This blog post is a translation from https://googology.wikia.org/ja/wiki/%E3%83%A6%E3%83%BC%E3%82%B6%E3%83%BC%E3%83%96%E3%83%AD%E3%82%B0:Kanrokoti/%E3%81%8F%E3%81%BE%E3%81%8F%E3%81%BE3%E5%A4%89%E6%95%B0%CF%88

Overview

EBOCF(Extended Buchholz's OCF) is 2 variables function. So, I decided to extend it like ψ_A(B) => ψ_0(A,B). Please be careful that this 3 variables ψ is neither OCF nor Ordinal notation. I named this notation "Kumakuma 3 variables ψ". "Kuma" means "bear" in Japanese🐻. In order to make the definiton, I referred to the following pages. I appreciate p進大好きbot corrected my definitons.

https://googology.wikia.org/wiki/User_blog:P%E9%80%B2%E5%A4%A7%E5%A5%BD%E3%81%8Dbot/Ordinal_Notation_Associated_to_Extended_Buchholz%27s_OCF

https://googology.wikia.org/wiki/Buchholz%27s_function#Extension


Definition

Notation

Here, I define the character string used for the notation.

I define sets T and PT of formal strings consisting of 0, ψ_, (, ), +, and commas in the following recursive way:

1. 0∈T.
2. For any X_1,X_2,X_3∈T, ψ_{X_1}(X_2,X_3)∈T and ψ_{X_1}(X_2,X_3)∈PT.
3. For any X_1,...,X_m∈PT (2≦m<∞), X_1+...+X_m∈T.

0 is abbreviated as $0, ψ_0(0,0) is abbreviated as $1, $1+...+$1 (n $1's) is abbreviated as $n for each \(n \in \mathbb{N}\) greater than 1, and ψ_0(0,$1) is abbreviated as $ω.


Ordering

Here, I define the magnitude relation between the notation.

For an X,Y∈T, I define the recursive 2-ary relation X<Y in the following recursive way:

1. If X=0, then X<Y is equivalent to Y≠0.
2. Suppose X=ψ_{X_1}(X_2,X_3) for some X_1,X_2,X_3∈T.
2-1. If Y=0, then X<Y is false.
2-2. Suppose Y=ψ_{Y_1}(Y_2,Y_3) for some Y_1,Y_2,Y_3∈T.
2-2-1. If X_1=Y_1 and X_2=Y_2, then X<Y is equivalent to X_3<Y_3.
2-2-2. If X_1=Y_1 and X_2≠Y_2, then X<Y is equivalent to X_2<Y_2.
2-2-3. If X_1≠Y_1, then X<Y is equivalent to X_1<Y_1.
2-3. If Y=Y_1+...+Y_{m'} for some Y_1,...,Y_{m'}∈PT (2≦m'<∞), then
   X<Y is equivalent to X=Y_1 or X<Y_1.
3. Suppose X=X_1+...+X_m for some X_1,...,X_m∈PT (2≦m<∞).
3-1. If Y=0, then X<Y is false.
3-2. If Y=ψ_{Y_1}(Y_2,Y_3) for some Y_1,Y_2,Y_3∈T, X<Y is equivalent to X_1<Y.
3-3. Suppose Y=Y_1+...+Y_{m'} for some Y_1,...,Y_{m'}∈PT (2≦m'<∞).
3-3-1. Suppose X_1=Y_1.
3-3-1-1. If m=2 and m'=2, then X<Y is equivalent to X_2<Y_2.
3-3-1-2. If m=2 and m'>2, then X<Y is equivalent to X_2<Y_2+...+Y_{m'}.
3-3-1-3. If m>2 and m'=2, then X<Y is equivalent to X_2+...+X_m<Y_2.
3-3-1-4. If m>2 and m'>2, then X<Y is equivalent to X_2+...+X_m<Y_2+...+Y_{m'}.
3-3-2. If X_1≠Y_1, then X<Y is equivalent to X_1<Y_1.


Cofinality

Here, I define the cofinality.

I define total recursive maps \begin{eqnarray*} \textrm{dom} \colon T & \to & T \\ X & \mapsto & \textrm{dom}(X) \\ \end{eqnarray*} in the following recursive way:

1. If X=0, then dom(X)=0.
2. Suppose X=ψ_{X_1}(X_2,X_3) for some X_1,X_2,X_3∈T.
2-1. Suppose dom(X_3)=0.
2-1-1. Suppose dom(X_2)=0.
2-1-1-1. If dom(X_1)=0 or dom(X_1)=$1, then dom(X)=X.
2-1-1-2. If dom(X_1)≠0,$1, then dom(X)=dom(X_1).
2-1-2. If dom(X_2)=$1, then dom(X)=X.
2-1-3. Suppose dom(X_2)≠0,$1.
2-1-3-1. If dom(X_2)<X, then dom(X)=dom(X_2).
2-1-3-2. Otherwise, then dom(X)=$ω.
2-2. If dom(X_3)=$1 or dom(X_3)=$ω, then dom(X)=$ω.
2-3. Suppose dom(X_3)≠0,$1,$ω.
2-3-1. If dom(X_3)<X, then dom(X)=dom(X_3).
2-3-2. Otherwise, then dom(X)=$ω.
3. If X=X_1+...+X_m for some X_1,...,X_m∈PT (2≦m<∞), then dom(X)=dom(X_m).


Fundamental Sequences

Here, I define the fundamental sequences using cofinality.

I define total recursive maps \begin{eqnarray*} [ \ ] \colon T \times T & \to & T \\ (X,Y) & \mapsto & X[Y] \end{eqnarray*} in the following recursive way:

1. If X=0, then X[Y]=0.
2. Suppose X=ψ_{X_1}(X_2,X_3) for some X_1,X_2,X_3∈T.
2-1. Suppose dom(X_3)=0.
2-1-1. Suppose dom(X_2)=0.
2-1-1-1. If dom(X_1)=0, then X[Y]=0.
2-1-1-2. If dom(X_1)=$1, then X[Y]=Y.
2-1-1-3. If dom(X_1)≠0,$1, then X[Y]=ψ_{X_1[Y]}(X_2,X_3).
2-1-2. If dom(X_2)=$1, then X[Y]=Y.
2-1-3. Suppose dom(X_2)≠0,$1.
2-1-3-1. If dom(X_2)<X, then X[Y]=ψ_{X_1}(X_2[Y],X_3).
2-1-3-2. Otherwise, then take a unique P,Q∈T such that dom(X_3)=ψ_{P}(Q,0).
2-1-3-2-1. Suppose Q=0.
2-1-3-2-1-1. If Y=$h (1≦h<∞) and X[Y[0]]=ψ_{X_1}(Γ,X_3) for unique Γ∈T, then
       X[Y]=ψ_{X_1}(X_2[ψ_{P[0]}(Γ,0)],X_3).
2-1-3-2-1-2. Otherwise, then X[Y]=ψ_{X_1}(X_2[ψ_{P[0]}(Q,0)],X_3).
2-1-3-2-2. Suppose Q≠0.
2-1-3-2-2-1. If Y=$h (1≦h<∞) and X[Y[0]]=ψ_{X_1}(Γ,X_3) for unique Γ∈T, then
       X[Y]=ψ_{X_1}(X_2[ψ_{P}(Q[0],Γ)],X_3).
2-1-3-2-2-2. Otherwise, then X[Y]=ψ_{X_1}(X_2[ψ_{P}(Q[0],0)],X_3).
2-2. Suppose dom(X_3)=$1.
2-2-1. If Y=$1, then X[Y]=ψ_{X_1}(X_2,X_3[0]).
2-2-2. If Y=$k (2≦k<∞), then
    X[Y]=ψ_{X_1}(X_2,X_3[0])+...+ψ_{X_1}(X_2,X_3[0]) (k's ψ_{X_1}(X_2,X_3[0])).
2-2-3. If neither of them, then X[Y]=0.
2-3. If dom(X_3)=$ω, X[Y]=ψ_{X_1}(X_2,X_3[Y]).
2-4. Suppose dom(X_3)≠0,$1,$ω.
2-4-1. If dom(X_3)<X, X[Y]=ψ_{X_1}(X_2,X_3[Y]).
2-4-2. Otherwise, then take a unique P,Q∈T such that dom(X_3)=ψ_{P}(Q,0).
2-4-2-1. Suppose Q=0.
2-4-2-1-1. If Y=$h (1≦h<∞) and X[Y[0]]=ψ_{X_1}(X_2,Γ) for unique Γ∈T, then
     X[Y]=ψ_{X_1}(X_2,X_3[ψ_{P[0]}(Γ,0)]).
2-4-2-1-2. Otherwise, then X[Y]=ψ_{X_1}(X_2,X_3[ψ_{P[0]}(Q,0)]).
2-4-2-2. Suppose Q≠0.
2-4-2-2-1. If Y=$h (1≦h<∞) and X[Y[0]]=ψ_{X_1}(X_2,Γ) for unique Γ∈T, then
     X[Y]=ψ_{X_1}(X_2,X_3[ψ_{P}(Q[0],Γ)]).
2-4-2-2-2. Otherwise, then X[Y]=ψ_{X_1}(X_2,X_3[ψ_{P}(Q[0],0)]).
3. Suppose X=X_1+...+X_m for some X_1,...,X_m∈PT (2≦m<∞).
3-1. If X_m[Y]=0 and m=2, then X[Y]=X_1.
3-2. If X_m[Y]=0 and m>2, then X[Y]=X_1+...+X_{m-1}.
3-3. If X_m[Y]≠0, then X[Y]=X_1+...+X_{m-1}+X_m[Y].


FGH

Here, I define the FGH.

I define total recursive maps \begin{eqnarray*} f \colon T \times \mathbb{N} & \to & \mathbb{N} \\ (X,n) & \mapsto & f_X(n) \end{eqnarray*} in the following recursive way:

1. If X=0, then \(f_X(n) = n+1\).
2. If X=$1 or X=Y+$1 for some Y∈T, then \(f_X(n) = f_{X[0]}^n(n)\).
3. If neither of them, then \(f_X(n) = f_{X[$n]}(n)\).


Large Function and Large Number

Here, I define large function and large number.

I define total recursive maps \begin{eqnarray*} g \colon \mathbb{N} & \to & PT \\ n & \mapsto & g(n) \end{eqnarray*} in the following recursive way:

1. If n=0, then \(g(n) = ψ_0(0,0)\).
2. Otherwise, then \(g(n) = ψ_{g(n-1)}(0,0)\).


I define total recursive maps \begin{eqnarray*} F \colon \mathbb{N} & \to & \mathbb{N} \\ n & \mapsto & F(n) \end{eqnarray*} as \(F(n) = f_{ψ_0(0,g(n))}(n)\).

I name \(F^{10^{100}}(10^{100})\) "Kumakuma 3 variables ψ number".


Naming

Here, I give names for ordinals.

I name an ordinal which correspond to \(ψ_0(0,ψ_{$2}(0,0))\) "KBHO" (Kuma-Bachmann–Howard Ordinal).

I name an ordinal which correspond to \(ψ_0(0,ψ_{$ω}(0,0))\) "KBO" (Kuma-Buchholz Ordinal).

Since ψ_0(ψ_{ψ_{ψ_{ψ_...}(0)}(0)}(0)) in EBOCF is called EBO (Extended Buchholz Ordinal) among Japanese googologists, I name an ordinal which correspond to the limitation of Kumakuma 3-ary ψ "EKBO" ( Extended Kuma- Buchholz Ordinal).

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