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くまくまψ関数 (English: KumaKuma ψ function) is a computable notation to generate large numbers created by a Japanese Googology Wiki user kanrokoti.[1] Its $$3$$-ary subsystem is called くまくま3変数ψ (English: 3 variables ψ),[2][3] and its full system, which is of $$4$$-ary, is called くまくま4変数ψ (English: 4 variables ψ).[4]

These are $$3$$- and $$4$$-ary extensions of the ordinal notation by p進大好きbot[5] associated to Extended Buchholz's function by Denis Maksudov.

## Significance

Since many people misunderstood Buchholz's function and Extended Buchholz's function in this community,[6][7] there were so many notations which people stated that were computable notations much stronger than the ordinal notation associated to Extended Buchholz's function but had serious problems: Many of them were ill-defined, had no actual algorithm to compute, or were much weaker than the ordinal notation associated to Extended Buchholz's function. Unlike them, くまくまψ関数 was created in a way precisely extending the ordinal notation associated to Extended Buchholz's function.

くまくまψ関数 is designed so that it can be easily used in analysis. We list examples of the benefits of くまくまψ関数:

1. This is an addition-based notation, which is useful in analysis because many of googological notations are addition-based.
2. This is compatible with Extended Buchholz's function, and hence analyses by Extended Buchholz's function can be continued by using this notation.
3. This notation is simple, because it employs only one function symbol ψ except for +, and only one constant 0.
4. Its $$3$$-ary subsystem is expected to reach $$\psi_{\chi_0(0)}(\psi_{\chi_{2}(0)}(0))$$, and the full $$4$$-ary system is expected to reach $$\psi_{\chi_0(0)}(\psi_{\chi_{3}(0)}(0))$$ with respect to Rathjen's psi.
5. This notation is extensible in various ways.[8][9][10]

## Definition

We explain the $$3$$-ary subsytem.[3]

### Notation

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

we 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, we define the magnitude relation between the notation.

For an X,Y∈T, we 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, we define the cofinality.

we 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, we define the fundamental sequences using cofinality. we 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, we define the FGH. we 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, we define large function and large number.

we 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 $$g(n) = ψ_{g(n-1)}(0,0)$$.

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

Kanrokoti coined $$F^{10^{100}}(10^{100})$$ as "Kumakuma 3 variables ψ number".

Similarly, Kanrokoti coined "Kumakuma 4 variables ψ number" using the $$4$$-ary full system.

### Naming

Here, we give names for ordinals.

Kanrokoti coined the ordinal which corresponds to $$ψ_0(0,ψ_{2}(0,0))$$ "KBHO" (Kuma-Bachmann–Howard Ordinal).

Kanrokoti coined the ordinal which corresponds 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, Kanrokoti coined the ordinal which corresponds to the limitation of Kumakuma 3-ary ψ "EKBO" ( Extended Kuma- Buchholz Ordinal).

Similarly, Kanrokoti coined "KKBHO" (KumaKuma-Bachmann–Howard Ordinal), "KKBO" (KumaKuma-Buchholz Ordinal), and "EKKBO" (Extended KumaKuma-Buchholz Ordinal).

## Analysis

Although there is no formal proof, it is expected that EKBO and EKKBO coincide with $$\psi_{\chi_0(0)}(\psi_{\chi_{2}(0)}(0))$$ and $$\psi_{\chi_0(0)}(\psi_{\chi_{3}(0)}(0))$$ respectively.