Side nesting (横ネスト in Japanese) is a method to generate a googological system introduced by a Japanese googologist mrna. It admits a generalisation called shifting definition.
Feature[]
One of the most characteristic and confusing feature of side nesting is that even if a notation employs \(+\) as a \(2\)-ary function symbol, it does not necessarily work as the addition. Namely, even if a valid expression \(a\) corresponds to a countable ordinal \(\alpha\) and \(a + a\) is also a valid expression, \(a + a\) does not necessarily corresponds to \(\alpha + \alpha\). The characteristic feature of side nesting on \(+\) is inherited by its generalisation shifting definition. If a notation based on side nesting does not employ \(+\), then a separator plays the role analogous to \(+\) in a notation based on side nesting.
Criterion[]
Since side nesting is a sort of a terminology for a description of a strategy to create a googology notation, there is no fixed formalised definition of "whether a notation admits a structure of side nesting or not". However, mrna is trying to show candidates of sufficient conditions, which helps us to understand what notations can be considered as typical side nesting.[1] Although the unformalised definition of "whether a notation admits a structure of side nesting or not" varies as time goes, such a trial will mildly fix the definition. For example, the original idea explained in #Feature refers only to separators such as the \(+\) symbol, and hence the ordinal notation associated to Buchholz's function was not literally regarded as a notation with side nesting. However, the recent trial allows wider structures and syntaxes, and hence the ordinal notation associated to Buchholz's function can be currently regarded as a notation with side nesting. Note that the notion of side nesting makes sense for a notation, but not for a function. Therefore Buchholz's function itself is not one with side nesting.
Examples[]
According to mrna, side nesting is a method which can be found in many notations, while there are few notations which intensionally focus on side nesting. The following three systems are intended to be based on side nesting:
Although none of them has been fully defined, many Japanese googologists get interested to their intended behaviour, because they are expected to be poweful. For the detail explanation of the strength, see the following subsections:
SSAN[]
- Main article: SSAN
SSAN is the first notation which employs side nesting introduced by a Japanese googologist mrna. It consists of infinite systems 0-SSAN, 1-SSAN, 2-SSAN, …, and ω-SSAN.[2][3][4]
S-σ[]
- Main article: S-σ
S-σ is a notation introduced by mrna as a system purely based on side nesting, while SSAN also employs another strategy than side nesting.[5]
Y function[]
- Main article: Y function
Y function has at least 10 versions, created by a Japanese googologist Yukito[6][7][8] None of 10 versions of Y function has been formalised yet.
References[]
- ↑ mrna, 横ネストについて, Japanese Googology Wiki user blog.
- ↑ mrna, 七星の剣数.
- ↑ mrna, 絶対秘神の七星剣数.
- ↑ mrna, 二童子達のバックナンバーズ.
- ↑ mrna, Yｶﾞﾁ解析, Google Spreadsheet.
- ↑ The user page of Yukito in the Japanese Googology Wiki.
- ↑ Yukito, Y関数, the Japanese Googology Wiki user blog.
- ↑ Yukito, 2019年11月18日版0-Y関数+1-Y関数, Google Document.
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