\(\omega\)-Y sequence is a difference sequence system introduced by a Japanese googologist Yukito in July 2021.[1][2] It is intended to be much stronger than the creator's past work Y sequence.
\(\omega\)-Y sequence has new difference sequences called galaxy and galaxy group (or super cluster), which are the extensions of Mt. Fuji for the multidimensional structure.
Structure of ω-Y(1,5,33)
Y sequence has "diagonal difference sequence" which consists of the numbers picked up from the top of the Mt. Fuji and it make a new Mt.Fuji (1,4,24 on the top of the Mt.Fuji based on (1,5,33) in the figure). On the other hand, \(\omega\)-Y sequence has not only diagonal difference sequences but also galactic difference sequence(or spiral difference sequence), which consists of the numbers picked up at the bottom-left corners of Mt.Fujis (1,3,14 on the (1,5,33)-based galaxy).
The galactic difference sequence (1,3,14) has an own difference sequence (2,11), and it makes a new galaxy based on the difference sequence (2,11) (in the right side of (1,5,33)). The galaxy difference sequence of the (2,1)-based galaxy is (2,7) and it makes another (5)-based galaxy. Now we got the 3 galaxies and it makes a new galaxy group based on (1,3) (in the bottom side of the (1,5,33)) by the difference sequence of the galaxy group difference sequence (1,2,5).
In this way, \(\omega\)-Y sequence can continue it infinitely, sequences, Mt.Fujis, galaxies, galaxy groups, 6th structures, 7th structures, 8th structures, ... and so on. The n-th structures consist of n-th dimensional structure. Especially, \(\omega\)-Y(1,n) consist of n-th dimensional simplex.
The difference of the structure of \(\omega\)-Y sequence to the one of Y sequence is that Y sequence makes a next Mt.Fuji by putting the diagonal sequence itself into the base of the new Mt.Fuji and \(\omega\)-Y sequence makes the next Mt.Fuji (or the next structure) by putting the "difference sequence" of the diagonal sequence. For example, the basic Mt.Fuji of Y(1,4) is ((1,4),(3)) and the diagonal sequence is (1,3), and the next Mt.Fuji becomes (1,3) itself in Y-sequence. On the other hand in \(\omega\)-Y sequence, the diagonal sequence of ((1,4),(3)) is (1,3) and the next Mt.Fuji becomes (2) which is the difference sequence of (1,3).
The first difference between Y sequence and \(\omega\)-Y sequence is the limit of (1,3,2,5,4,9,8,17,\(\cdots,2^k,2^{k+1}+1,\cdots\)), Y(1,3,3) on Y sequence and \(\omega\)-Y(1,3,2,5,5) on \(\omega\)-Y sequence.
Yukito said[3] he had tried to make Y^n sequences whose limit Y^n(1,\(\omega\)) is equal to Y^n+1(1,3) at first time, however, it had critical bug and he changed the way. While Y^ns unlock the new dimensions, \(\omega\)-Y sequence unlocks all the dimensions at once.
Definitions[]
Original definition[]
Yukito declared the completion of the \(\omega\)-Y sequence and it was defined as the User:Naruyoko's program in twitter[4]
Yukito said that he coined \(\omega\)-Y sequence number using \(\omega\)-Y sequence as a masterpiece of Yukito by giving the detailed definition on a user blog later[5][6] also.
Expansion rule[]
The expansion rule of \(\omega\)-Y sequence to solve an expression like \(\omega\)-Y(s)[n] for a valid sequence \(s\) and a natural number \(n\) is defined by the function expand(s,n,stringify) on the code by Naruyoko.
Large number[]
Yukito said that he named \(f^{2000}(1)\) using f(n)=ω-Y\((1,\omega)[n]\) as \(\omega\)-Y sequence number[5].
Programs[]
Stacked drawing of ω-Y(1,5,33)
User:Naruyoko made a program "Study and Expand Sequence(仮)"[2] to expand \(\omega\)-Y sequences.
User:Naruyoko made a program MEGA whY mountain to draw the structure of \(\omega\)-sequence. It draws a multi-dimensional Mt. Fuji with a structure that looks like a pile of Mt.Fuji on top of Mt.Fuji, one after another. Yukito calls it Mega-Mt.Fuji.
See also[]
By Aeton: Okojo numbers · N-growing hierarchy
By 新井 (Arai): Arai's psi function
By aster: White-aster notation · White-aster
By バシク (BashicuHyudora): Primitive sequence number · Pair sequence number · Bashicu matrix system 1/2/3/4 original idea
By ふぃっしゅ (Fish): Fish numbers (Fish number 1 · Fish number 2 · Fish number 3 · Fish number 4 · Fish number 5 · Fish number 6 · Fish number 7 · S map · SS map · s(n) map · m(n) map · m(m,n) map) · Bashicu matrix system 1/2/3/4 formalisation · TR function (I0 function)
By Gaoji: Weak Buchholz's function
By じぇいそん (Jason): Irrational arrow notation · δOCF · δφ · ε function
By 甘露東風 (Kanrokoti): KumaKuma ψ function
By koteitan: Bashicu matrix system 2.3
By mrna: 段階配列表記 · 降下段階配列表記 · 多変数段階配列表記 · SSAN · S-σ
By Naruyoko Naruyo: Y sequence formalisation · ω-Y sequence formalisation
By Nayuta Ito: N primitive · Flan numbers (Flan number 1 · Flan number 2 · Flan number 3 · Flan number 4 version 3 · Flan number 5 version 3) · Large Number Lying on the Boundary of the Rule of Touhou Large Number 4 · Googology Wiki can have an article with any gibberish if it's assigned to a number
By Okkuu: Extended Weak Buchholz's function
By p進大好きbot: Ordinal notation associated to Extended Weak Buchholz's function · Ordinal notation associated to Extended Buchholz's function · Naruyoko is the great · Large Number Garden Number
By たろう (Taro): Taro's multivariable Ackermann function
By ゆきと (Yukito): Hyper primitive sequence system · Y sequence original idea · YY sequence · Y function · ω-Y sequence original idea
By バシク (BashicuHyudora): Bashicu matrix system as a notation template
By じぇいそん (Jason): Shifting definition
By mrna: Side nesting
By Nayuta Ito and ゆきと (Yukito): Difference sequence system
By ふぃっしゅ (Fish): Ackermann function
By koteitan: Ackermann function · Beklemishev's worms · KumaKuma ψ function
By Mitsuki1729: Ackermann function · Graham's number · Conway's Tetratri · Fish number 1 · Fish number 2 · Laver table
By みずどら: White-aster notation
By Naruyoko Naruyo: p進大好きbot's Translation map for pair sequence system and Buchholz's ordinal notation · KumaKuma ψ function · Naruyoko is the great
By 猫山にゃん太 (Nekoyama Nyanta): Flan number 4 version 3 · Fish number 5 · Laver table
By Okkuu: Fish number 1 · Fish number 2 · Fish number 3 · Fish number 5 · Fish number 6
By rpakr: p進大好きbot's ordinal notation associated to Extended Weak Buchholz's function · Standardness decision algorithm for Taranovsky's ordinal notation
By ふぃっしゅ (Fish): Computing last 100000 digits of mega · Approximation method for FGH using Arrow notation · Translation map for primitive sequence system and Cantor normal form
By Kihara: Proof of an estimation of TREE sequence · Proof of the incomparability of Busy Beaver function and FGH associated to Kleene's \(\mathcal{O}\)
By koteitan: Translation map for primitive sequence system and Cantor normal form
By Naruyoko Naruyo: Translation map for Extended Weak Buchholz's function and Extended Buchholz's function
By Nayuta Ito: Comparison of Steinhaus-Moser Notation and Ampersand Notation
By Okkuu: Verification of みずどら's computation program of White-aster notation
By p進大好きbot: Proof of the termination of Hyper primitive sequence system · Proof of the termination of Pair sequence number · Proof of the termination of segements of TR function in the base theory under the assumption of the \(\Sigma_1\)-soundness and the pointwise well-definedness of \(\textrm{TR}(T,n)\) for the case where \(T\) is the formalisation of the base theory
By 小林銅蟲 (Kobayashi Doom): Sushi Kokuu Hen
By koteitan: Dancing video of a Gijinka of Fukashigi · Dancing video of a Gijinka of 久界 · Storyteller's theotre video reading Large Number Garden Number aloud
See also: Template:Googology in Asia
Sources[]
- ↑ Yukito, twitter, July 17, 2021.
- ↑ 2.0 2.1 Naruyoko, "Study and Expand Sequence(仮)", github io, July 17, 2021
- ↑ Yukito, twitter, Aug. 17, 2021
- ↑ Yukito, twitter, Sept. 1, 2021
- ↑ 5.0 5.1 Yukito twitter, Sept. 1, 2021
- ↑ Yukito twitter, Sept. 1, 2021