Ω^ψ(Ω))
φ(1#φ(1#φ(1#1)))=ψ(Ω^ψ(Ω^ψ(Ω)))
φ(1#1,0)=ψ(Ω^Ω)
φ(1#1,1)=ψ(Ω^Ω+1)
φ(1#1,2)=ψ(Ω^Ω+2)
φ(1#1,φ(1#1,0))=ψ(Ω^Ω+ψ(Ω^Ω))
φ(1#2,0)=ψ(Ω^Ω*2)
φ(1#3,0)=ψ(Ω^Ω*3)
φ(1#φ(1#1,0),0)=ψ(Ω^Ω*ψ(Ω^Ω))
φ(1#1,0,0)=ψ(Ω^Ω^2)
φ(1#1,0,0,0)=ψ(Ω^Ω^3)
φ(2#0)=φ(1#0)=φ(0)
φ(2#1)=φ(1#1,0)
φ(2#2)=φ(1#1,0,0)
φ(2#ω)=ψ(Ω^Ω^ω)
φ(2#φ(2#1))=ψ(Ω^Ω^ψ(Ω^Ω))
φ(2#1,0)=ψ(Ω^Ω^Ω)
φ(2#1,1)=ψ(Ω^Ω^Ω+1)
φ(2#2,0)=ψ(Ω^Ω^Ω*2)
φ(2#1,0,0)=ψ(Ω^Ω^Ω^2)
φ(2#1,0,0,0)=ψ(Ω^Ω^Ω^3)
φ(3#ω)=ψ(Ω^Ω^Ω^ω)
φ(3#1,0)=ψ(Ω^Ω^Ω^Ω)
φ(4#1,0)=ψ(Ω^Ω^Ω^Ω^Ω)
before ψ(Ω2^ω)
φ(ω#0)=ψ(Ω2)
φ(ω#1)=ψ(Ω2+1)
φ(ω#2)=ψ(Ω2+2)
φ(ω#1,0)=ψ(Ω2+Ω)
φ(ω#1,0,0)=ψ(Ω2+Ω^2)
φ(ω+1#ω)=ψ(Ω2+Ω^ω)
φ(ω+1#1,0)=ψ(Ω2+Ω^Ω)
φ(ω+2#1,0)=ψ(Ω2+Ω^Ω^Ω)
φ(ω*2#0)=ψ(Ω2*2)
φ(ω*2#1,0)=ψ(Ω2*2+Ω)
φ(ω*2+1#1,0)=ψ(Ω2*2+Ω^Ω)
φ(ω*3#0)=ψ(Ω2*3)
φ(ω*4#0)=ψ(Ω2*4)
φ(ω…
BO without ordinal-like objects
This is a function which (at least when written down) does not use any ordinal-like objects. That is, it is written entirely using numbers. However, the definition does use ordinals. Assume FSes are the ones for ExBuchholz. \begin{eqnarray*} t(k,n) &=& \max\{α \mid g_α(n) = k\} \\ λ(k,n) &=& \begin{cases}n+1 & [k = 0] \\ λ(g_{t(k,n)}(n+1)-1,n+1) & [t(k,n) \in \text{Suc}] \\ λ(g_{t(k,n)[n]}(n+1),n+1) & [t(k,n) \in \text{Lim}]\end{cases} \\ N_0 &=& 2 \\ N_{n+1} = λ(N_n,2) \\ \end{eqnarray*} \(N_n\) grows at the level of Buchholz's ordinal.
Simplified JS Code for Hyper N-ary System
hns([0,1,2,3,4,5,6,7,8,9,...],n) \(\leq f_{\varepsilon_0}(n)\)
362 characters, originally used to be 1021 characters
Introduction of post-EBO
Japanese version
- 1 Overview
- 2 But why 3-var ?
- 3 Their difference in expansion
- 4 Correspondences to cardinals
- 5 Cofinality of terms
If you view this from a smartphone, enabling desktop mode of your browser will allows it to display formulas. Also, in case you find ads are annoying, you can turn them off from personal setting, which is available after logging in.
I introduce the world of post-EBO. In this blog post, I use OFP 3-var ψ and 3-var ψ as post-EBO notations. OFP 3-var ψ goes beyond EBO by extending the ON of EBOCF to 3-var simply. 3-var ψ goes beyond EBO by extending the ON of EBOCF to 3-var naturally.
Therefore, I assume the reader has sufficient understanding to the ON of EBOCF hereafter.
First of all, why do both notation form 3-var ? - That'…
SVO, LVO and Beyond
Veblen Notation that builds a string of ordinals are simple and elegant. On this blog, I would like to explore the limit it can go, in layman's language.
- 1 Basic Concept
- 2 Small Veblen Ordinal (SVO) and Large Veblen Ordinal (LVO)
- 3 Beyond LVO
- 4 Examples
- 5 Weird Notes
First, let's start with φ(α,0), where α denotes any ordinal. Then we have
φ(α) = ωα
and φ(1,0) = ε0 = ωωω•••ωω = φ(φ(φ ... (φ(φ(0))) ... )) (with ω copies of φ's)
and φ(1,α) = εα
Next we have
and φ(2,0) = ζ0 = εε••εε0 = φ(1,φ(1,φ( ... 1,φ(1,φ(1,0)) ... ))) (with ω copies of φ's)
and φ(2,α) = ζα
and so on.
We define φ(α@ω) = φ(α,0,0, ... ,0,0) (with ω copies of 0's), so
SVO = φ(1@ω)
= {φ(1), φ(1,0), φ(1,0,0), φ(1,0,0,0), ...}
= {ω, ε0 , Γ0 , Ackermann ordinal, ...}
Next we have
LVO = φ(1@@ω)
= φ(1@φ(1@φ( ...…
k array part of Vll and Vlll
part Vll
k{1(2*(1))2}=k{1(2*)2}
k{1(2*(2))2}=k{1(2**)2}
k{1(2*(3))2}=k{1(2***)2}
k{1(2*(1,2))2}
k{1(2*(2(2)2))2}
k{1(2*(2(2*(1,2))2))2}
k{1(2*(2(2*(1(2*(1,2))2))2))2}
k{1(2*(1*))2}=k{1(2*(1))2}
k{1(2*(2(1*))2}=k{1(2*(2))2}
k{1(2*(1,2(1*))2}=k{1(2*(1,2))2}
k{1(2*(1(2*(1,2(1*))2(1*))2}=k{1(2*(1(2*(1,2))2))2}
k{1(2*(2*))2}=k{1(2*(…(1(2*(1,2(1*))2*(1*))2(1*))…))2(1*))2}
k{1(2*(3*))2}=k{1(2*(…(1(2*(1,2(2*))2*(2*))2(2*))…))2(2*))2}
k{1(2*(1,2*))2}
k{1(2*(1(2*(1,2*))2*))2}
k{1(2*(1(2*(1(2*(1,2*))2*))2*))2}
k{1(2*(2**))2}=k{1(2*(…(1(2*(1,2*))2*))2*))…))2*))2}
k{1(2*(2***))2}=k{1(2*(…(1(2*(1,2**))2**))2**))…))2**))2}
k{1(2*(2****))2}=k{1(2*(…(1(2*(1,2***))2***))2***))…))2***))2}
k{1(2*(2*(1,2)))2}=k{1(2*(2**…**))2}
k{1(2*(2*(2*(1,2))))2}
k{1(2*(2*(2*(2*(1,2)))))2}
…
…
Repetition notation
(1)[0]=0
(2)[0]=0
(n)[0]=0
(n)[h]=h
(N)[]=(n)[n]
(n)[10]=(n)[n+1]
(n)[20]=(n)[10][10]
(n)[30]=(n)[20][20][20]
(n)[h+10]=(...(n)...)[h0]...h+1...[h0]
(n)[11]=(n)[n0]
(n)[21]=(n)[n+10]
(n)[12]=(n)[n1]
etc...
(n)[]=(n)[nn]
(n)[]=(n)[nn]
etc...
1018-101001100101010101011010100110101010101010100100000
Foohol Sesexagintacentillion Pampena's Prime Septensexagintacentillion Googoocix 10^512 Astrigol Septuagintacentillion some random plex Googocci Googoocxij Quinseptuagintacentillion Googocciv … Maximusquadrillion Pentlastillion Femtillion Googolquintigong Decans Dupixul
Ultra N-ary Sequence System
The following definition is an extension of the Hyper N-ary Sequence System, whose growth rate is \(\varepsilon_0\).
Take an array of natural numbers whose first number is 1, and assign an index to it. The full format should look like [index]. Let l(A) represent all but the last element of A. Now do the following:
If the array you took is [1], return the index+1.
If the array isn't [1], but ends with 1, then return ^i[i], where f^x represents function iteration. For example, [4] = [[4]] and [4] = [[4]]
Otherwise, do the following expansion process until it ends with a 1:
- Let p be the 2nd entry in a sequence, and v be the first line of the number p before another number that isn't p. If p=3:
- If there are only 2 numbers in the sequence, then is t…
M.O.T
3
9 27 81 243 729 2187 6561 19683 59049 177147 531441 1594323 4782969 14348907 43046721 129140163 Fznine Ternary-guppychunk Ternary-guppy 3^21
Epsilon-0 in javascript
This is a new system I call the Hyper N-ary Sequence System. Valid arrays in this system are called Hyper N-ary Sequences.
The ascending factorial notation
I can explain it more in the comments if you do not understand.
- 1 Base Pattern
- 2 The extended pattern
- 3 Fully Extended
- 4 Pure extension
- 5 Absolute extension
- 6 Inferniation
- a[!]b=a+(a+1)+(a+2)+(a+3)...a+(b-1)+(a+b)=a[![1]b
- a[![2]b=a×(a+1)×(a+2)×(a+3)...a+(b-1)×(a+b)
- a[![3]b=a↑(a+1)↑(a+2)↑(a+3)...a+(b-1)↑(a+b)
- a[![4]b=a↑b(a+1)↑b(a+2)↑b(a+3)...a+(b-1)↑b(a+b)
- a[![5]b=a{b}b(a+1){b}b(a+2){b}b(a+3)...a+(b-1){b}b(a+b)
- a[![6]b=a{b}a[![5]b(a+1){b}a[![5]ba+2){b}a[![5]ba+3)...a+(b-1){b}a[![5]b(a+b)
- a[![6]b=a{b}a[![6]b(a+1){b}a[![6]ba+2){b}a[![6]ba+3)...a+(b-1){b}a[![6]b(a+b)
- a[![b]c=a{c}a[![b-1]b(a+1){c}a[![b-1]ba+2){c}a[![6]ba+3)...a+(c-1){c}a[![b-1]b(a+c) if b>6
- a[![b[d]c=a{c}a[![b-1]b(a+1){c}a[![b-1]ba+2){c}a[![6]ba+3)...a+(c-1){c}a[![b-1]b(a+c){c}a[![b-1]b(a+1){c}a[![…
Warlter's Square Brackets Notation
Warlter's Square Brackets Notation is a notation defined by Warlter545.
Define recursively as below
- \([][n]=n\)
- \([a+1,_1X][n]=[a,_1X][f(n)]\)
- \([X,_10,_1b+1,_1X][n]=[X,_1n,_1b,_1X][n]\)
- \([X,_m0][n]=[X,_m][n]\)
- \([X,_1c,_{d+1},X][n]=[X,_1\underbrace{c,_dc,_d…c,_d}_n,X][n]\)
If 1 is not suitable, check 2, if 2 is not suitable, check 3, if 3 is not suitable, check 4, if 4 is not suitable, check 5
Check 3 and 5 from the left, and calculate only the first place that satisfies the rule.
Also, \(X\) are numbers greater than or equal to 0 and greater than or equal to 0.
10729-1010100
Duoquadragintaducentillion Sesquinquagintaducentillion Ogolchime Novenonagintaducentillion Trecentillion Octal-googolchime Astrapengol 10^999 Phichime Eulerchime Pichime Googolchime Octyillion Googgool 10^1200 Quadringentillion Hexadecimal-Goomol Quingentillion Ecetonding Duodeciquingentillion 10^1600 Sescentillion
an OCF
Attempt No. 3 is at least my 4th or 5th attempt to making an ordinal collapsing function and is my 3rd attempt to making an OCF more powerful than Rathjen's Psi (hence the name Attempt No. 3)
Let L be equal to c in the base booster system at nlevels=2
\(\{L\}\cup(L\cap(enum[\{\alpha|\alpha\in S\land\forall\beta\in\alpha\exists\gamma\in\alpha\cap S\land\beta\in\gamma\}](\delta)\bigcup_{\epsilon\in\delta}St_{0} ^{\epsilon}(S)))\subseteq St_{0} ^{\delta}(S)\)
\(\beta\not\in L\land\beta,\gamma\in St_{\alpha} ^{\delta}(S)\cup sup(St_{\alpha} ^{\delta}(S)\cap L)\rightarrow\beta+\gamma,\omega^{L+\gamma}\in St_{\alpha} ^{\delta}(S)\)
\(\epsilon\in S \land(\forall\gamma\in\delta(\forall(\beta\in\alpha\cap(St_{\alpha} ^{\gamma}(S)\cup\epsilon)\forall(T\…
Hash notation
A#b=a^^...a... a^^...a... b layers ...a...^^a ...a...^^a
a##1=a#a#a...a...a
a##b=(a##(b-1))##(b-1)
a###1=a##a##...a...a
a###b=(A###(b-1))###(b-1)
etc...
a(#^c)b=a###...c...###b
a#-[2]b=a## ...a... ##... b layers ...## ...##a
a##-[2]b=a##-[2]a#-[2]...b...a
etc...
a#-[3]b=a##...a##...b layers... ##a-[2]...##-[2]a
a#-[4]b=a##...a##...b layers... ##a-[3]...##-[3]a
etc...
a#-[#]b=a#-[a#-[...b layers...[a]...b layers..]a]a
a##-[#]1=a#-[#]...b...a
a##-[#]b=(A##-[#](b-1))##-[#](b-1)
A New Array Notation Inspired By Lawrence Hollom
Hey, I've come out of hibernation to do something googology-related! This post has a link to a document featuring the first part of my new array notation, heavily inspired by Lawrence Hollom's Hyperfactorial Array Notation, or the linear part of it at least. https://docs.google.com/document/d/1AUVnf4EnwGSAWAB5cKYiB2d3QrChL1scjlEqVouwIwM/edit?usp=sharing
Illions
1 10 100 Thousand Myriad Lakh Million Crore 100000000 Billion 10^10 10^11 10^12 10^13 10^14 Quadrillion 10^16 10^17 10^18 Guppychunk Guppy Sextillion 10^22 10^23 10^24 10^25 10^26 Octillion 10^28 10^29 10^30 10^31 10^32 Decillion 10^34 Goby Undecillion 10^37 10^38 Duodecillion 10^40 10^41 10^42 10^43 10^44 Quattuordecillion 10^46 10^47 Quindecillion 10^49 Lcillion 10^51 10^52 Tallakshana 10^54 10^55 Asougi Octodecillion 10^58 10^59 10^60 10^61 10^62 Vigintillion 10^64 10^65 Unvigintillion 10^67 Muryoutaisuu Duovigintillion Ogolspeck 10^71 Tresvigintillion 10^73 10^74 Quattuorvigintillion 10^76 10^77 Quinvigintillion Ogolchunk Ogol
The gianting number ultra forever all multiversal extended part 1
20 21 22 23 23.6666666/gaz 24 25 27 30 32 36 40 phiplex 49 50 55 60 61/gagthree 64 70 80 81 85 90 100! 110 111 120 121 123 125 128 130 140 144 150 160 162 169 170 180 190 196 200 210 216 220 222 223 225 230 240 243 250 255 256 260 270 280 289 290 299 300 310 320 321 324 325 330 333 340 341 350 360 361 370 378 380 390 399 400 405 410 420 430 435 440 441 450 456 460 470 480 484 486 490 496 499 500! 510 520 eplex 529 543 550 567 599 600 616 625 648 666 700 753 777 800 847 894 900 909 990 999 999.9999999999999999999999 1,000! 1,001 1,024 Lily Gartreys Long thousand Unexian Maha Piplex Great gross Hardy-Ramanujan Number Eyelash mite-chunk 2048 Ternary-pipsqueak Great Baker's gross Planus Heads-Pentprimol Erbe Poulter's great gross 3000 Garking…
new version of the i array
A spontaneous idea occurred to me how this array can be made more beautiful and concise.
The first 2 parts are the same as before
The estimated growth rates: ε_ω in FGH
part lll
i[n,,2]=i[n,n]
i[n,,3]=i[n,n,n]
i[n,,n]=i[n(2)n]
i[n,,n,2]=i[n(2)n(2)n]
i[n,,n,3]=i[n(2)n(2)n(2)n]
i[n,,n,n]=i[n(3)n]
i[n,,n,n,2]=i[n(3)n(3)n]
i[n,,n,n,n]=i[n(4)n]
i[n,,n(2)n]=i[n,,n,n,…n]
i[n,,n(2)n(2)n]=i[n,,n(2)n,n,…n]
i[n,,n(3)n]=i[n,,n(2)n(2)…n]
i[n,,n(4)n]=i[n,,n(3)n(3)…n]
i[n,,n,,1]=i[n,,n]
i[n,,n,,2]=i[n,,n,n]
i[n,,n,,3]=i[n,,n,n,n]
i[n,,n,,n]=i[n,,n(2)n]
i[n,,n,,n,n]=i[n,,n(3)n]
i[n,,n,,n,n,n]=i[n,,n(4)n]
i[n,,n,,n,,n]=i[n,,n,,n(2)n]
i[n,,n,,n,,n,,n]=i[n,,n,,n,,n(2)n]
i[n(2,,)n]=i[n,,n,,…n]
i[n(2,,)n,2]=i[n(2,,)i[n(2,,)n]]
i[n(2,,)n,,2]=i[n(2,,)n,n]
i[n(2,,)n,,3]=i[n(2,,)n,n,n]
i[n(2…
Analysis of the "i" array
I'm not sure that the analysis is correct, so you can point out errors if there are any
before ε0:
i[n]=n
i[n,n]≈f2(n)
i[n,n,n]≈f3(n)
i[n,n,n,n]=f4(n)
i[n(2)n]=fω(n)
i[n(2)n+1]=fω(n+1)
i[n(2)n,2]=fω(fω(n))
i[n(2)n,n]=fω+1(n)
i[n(2)n,n,n]=fω+2(n)
i[n(2)n(2)n]=ω*2
i[n(2)n(2)n,n]=ω*2+1
i[n(2)n(2)n(2)n]=ω*3
i[n(3)n]=ω^2
i[n(3)n,n]=ω^2+1
i[n(3)n(2)n]=ω^2+ω
i[n(3)n(2)n(2)n]=ω^2+ω*2
i[n(3)n(3)n]=ω^2*2
i[n(3)n(3)n(3)n]=ω^2*3
i[n(4)n]=ω^3
i[n(5)n]=ω^4
i[n,,n]=ω^ω
i[n,,n,n]=(ω^ω)+1
i[n,,n(2)n]=(ω^ω)+ω
i[n,,n(2)n(2)n]=(ω^ω)+ω*2
i[n,,n(3)n]=(ω^ω)+ω^2
i[n,,n,,n]=(ω^ω)*2
i[n,,n,,n,,n]=(ω^ω)*3
i[n(2,,)n]=ω^ω+1
i[n(3,,)n]=ω^ω+2
i[n(4,,)n]=ω^ω+3
i[n,,,n]=ω^ω*2
i[n(2,,,)n]=ω^ω*2+1
i[n,,,,n]=ω^ω*3
i[n,,,,,n]=ω^ω*4
i[n,,,…,,,n]=ω^ω^2
n(3)n#n=ω^ω^2
n(3)n,2#n=(ω^ω^2)*2
n(3)n,3#n=(ω^ω^2)*3
n(3)n,n#n=ω…
good references
Project:Policy said all main namespace article need a external source. But how to find to a good source? Today we have find some website to let you find sources fast:
- MathWorld is Eric Weisstein created a website. When find the CITE AS is a method to referencing.
- arXiv is a free distribution service and an open-access archive for nearly 2.4 million scholarly articles in the fields of physics, mathematics, computer science, quantitative biology, quantitative finance, statistics, electrical engineering and systems science, and economics. Materials on this site are not peer-reviewed by arXiv.(Website introduction) It's a good reference and have a large number of the references.