X-Sequence Nested Exponential Notation

The X-Sequence Nested Exponential Notation is a custom notation based on the Fast Growing Hierarchy.

B
{
S
}
# C

f

S

(
B
C
)

{\displaystyle B\{S\}C=f\_{S}(B+C)}

A
{
S
}
# B

f

S

(
A
B
)

{\displaystyle A\{S\}B=f\_{S}(A+B)}

B
{
X
}
# C

f

ω

(
B
C
)
,

{\displaystyle B\{X\}C=f\_{\omega }(B+C),}

B and C are positive finite integers and X is for X sequence nested Exponential notation.

B
{
X
1
}
# C

f

ω
1

(
B
{
X
}
C
)

{\displaystyle B\{X+1\}C=f\_{\omega +1}(B\{X\}C)}

B
{
X
2
}
# C

f

ω
2

(
B
{
X
}
C
)

{\displaystyle B\{X+2\}C=f\_{\omega +2}(B\{X\}C)}

B
{
X
X
}
# C
B
{
2
X
}
# C

f

ω
2

(
B
{
X
}
C
)

{\displaystyle B\{X+X\}C=B\{2X\}C=f\_{\omega 2}(B\{X\}C)}

B
{
X
X
X
}
# C
B
{
2
X
X
}
# C
B
{
3
X
}
# C

f

ω
3

(
B
{
X
}
C
)

{\displaystyle B\{X+X+X\}C=B\{2X+X\}C=B\{3X\}C=f\_{\omega 3}(B\{X\}C)}

B
{
X
X
}
# C
B
{

X

2

}
# C

f

ω

2

(
B
{
X
}
C
)

{\displaystyle B\{XX\}C=B\{X^{2}\}C=f\_{\omega ^{2}}(B\{X\}C)}

B
{

X

2

X
}
# C
B
{

X

3

}
# C

f

ω

3

(
B
{
X
}
C
)

{\displaystyle B\{X^{2}X\}C=B\{X^{3}\}C=f\_{\omega ^{3}}(B\{X\}C)}

B
{

X

X

}
# C

f

ω

ω

(
B
{
X
}
C
)

{\displaystyle B\{X^{X}\}C=f\_{\omega ^{\omega }}(B\{X\}C)}

B
{

X

X
1

}
# C
B
{

X

X

X
}
# C

f

ω

ω
1

(
B
{
X
}
C
)

{\displaystyle B\{X^{X+1}\}C=B\{X^{X}X\}C=f\_{\omega ^{\omega +1}}(B\{X\}C)}

B
{

X

X
X

}
# C

f

ω

ω
ω

(
B
{
X
}
C
)

{\displaystyle B\{X^{X+X}\}C=f\_{\omega ^{\omega +\omega }}(B\{X\}C)}

B
{

X

X

2

}
# C

f

ω

ω

2

(
B
{
X
}
C
)

{\displaystyle B\{X^{X^{2}}\}C=f\_{\omega ^{\omega ^{2}}}(B\{X\}C)}

B
{

X

X

X

}
# C

f

ω

ω

ω

(
B
{
X
}
C
)

{\displaystyle B\{X^{X^{X}}\}C=f\_{\omega ^{\omega ^{\omega }}}(B\{X\}C)}

B
{

X

X

X

X

}
# C

f

ω

ω

ω

ω

(
B
{
X
}
C
)

{\displaystyle B\{X^{X^{X^{X}}}\}C=f\_{\omega ^{\omega ^{\omega ^{\omega }}}}(B\{X\}C)}

B
{
X
↑↑

X

}
# C
B
{
X
(
1
,
0
)
}
# C

f

ε

0

(
B
{
X
}
C
# )

f

φ
(
1
,
0
)

(
B
{
X
}
C
)

{\displaystyle B\{X\uparrow \uparrow {X}\}C=B\{X(1,0)\}C=f\_{\varepsilon \_{0}}(B\{X\}C)=f\_{\varphi (1,0)}(B\{X\}C)}

B
{
X
(
1
,
0
,
0
)
}
# C

f

φ
(
1
,
0
,
0
)

(
B
{
X
}
C
# )

f

Γ

0

(
B
{
X
}
C
)

{\displaystyle B\{X(1,0,0)\}C=f\_{\varphi (1,0,0)}(B\{X\}C)=f\_{\Gamma \_{0}}(B\{X\}C)}

B
{
X
(
1
,
0
,
0
,
0
)
}
# C

f

φ
(
1
,
0
,
0
,
0
)

(
B
{
X
}
C
# )

f

A
c
k

(
B
{
X
}
C
)

{\displaystyle B\{X(1,0,0,0)\}C=f\_{\varphi (1,0,0,0)}(B\{X\}C)=f\_{Ack}(B\{X\}C)}

B
{
G
(

Γ

Γ

X

)
}
# C
B
{
X
(
1
,
0
,
0....0
,
0
,
0
)
}
# C

f

ψ
(

Ω

Ω

ω

)

(
B
{
X
}
C
)

{\displaystyle B\{G(\Gamma ^{\Gamma ^{X}})\}C=B\{X(1,0,0....0,0,0)\}C=f\_{\psi (\Omega ^{\Omega ^{\omega }})}(B\{X\}C)}

B
{
G
(

Γ

Γ

Γ

)
}
# C

f

ψ
(

Ω

Ω

Ω

)

(
B
{
X
}
C
)

{\displaystyle B\{G(\Gamma ^{\Gamma ^{\Gamma }})\}C=f\_{\psi (\Omega ^{\Omega ^{\Omega }})}(B\{X\}C)}

B
{
G
(

Γ

Γ

)
}
# C

f

ψ
(

Ω

Ω

)

(
B
{
X
}
C
)

{\displaystyle B\{G(\Gamma \_{\Gamma })\}C=f\_{\psi (\Omega \_{\Omega })}(B\{X\}C)}

B
{
G
(

G

I

(
0
)
)
}
# C

f

ψ
(

ψ

I

(
0
)
)

(
B
{
X
}
C
)

{\displaystyle B\{G(G\_{I}(0))\}C=f\_{\psi (\psi \_{I}(0))}(B\{X\}C)}

B
{
G
(
I
)
}
# C

f

ψ
(
I
)

(
B
{
X
}
C
)

{\displaystyle B\{G(I)\}C=f\_{\psi (I)}(B\{X\}C)}

B
{
G
(

I

I

)
}
# C

f

ψ
(

I

I

)

(
B
{
X
}
C
)

{\displaystyle B\{G(I\_{I})\}C=f\_{\psi (I\_{I})}(B\{X\}C)}

B
{
G
(

G

I
(
1
,
0
)

(
0
)
)
}
# C

f

ψ
(

ψ

I
(
1
,
0
)

(
0
)
)

(
B
{
X
}
C
)

{\displaystyle B\{G(G\_{I(1,0)}(0))\}C=f\_{\psi (\psi \_{I(1,0)}(0))}(B\{X\}C)}

B
{
Ψ
(

Γ

X

X

)
}
# C

f

C
(

Ω

ω

ω

)

(
B
{
X
}
C
)

{\displaystyle B\{\Psi (\Gamma ^{X^{X}})\}C=f\_{C(\Omega ^{\omega ^{\omega }})}(B\{X\}C)}

, The C in the subscript is the Catching Function.

B
{
Ψ
(

Ψ

1

(
Γ
)
)
}
# C

f

C
(

C

1

(
Ω
)
)

(
B
{
X
}
C
)

{\displaystyle B\{\Psi (\Psi \_{1}(\Gamma ))\}C=f\_{C(C\_{1}(\Omega ))}(B\{X\}C)}

B
{
Ψ
(

Ψ

1

(
Γ

)

Γ

)
}
# C

f

C
(

C

1

(
Ω

)

Ω

)

(
B
{
X
}
C
)

{\displaystyle B\{\Psi (\Psi \_{1}(\Gamma )^{\Gamma })\}C=f\_{C(C\_{1}(\Omega )^{\Omega })}(B\{X\}C)}

B
{
℧
(
℧
(
℧
(

Γ

100

2
,
0
)
,
0
)
,
0
)
}
# C

f

C
(
C
(
C
(

Ω

100

2
,
0
)
,
0
)
,
0
)

(
B
{
X
}
C
)

{\displaystyle B\{\mho (\mho (\mho (\Gamma \_{100}2,0),0),0)\}C=f\_{C(C(C(\Omega \_{100}2,0),0),0)}(B\{X\}C)}

Using Taranovsky's C.

f

Σ

0

(
n
# )

f

C
(
C
(
C
.
.
.
.
.
C
(
C
(
C
(
C
(

Ω

ω

,

Ω

ω

)
,
0
)
,
0
)
,
0
)
,
0.......
)

⏟

n C's

(
n
)

{\displaystyle f\_{\Sigma \_{0}}(n)=f\_{\underbrace {C(C(C.....C(C(C(C(\Omega \_{\omega },\Omega \_{\omega }),0),0),0),0.......)} \_{\text{n C's}}}(n)}

, There are simply n C's.

B
{
M
(
0
)
}
# C

f

P
(
0
)

(
B
{
X
}
C
)

{\displaystyle B\{M(0)\}C=f\_{P(0)}(B\{X\}C)}

B
{
M
(
0
)
2
}
# C

f

P
(
0
)
2

(
B
{
X
}
C
)

{\displaystyle B\{M(0)2\}C=f\_{P(0)2}(B\{X\}C)}