X-Sequence Nested Exponential Notation

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

$$B\{S\}C = f_{S}(B+C)$$

$$A\{S\}B = f_{S}(A+B)$$

$$B\{X\}C = f_{\omega}(B+C), $$B and C are positive finite integers and X is for X sequence nested Exponential notation.

$$ B\{X+1\}C = f_{\omega+1}(B\{X\}C)$$

$$B\{X+2\}C = f_{\omega+2}(B\{X\}C)$$

$$B\{X+X\}C = B\{2X\}C = f_{\omega2}(B\{X\}C)$$

$$B\{X+X+X\}C = B\{2X+X\}C = B\{3X\}C = f_{\omega3}(B\{X\}C)$$

$$B\{XX\}C = B\{X^{2}\}C = f_{\omega^{2}}(B\{X\}C)$$

$$B\{X^{2}X\}C = B\{X^{3}\}C = f_{\omega^{3}}(B\{X\}C)$$

$$B\{X^{X}\}C = f_{\omega^{\omega}}(B\{X\}C)$$

$$B\{X^{X+1}\}C = B\{X^{X}X\}C = f_{\omega^{\omega+1}}(B\{X\}C)$$

$$B\{X^{X+X}\}C = f_{\omega^{\omega+\omega}}(B\{X\}C)$$

$$B\{X^{X^2}\}C = f_{\omega^{\omega^2}} (B\{X\}C)$$

$$B\{X^{X^{X}}\}C = f_{\omega^{\omega^\omega}}(B\{X\}C)$$

$$B\{X^{X^{X^X}}\}C = f_{\omega^{\omega^{\omega^\omega}}}(B\{X\}C)$$

$$B\{X\uparrow\uparrow{X}\}C = B\{X(1,0)\}C = f_{\varepsilon_{0}}(B\{X\}C) = f_{\varphi(1,0)}(B\{X\}C)$$

$$B\{X(1,0,0)\}C = f_{\varphi(1,0,0)}(B\{X\}C) = f_{\Gamma_0}(B\{X\}C)$$

$$B\{X(1,0,0,0)\}C = f_{\varphi(1,0,0,0)}(B\{X\}C) = f_{Ack}(B\{X\}C)$$

$$B\{G(\Gamma^{\Gamma^X})\}C = B\{X(1,0,0....0,0,0)\}C = f_{\psi(\Omega^{\Omega^\omega})}(B\{X\}C)$$

$$ B\{G(\Gamma^{\Gamma^{\Gamma}})\}C = f_{\psi(\Omega^{\Omega^\Omega})}(B\{X\}C)$$

$$B\{G(\Gamma_\Gamma)\}C = f_{\psi(\Omega_\Omega)}(B\{X\}C)$$

$$B\{G(G_I(0))\}C = f_{\psi(\psi_{I}(0))}(B\{X\}C)$$

$$B\{G(I)\}C = f_{\psi(I)}(B\{X\}C)$$

$$B\{G(I_I)\}C = f_{\psi(I_I)}(B\{X\}C)$$

$$B\{G(G_{I(1,0)}(0))\}C = f_{\psi(\psi_{I(1,0)}(0))}(B\{X\}C)$$

$$B\{\Psi(\Gamma^{X^X})\}C = f_{C(\Omega^{\omega^\omega})}(B\{X\}C)$$, The C in the subscript is the Catching Function.

$$B\{\Psi(\Psi_{1}(\Gamma))\}C = f_{C(C_1(\Omega))}(B\{X\}C)$$

$$B\{\Psi(\Psi_{1}(\Gamma)^\Gamma)\}C = f_{C(C_{1}(\Omega)^\Omega)}(B\{X\}C)$$

$$B\{\mho(\mho(\mho(\Gamma_{100}2,0),0),0)\}C = f_{C(C(C(\Omega_{100}2,0),0),0)}(B\{X\}C)$$Using Taranovsky's C.

$$f_{\Sigma_0}(n) = f_{\underbrace{C(C(C.....C(C(C(C(\Omega_{\omega},\Omega_{\omega}),0),0),0),0.......)}_{\text{n C's}}}(n)$$, There are simply n C's.

$$B\{M(0)\}C = f_{P(0)}(B\{X\}C)$$

$$B\{M(0)2\}C = f_{P(0)2}(B\{X\}C)$$