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Bracket Notation

Bracket Notation FGH Exact value
a$2 $$f_1(a)$$ $$a^2$$ a$[0] $$f_2(a)$$ $$a^a$$
a$[1] $$f_3(a)$$ a$[2] $$f_4(a)$$
a$[[0]] $$f_{\omega}(a)$$ a$[0][[0]] = a$a[[0]] $$f_{\omega}(f_1(a))$$ a$[1][[0]] $$f_{\omega}(f_2(a))$$
a$[[0]][[0]] $$f_{\omega}(f_\omega(a))$$ a$[1[0]] $$f_{\omega+1}(a)$$
a$[[0][0]] $$f_{\omega*2}(a)$$ a$[[1]] $$f_{\omega^2}(a)$$
a$[[2]] $$f_{\omega^3}(a)$$ a$[[[0]]] $$f_{\omega^{\omega}}(a)$$
a$[[[[0]]]] $$f_{\omega^{\omega^{\omega}}}(a)$$ a$[[[...[[0]]...]] $$f_{\varepsilon_{0}}(a)$$

Extended Bracket Notation

Extended Bracket Notation FGH
$$[0]_2$$ $$\varepsilon_{0}$$
$$[[0]_2]$$ $$\varepsilon_{0}+1$$
$$[1[0]_2]$$ $$\varepsilon_{0}+2$$
$$[[0][0]_2]$$ $$\varepsilon_{0}+\omega$$
$$[[0]_2[0]_2]$$ $$\varepsilon_{0}*2$$
$$[[[0]_2]]$$ $$\omega^{\varepsilon_{0}+1}$$
$$[[[[0]_2]]]$$ $$\omega^{\omega^{\varepsilon_{0}+1}}$$
$$[1]_2$$ $$\varepsilon_{1}$$
$$[2]_2$$ $$\varepsilon_{2}$$
$$[[0]]_2$$ $$\varepsilon_{\omega}$$
$$[[0]_2]_2$$ $$\varepsilon_{\varepsilon_{0}}$$
$$[0]_3$$ $$\zeta_{0}$$
$$[[0]_3]$$ $$\zeta_{0}+1$$
$$[[[0]_3]]$$ $$\omega^{\zeta_{0}+1}$$
$$[[[[0]_3]]]$$ $$\omega^{\omega^{\zeta_{0}+1}}$$
$$[[0]_3]_2$$ $$\varepsilon_{\zeta_{0}+1}$$
$$[[[0]_3]_2]_2$$ $$\varepsilon_{\varepsilon_{\zeta_{0}+1}}$$
$$[1]_3$$ $$\zeta_{1}$$
$$[[0]_3]_3$$ $$\zeta_{\zeta_{1}}$$
$$[0]_4$$ $$\eta_{0}$$
$$[0]_{[0]}$$ $$\psi(\Omega^{\omega})$$
$$[0]_{[0]_{[0]}}$$ $$\psi(\Omega^{\vartheta(\Omega^{\omega})})$$
$$[0,1]$$

$$\psi(\Omega^{\Omega})$$

Linear Array Notation

Linear Array Notation FGH
$$[0,1]$$ $$\psi(\Omega^{\Omega})$$
$$[1,1]$$ $$\psi(\Omega^{\Omega})+1$$
$$[[0,1],1]$$ $$\omega^{\psi(\Omega^{\Omega})+1}$$
$$[[[0,1],1],1]$$ $$\omega^{\omega^{\psi(\Omega^{\Omega})+1}}$$
$$[[0,1]_2,1]$$ $$\psi(\Omega^{\Omega}+1)$$
$$[[0,1]_3,1]$$ $$\psi(\Omega^{\Omega}+2)$$
$$[[0,1]_{[0,1]},1]$$ $$\psi(\Omega^{\Omega}+\psi(\Omega^{\Omega}))$$
$$[0,2]$$ $$\psi(\Omega^{\Omega}+\Omega)$$
$$[0,[0]]$$ $$\psi(\Omega^{\Omega}+\Omega\omega)$$
$$[0,[0,1]]$$ $$\psi(\Omega^{\Omega}+\Omega\psi(\Omega^{\Omega}))$$
$$[0,[0,1]_2]$$ $$\psi(\Omega^{\Omega}+\Omega^{2})$$
$$[0,[0,1]_3]$$ $$\psi(\Omega^{\Omega}+\Omega^{3})$$
$$[0,[0,1]_4]$$ $$\psi(\Omega^{\Omega}+\Omega^{4})$$
$$[0,[0,1]_5]$$ $$\psi(\Omega^{\Omega}+\Omega^{5})$$
$$[0,[0,1]_{[0]}]$$ $$\psi(\Omega^{\Omega}+\Omega^{\omega})$$
$$[0,[0,1]_{[0,1]}]$$ $$\psi(\Omega^{\Omega}+\Omega^{\psi(\Omega^{\Omega})})$$
$$[0,0,1]$$ $$\psi(\Omega^{\Omega}2)$$
$$[0,1,1]$$ $$\psi(\Omega^{\Omega}2+\Omega)$$
$$[0,[0,0,1]_2,1]$$ $$\psi(\Omega^{\Omega}2+\Omega\psi(\Omega^{\Omega}2))$$
$$[0,0,2]$$ $$\psi(\Omega^{\Omega}3)$$
$$[0,0,[0,0,1]_2]$$ $$\psi(\Omega^{\Omega+1})$$
$$[0,0,[0,0,1]_3]$$ $$\psi(\Omega^{\Omega+2})$$
$$[0,0,[0,0,1]_{[0]}]$$ $$\psi(\Omega^{\Omega+\omega})$$
$$[0,0,[0,0,1]_{[0,0,1]}]$$ $$\psi(\Omega^{\Omega+\psi(\Omega^{\Omega}2)})$$
$$[0,0,0,1]$$ $$\psi(\Omega^{\Omega2})$$
$$[0,0,1,1]$$ $$\psi(\Omega^{\Omega2}2)$$
$$[0,0,[0,0,0,1]_2,1]$$ $$\psi(\Omega^{\Omega2+1})$$
$$[0,0,0,2]$$ $$\psi(\Omega^{\Omega3})$$
$$[0,0,0,[0,0,0,1]_2]$$ $$\psi(\Omega^{\Omega^2})$$
$$[0,0,0,[0,0,0,2]_2]$$ $$\psi(\Omega^{\Omega^22})$$
$$[0,0,0,[0,0,0,[0,0,0,1]_2]_2]$$ $$\psi(\Omega^{\Omega^2\psi(\Omega^{\Omega^2})})$$
$$[0,0,0,[0,0,0,1]_3]$$ $$\psi(\Omega^{\Omega^3})$$
$$[0,0,0,0,1]$$ $$\psi(\Omega^{\Omega^{\Omega}})$$
$$[0,0,0,1,1]$$ $$\psi(\Omega^{\Omega^{\Omega}2})$$
$$[0,0,0,[0,0,0,0,1],1]$$ $$\psi(\Omega^{\Omega^{\Omega}\psi(\Omega^{\Omega^{\Omega}})})$$
$$[0,0,0,[0,0,0,0,1]_2,1]$$ $$\psi(\Omega^{\Omega^{\Omega+1}})$$
$$[0,0,0,0,2]$$ $$\psi(\Omega^{\Omega^{\Omega2}})$$
$$[0,0,0,0,[0,0,0,0,1]_2]$$ $$\psi(\Omega^{\Omega^{\Omega^2}})$$
$$[0,0,0,0,0,1]$$ $$\psi(\Omega^{\Omega^{\Omega^{\Omega}}})$$
$$[0 \rightarrow 1]$$ $$\psi(\varepsilon_{\Omega+1})$$

Extended Array Notation

Extended Array Notation FGH
$$[0 \rightarrow 1]$$ $$\psi(\varepsilon_{\Omega+1})$$
$$[1 \rightarrow 1]$$ $$\psi(\varepsilon_{\Omega+1})+1$$
$$[[0 \rightarrow 1] \rightarrow 1]$$ $$\omega^{\psi(\varepsilon_{\Omega+1}+1)}$$
$$[[0 \rightarrow 1]_2 \rightarrow 1]$$ $$\psi(\varepsilon_{\Omega+1}+1)$$
$$[[0 \rightarrow 1]_{[0 \rightarrow 1]} \rightarrow 1]$$ $$\psi(\varepsilon_{\Omega+1}+\psi(\varepsilon_{\Omega+1}))$$
$$[0,1 \rightarrow 1]$$ $$\psi(\varepsilon_{\Omega+1}2)$$
$$[0,2 \rightarrow 1]$$ $$\psi(\varepsilon_{\Omega+1}3)$$
$$[0,[0 \rightarrow 1]_2 \rightarrow 1]$$ $$\psi(\varepsilon_{\Omega+1}^2)$$
$$[0,0,1 \rightarrow 1]$$ $$\psi(\varepsilon_{\Omega+1}^{\Omega})$$
$$[0,0,0,1 \rightarrow 1]$$ $$\psi(\varepsilon_{\Omega+1}^{\Omega^{\Omega}})$$
$$[0 \rightarrow 2]$$ $$\psi(\varepsilon_{\Omega+1}^{\varepsilon_{\Omega+1}})$$
$$[0 \rightarrow [0]]$$ $$\psi(\varepsilon_{\Omega+2})$$
$$[0 \rightarrow [0\rightarrow 1]]$$ $$\psi(\varepsilon_{\Omega+\psi(\varepsilon_{\Omega+1})})$$
$$[0 \rightarrow [0\rightarrow 1]_2]$$ $$\psi(\varepsilon_{\Omega*2})$$
$$[0 \rightarrow [0\rightarrow 1]_3]$$ $$\psi(\varepsilon_{\Omega*3})$$
$$[0 \rightarrow [0\rightarrow 1]_{[0\rightarrow 1]}]$$ $$\psi(\varepsilon_{\Omega*\psi(\varepsilon_{\Omega+1})})$$
$$[0 \rightarrow 0,1]$$ $$\psi(\varepsilon_{\Omega^{2}})$$
$$[0 \rightarrow 0,0,1]$$ $$\psi(\varepsilon_{\Omega^{\Omega}})$$
$$[0 \rightarrow 0,0,0,1]$$ $$\psi(\varepsilon_{\Omega^{\Omega^{\Omega}}})$$
$$[0 \rightarrow 0 \rightarrow 1]$$ $$\psi(\varepsilon_{\varepsilon_{\Omega+1}+1})$$
$$[0 \rightarrow_{\{2\}}1]$$ $$\vartheta(\zeta_{\Omega+1})$$
$$[0 \rightarrow_{\{3\}} 1]$$ $$\vartheta(\eta_{\Omega+1})$$
$$[0 \rightarrow_{\{[0]\}} 1]$$ $$\vartheta(\varphi(\omega,\Omega+1))$$
$$[0 \rightarrow_{\{\{[0]\}\}} 1]$$ $$\vartheta(\varphi(\varphi(\omega,\Omega+1),\Omega+1))$$
$$[0 \rightarrow_{\{0\}_2} 1]$$

$$\vartheta(\Omega_2)$$

Extended Array Notation part 2

Dollars Function FGH
$$[0 \rightarrow_{\{0\}_2} 1]$$ $$\vartheta(\Omega_2)$$
$$[1 \rightarrow_{\{0\}_2} 1]$$ $$\vartheta(\Omega_2)+1$$
$$[[0 \rightarrow_{\{0\}_2} 1] \rightarrow_{\{0\}_2} 1]$$ $$\vartheta(\Omega_2)*2$$
$$[[0 \rightarrow_{\{0\}_2} 1]_2 \rightarrow_{\{0\}_2} 1]$$ $$\varepsilon_{\vartheta(\Omega_2)}$$
$$[0,1 \rightarrow_{\{0\}_2} 1]$$ $$\Gamma_{\vartheta(\Omega_2)}$$
$$[0 \rightarrow_{\{0\}_2} 2]$$ $${\vartheta(\Omega_2,1)}$$
$$[0 \rightarrow_{\{0\}_2} 3]$$

$${\vartheta(\Omega_2,2)}$$

$$[0 \rightarrow_{\{0\}_2}[0 \rightarrow_{\{0\}_2}1]_2]$$

$$\vartheta(\Omega_2+1)$$

$$[0 \rightarrow_{\{0\}_2}[0 \rightarrow_{\{0\}_2}1]_3]$$ $$\vartheta(\Omega_2+2)$$
$$[0 \rightarrow_{\{0\}_2}0,1]$$ $$\vartheta(\Omega_2+\Omega)$$
$$[0 \rightarrow_{\{0\}_2}0 \rightarrow 1]$$ $$\vartheta(\Omega_2+\varepsilon_{\Omega+1})$$
$$[0 \rightarrow_{1\{0\}_2}1]$$ $$\vartheta(\Omega_2*2)$$
$$[0 \rightarrow_{2\{0\}_2}1]$$ $$\vartheta(\Omega_2*3)$$
$$[0 \rightarrow_{\{1\}_2}1]$$ $$\vartheta(\Omega_2^2)$$
$$[0 \rightarrow_{\{\{[0]\}\}_2}1]$$ $$\vartheta(\Omega_2^\Omega)$$
$$[0 \rightarrow_{\{0\}_3} 1]$$ $$\vartheta(\Omega_2^{\varepsilon_{\Omega+1}})$$
$$[0 \rightarrow_{\{0\}_4} 1]$$ $$\vartheta(\Omega_2^{\zeta_{\Omega+1}})$$
$$[0 \rightarrow_{\{0\}_[0]} 1]$$ $$\vartheta(\Omega_2^{\varphi(\omega, \Omega+1)})$$
$$[0 \rightarrow_{\{0\}_[0 \rightarrow_{\{0\}} 1]} 1]$$ $$\vartheta(\Omega_2^{\varphi(\varphi(\omega,\Omega+1), \Omega+1)})$$
$$[0 \rightarrow_{\{0,1\}} 1]$$ $$\vartheta(\Omega_2^{\Omega_2})$$
$$[0 \rightarrow_{\{0\rightarrow 1\}} 1]$$ $$\vartheta(\varepsilon_{\Omega_2+1})$$
$$[0 \rightarrow_{\{0\rightarrow_2 1\}} 1]$$ $$\vartheta(\zeta_{\Omega_2+1})$$
$$[0 \rightarrow_{\{0\rightarrow_{\{0\rightarrow 1\}} 1\}} 1]$$ $$\vartheta(\varphi(\omega,\Omega_2+1))$$
$$[0 \rightarrow_{\{0\rightarrow_{\{0\rightarrow_{\{0\rightarrow 1\}} 1\}} 1\}} 1]$$

$$\vartheta(\varphi(\varphi(\omega,\Omega_2+1),\Omega_2+1))$$

limit of Extended Array Notation $$\vartheta(\Omega_3)$$