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Not to be confused with SAN.

SSAN is a notation introduced by a Japanese Googology Wiki user mrna,[1] and is the first notation which employs side nesting. It consists of infinite systems 0-SSAN, 1-SSAN, 2-SSAN, …, and ω-SSAN.

There are three large numbers defined by SSAN: 七星の剣数, 絶対秘神の七星剣数, and 二童子達のバックナンバーズ. The names of the first two numbers come from spell cards of a character 摩多羅隠岐奈[2] in Japanese famous video game series 東方Project, and the name of the third number comes from nicknames of characters 爾子田里乃[3] and 丁礼田舞[4] in the series.


Although the full system is currently ill-defined, 0-SSAN has been defined.


The first system 0-SSAN has already been formalised, and was used to define a large number 七星の剣数 submitted to a Japanese googological event.[5] This system can be roughly given by interpreting pair sequences with respect to Bashicu matrix system version 2.3 into labeled brackets. In particular, the limit of 0-SSAN corresponds to \(\psi_0(\Omega_{\omega})\) with respect to Buchholz's function.


The second system 1-SSAN was partially defined, and was used to define two large numbers 絶対秘神の七星剣数 and 二童子達のバックナンバーズ submitted to a Japanese googology event.[6][7] However, many errors were found in the original definition of 1-SSAN, and hence the two large numbers were rejected by the referees of the event. Although 1-SSAN is intended to go beyond \(\psi(\Omega_{K+1})\) with respect to an undefined ordinal collapsing function \(\psi\) and the least weakly compact cardinal \(K\), it has not been completely defined yet.

Higher SSANs

The latter systems such as 2-SSAN and ω-SSAN have not been formalised, either. According to Yukito, the limit of ω-SSAN is expected to go beyond the limit of SY sequence, which is Yukito's unformalised notation expected to go beyond the limit of Y sequence, which is expected to go beyond the limit of Bashicu matrix system version 2.3 under the assumption of the terminations of their formalisations.


  1. The user page of mrna in Japanese Googology Wiki.
  2. 摩多羅隠岐奈 in pixiv dictionary.
  3. 爾子田里乃 in piciv dictionary.
  4. 丁礼田舞 in pixiv dictionary.
  5. mrna, 七星の剣数.
  6. mrna, 絶対秘神の七星剣数.
  7. mrna, 二童子達のバックナンバーズ.

See also

By Aeton: Okojo numbers · N-growing hierarchy
By 新井 (Arai): Arai's \(\psi\)
By バシク (BashicuHyudora): Primitive sequence number · Pair sequence number · Bashicu matrix system 1/2/3/4
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 computation programmes · TR function (I0 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 小林銅蟲 (Kobayashi Doom): Sushi Kokuu Hen
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 computation programme · ω-Y sequence computation programme
By Nayuta Ito: N primitive · Flan numbers · Large Number Lying on the Boundary of the Rule of Touhou Large Number 4
By p進大好きbot: 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 · YY sequence · Y function · ω-Y sequence
See also: Template:Googology in Asia