FANDOM


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 \(\vartheta(\Omega_3)\)

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