1. -Phylogenetics of Googology-
size
Preinternet Era
𝑓 𝜔 𝑛
𝑓 𝜔 2 𝑛
𝑓 𝜔 𝜔 𝑛
𝑓 𝜀 0 𝑛
𝑓 Γ 0 𝑛
𝑓 𝜓 𝜖 Ω+1 (𝑛)
𝑓 𝜓 Ω 𝜔 (𝑛)
𝑓 𝜓 𝜖 Ω 𝜔 +1 (𝑛)
𝑓 𝜓 Ω Ω (𝑛)
𝑓 𝜔 1 CK 𝑛
Uncomputable Functions
𝑛< 𝑓 3 (𝑛)
Tetration
(Rucker, 1901) 𝑛
G(𝑛)< 𝑓 𝜀 0 𝑛
Goodstein sequence
(Goodstein, 1944)
𝐴(𝑛,𝑛)< 𝑓 𝜔 (𝑛)
Ackermann Function
(Ackermann,1928)
Feferman–Schütte ordinal
Bachmann-Howard ordinal
BB 𝑛 < 𝑓 𝜔 1 CK 𝑛
Busy Beaver Function
(Tibor Radó, 1961)
time
𝑛 ↑ 𝑛 𝑛< 𝑓 𝜔 (𝑛)
Knuth’s
Arrow Notation
(Knuth, 1976)
𝑓 64 (4)< 𝑓 𝜔+1 (64)
Graham’s Number
(Gardner, 1977)
Takeuchi-Feferman-Buchholz ordinal
𝑛(𝑘)< 𝑓 𝜀 𝑛
Kirby-Paris hydra
(Kirby & Paris, 1982)
≈ 𝑓 𝜓 0 ( 𝜀 Ω 𝜔 +1 ) (𝑛)
Buchholz hydra
(Buchholz, 1987)
𝑎,𝑏,𝑐 < 𝑓 𝜔 (𝑛)
Poly-Cell Notation
(Saibian, 1991)
E-Notations
𝑆 𝑛 < 𝑓 𝜔 1 CK 𝑛
Frantic frog function
(Tibor Radó, 1961)
Netcommunity Era
1996 :Start of Large Number Site
𝑐𝑔 𝑛 = n→𝑛→⋯→𝑛 𝑛
<𝑓 𝜔 2 (𝑛)
Conway’s chained arrow notation
(Conway, 1996)
𝑛(𝑘)< 𝑓 𝜔 𝜔 (𝑘)
Block subsequence theorem
(Friedman, 1998)
𝑛↑𝑛 𝑛 𝑛< 𝑓 𝜔 3 (𝑛)
Bird’s rotating Arrow Notation
(Bird, 2000)
Circle 𝑘 < 𝑓 𝜖 0 (𝑛)
Circle Theorem
(Friedman, 2000)
Friedman’s Graph Approaches
𝐷 5 (99)≈CoC
Loader Number
(Loader, 2001)
2002 : (Start of 2ch Googology Thread in Japan)
𝑆𝑆 63 3,𝑥+1,𝑆
<𝑓 𝜔 2 +1 (63)
Fish’s Number Version 1
(Fish, 2002)
𝑆𝑆 63 3,𝑥+1,𝑆
<𝑓 𝜔 3 (63)
Fish’s Number Version 2
(Fish, 2002)
𝐹 < 𝐹 𝜔 𝜔 𝜔 CK +1 (63)
Fish Number Version 4
(Fish, 2002)
𝑠𝑠(2) 63 𝑓 <𝑓 𝜔 𝜔+1 ×63+1 (63)
Fish’s Number Version 3
(Fish, 2002)
Fish’s Mapping Systems
𝐹 5 63 (3)< 𝑓 𝜀 0 (𝑛−2)
Fish’s Number Version 5
(Fish, 2003)
𝑓 𝜓 0 Ω 𝜔 (𝑘)<SCG(𝑘)< 𝑓 𝜓 0 ( 𝜖 Ω 𝜔 +1 ) (𝑘)
Subcubic Graph Number
(Friedman, 2006)
TREE 𝑛
< 𝑓 𝜗( Ω 𝜔 𝜔) (𝑛)
Tree Sequence
(Friedman, 2006)
𝑛,𝑛,⋯,𝑛 < 𝑓 𝜔 𝜔 (𝑛)
Bird’s Linear Array
(Bird, 2006)
𝑛,𝑛 𝑘+1 2 < 𝑓 𝜔 𝜔 𝜔 (𝑛)
Bird’s Multi-Dimensional Array
(Bird, 2006)
{𝑛,𝑛 1,1,⋯,1,2 2}< 𝑓 𝜔 𝜔 𝜔 𝜔 (𝑛)
Bird’s Hyper-Dimensional Array
(Bird, 2006)
Bird’s
Array Systems
{n, n [1 [1 [ ... [1 [1, 2] 2] ... ] 2] 2] 2}< 𝑓 𝜀 0 𝑛
Bird’s Nested Array
(Bird, 2006)
𝐴𝑘 [[… 1,…,1 …],[… 3 …] < 𝑓 𝜀 0 (𝑛)
Taro’s Multiple List Ackermann function
(Taro, 2007)
𝐹 6 63 (3)≈ 𝑓 𝜁 0 +1 (63)
Fish’s Number Version 6
(Fish, 2007)
𝐴 1,1,⋯,1 < 𝑓 𝜔 𝜔 (𝑛)
Taro’s Multivariable
Ackermann function
(Taro, 2007)
𝐴𝑘 1,1,⋯,1 , 𝑛 < 𝑓 𝜔 𝜔 𝜔 (𝑛)
Taro’s Double List
Ackermann function
(Taro, 2007)
Rayo
Rayo’s Number
(Agustín Rayo, 2007)
Taro’s modified
Ackermann Systems
Worldwide War Era
2008 :Start of Googology Wiki
E100#100#...#100< 𝑓 𝜔 99 (100)
Hyper-E Notation
(Saibian, 2008)
FPLCI 𝑎 < SMAH +
Finite Promise Games
(Friedman, 2009)
𝐶 𝑎 =𝑎 → 𝑎 𝑎< 𝐹 𝜔 3 (𝑎)
Peter Hurford’s extensions of
chained arrow notation
(Hurford, 2011)
USGDCS 2 𝑘 < 𝐻𝑈𝐺𝐸 +
Greedy clique sequence
(Friedman, 2010)
3,3,3/2
≈𝑓 𝜗 Ω 𝜔 +2 (3)
big boowa
With BEAF Legion Array
(Bowers, 2013) 𝐹
≈ 𝑅 𝜁
Fish Number Version 7
(Fish, 2013)
𝐿100,10 10,10 &,𝐿,10 ≈ 𝑓 𝜗 Ω 𝜔 ~ 𝑓 𝜓 𝜓 𝐼
Meameamealokkapoowa oompa
(Bowers, 2013)
Ξ 𝑛
Xi function
(Adam Goucher, 2013)
10,
≈𝑓 𝜔 𝜔 (10)
Gongulus with BEAF Multi
Dimensional Array
(Bowers, 2013)
3,3 0,2 2
≈𝑓 𝜔 𝜔 𝜔 2 (3)
Duratri with BEAF Super Dimensional Array
(Bowers, 2013)
10↑↑100&10≈𝑓 𝜀 0 (101)
Goppatoth with
BEAF Higher Tetrational Array
(Bowers, 2013)
𝑛,𝑎,𝑏,𝑐,𝑑,… ≈ 𝑓 𝜔 𝜔 𝑎 (𝑛)
Array Notation
(Bowers, 2013)
BEAF Array
Systems
𝑎$ [0] [0] 2 < 𝑓 𝜗 Ω Ω (𝑛)
Extended Bracket Notation of Dollar Function
(Whytagoras, 2013)
𝑎$ < 𝑓 𝜀 0 (𝑎)
Bracket Notation of Dollar Function
(Whytagoras, 2013)
𝑆(𝑛)< 𝑓 𝜗 𝜃 1 Ω (𝑛)
Bird’s H Function
(Bird, 2013)
1 0, ,1 2 ,1 < 𝑓 ≫𝜓( 𝜓 𝐼 ( 𝐼 𝜔 )) (𝑛)
Linear Array Notation of Dollar Function
(Whytagoras, 2013)
Dollar Functions
E100#^(100)100< 𝑓 𝜔 𝜔 (100)
Hyper-E Notation
(Saibian, 2013)
E100#^#^#100< 𝑓 𝜀 0 (100)
Cascading-E Notation
(Saibian, 2013)
𝑛R … 𝑛 nests … 0 … < 𝑓 𝜀 0 (𝑎)
Brace Notation
(Hyp cos, 2013)
𝑛R𝑚
R Function
(Hyp cos, 2013)
{𝑛, 𝑛 [1 [1 [1 [1
[1 / / 5 2]
2] 2] 2] 2] 2]< 𝑓 𝜗 Ω 𝜔 (𝑛)
Bird’s Hierarchical Hyper-Nested Array Notation
(Bird, 2014)
FOOT
BIG FOOT
(Wojowu, 2014)
1 2 / 𝑆 𝑛 2 2 < 𝑓 𝜗 Ω Ω (𝑛)
Bird’s Hierarchical Hyper-Nested Array Notation
(Bird, 2014)
TR ZFC, <ZFC
Transcendental Integer
(Friedman, 2014)
𝐻(𝑛)< 𝑓 𝜗( 𝜖 Ω +1)
Bird’s H Function
(Bird, 2014)
𝑈(𝑛)< 𝑓 𝜗 Ω 𝜔 (𝑛)
Bird’s U Function
(Bird, 2014)
Bashicu’s System
𝑆 𝑛 ≈𝑓 𝜗 Ω Ω (𝑛)
Bird’s H Function
(Bird, 2014)
< 𝑓 𝜔+1 (10)
Small primitive sequence number
(Bashicu, 2014)
<𝑓 𝜀 0 +1 (10)
Primitive sequence number
(Bashicu, 2014)
Pair(𝑛)≈𝑓 𝜗( Ω 𝜔 ) (𝑛)
Pair sequence number
(Bashicu, 2014)
Bashicu Matrix System
(Bashicu, 2014)
𝑠(𝑛,𝑛{1{1⋯
1,2` ⋯2`}2}2)< 𝑓 𝜗 𝜖 Ω +1 (𝑛)
Expanding Array Notation
(Hyp cos, 2015)
s(n,n {1 ,, 1 ,, 2} 2)< 𝑓 𝜓 𝜓 𝐼 𝑛
Primary dropping array notation
(Hyp cos, 2015)
𝑠(𝑛,𝑛 {1` `…`2} 2)
< 𝑓 𝜗 Ω 𝜔 (𝑛)
Multiple Expanding Array Notation
(Hyp cos, 2015)
𝑠 𝑛,𝑛 1 1 1,
< 𝑓 𝜀 0 (𝑛)
Expanded Array Notation
(Hyp cos, 2015)
𝑠 𝑛,⋯,𝑛 < 𝑓 𝜔 𝜔 (𝑛)
Linear Array Notation
(Hyp cos, 2015)
𝑛,𝑛 1 ⋅ 𝑛 2 < 𝑓 𝜀 0 (𝑛)
Expanding Array Notation
(Hyp cos, 2015)
FOFT googol (googol)
BIG FOFT
(deedlit, 2016)
Googol Maps
-Phylogenetics of Googology-
Strong Array
Notations 𝑘
Little Bigeddon
(Emlightened, 2017)
Edited by koteitan 11 Feb 2017