Y function is a sort of functions introduced by a Japanese googologist Yukito, and is a function which employs side nesting.[1][2][3]


It has at least 10 versions. One version of Y function is expected to roughly reach (0,0,0)(1,1,1)(2,2,1)(3,1,0)(2,2,1) in Bashicu matrix system version 2.3 under the assumption of its termination.

Another version of Y function called 0-Y is unexpectedly broken, and hence the limit was considered to correspond to \(\zeta_0\) in fast-growing hierarchy, which is not as big as Yukito had first expected. However, it later turned out to correspond to \(2\) in fast-growing hierarchy.

The limit of another version of Y function called 1-Y is expected to correspond to \(\psi(I)\) with respect to an unspecified ordinal collapsing function in fast-growing hierarchy with respect to an unspecified system of fundamental sequences.

None of those versions of Y function has been formalised yet.


Yukito conjectured that \(Y(\Omega)+Y_{\omega^3}(Y(\Omega))\) in one version of Y function safely outgrows the limit of Y sequence on 27/02/2020, and hence Y function was supposed to be a quite strong system if it would be completely formalised. However, Yukito noticed that the method applied to S-σ to avoid a certain problem is not applicable to Y function, and hence clarified that he decided to terminate the project to complete the study of Y function on 20/07/2020.


  1. The user page of Yukito in the Japanese Googology Wiki.
  2. Yukito, Y関数, the Japanese Googology Wiki user blog.
  3. Yukito, 2019年11月18日版0-Y関数+1-Y関数, Google Document.

See also

Fish numbers: Fish number 1 · Fish number 2 · Fish number 3 · Fish number 4 · Fish number 5 · Fish number 6 · Fish number 7
Mapping functions: S map · SS map · S(n) map · M(n) map · M(m,n) map
By Aeton: Okojo numbers · N-growing hierarchy
By BashicuHyudora: Primitive sequence number · Pair sequence number · Bashicu matrix system
By Kanrokoti: KumaKuma ψ function
By 巨大数大好きbot: Flan numbers
By Jason: Irrational arrow notation · δOCF · δφ · ε function
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 Nayuta Ito: N primitive
By p進大好きbot: Large Number Garden Number
By Yukito: Hyper primitive sequence system · Y sequence · YY sequence · Y function
Indian counting system: Lakh · Crore · Tallakshana · Uppala · Dvajagravati · Paduma · Mahakathana · Asankhyeya · Dvajagranisamani · Vahanaprajnapti · Inga · Kuruta · Sarvanikshepa · Agrasara · Uttaraparamanurajahpravesa · Avatamsaka Sutra · Nirabhilapya nirabhilapya parivarta
Chinese, Japanese and Korean counting system: Wan · Yi · Zhao · Jing · Gai · Zi · Rang · Gou · Jian · Zheng · Zai · Ji · Gougasha · Asougi · Nayuta · Fukashigi · Muryoutaisuu
Other: Taro's multivariable Ackermann function · TR function · Arai's \(\psi\) · Sushi Kokuu Hen

Community content is available under CC-BY-SA unless otherwise noted.