FANDOM


Number and Bracket Notation

Order Type Approx Order Type using \(\omega\) FGH Notation using second entry ExE, \(\alpha\) for anything Dollar Function
1 1 1 \(2n\) \(\alpha\$[0]\)
n n n \(E2\text{#}\text{#}(n-1)\) \(\alpha\$[0]...[0]\) (n \([0]\)'s)
\(\omega\) \(\omega\) \([1]\) \(E\alpha\text{#}\text{#}\alpha\) \(\alpha\$[1]\)
\(\omega + n\) \(\omega + n\) \(n[1]\) \(E\alpha\text{#}\text{#}\alpha\text{#}\text{#}(n+1)\) \(\alpha\$[0]...[0][1]\) (n \([0]\)'s)
\(\omega 2\) \(\omega 2\) \([1][1]\) \(E\alpha\text{#}\text{#}\alpha\text{#}\text{#}\alpha\) \(\alpha\$[1][1]\)
\(\omega 2 + n\) \(\omega 2 + n\) \(n[1][1]\) \(E\alpha\text{#}\text{#}\alpha\text{#}\text{#}\alpha\text{#}\text{#}(n+1)\) \(\alpha\$[0]...[0][1][1]\) (n \([0]\)'s)
\(\omega 3\) \(\omega 3\) \([1][1][1]\) \(E\alpha\text{#}\text{#}\alpha\text{#}\text{#}\alpha\text{#}\text{#}\alpha\) \(\alpha\$[1][1][1]\)
\(\omega n\) \(\omega n\) \([1]...[1]\) (n [1]'s) \(E\alpha\text{#}\text{#}\text{#}(n+1)\) \(\alpha\$[1]...[1]\) (n [1]'s)
\(\omega n + m\) \(\omega n + m\) \(m[1]...[1]\) (n [1]'s) \(E\alpha\text{#}\text{#}\alpha ... \alpha\text{#}\text{#}\alpha\text{#}\text{#}(n+1)\) (n+1 ##) \(\alpha\$[0]...[0][1]...[1]\) (n [1]'s and m [0]'s)
\(\omega^2\) \(\omega^2\) \([2]\) \(E\alpha\text{#}\text{#}\text{#}\alpha\) \(\alpha\$[1]\)
\(\omega^2 n\) \(\omega^2 n\) \([2]...[2]\) (n [2]'s) \(E\alpha\text{#}\text{#}\text{#}\text{#}(n+1)\) \(\alpha\$[1]...[1]\) (n [1]'s)
\(\omega^3\) \(\omega^3\) \([3]\) \(E\alpha\text{#}\text{#}\text{#}\text{#}\alpha\) \(\alpha\$[2]\)
\(\omega^n\) \(\omega^n\) \([n]\) \(E\alpha\text{#}\text{^}\text{#}(n+1)\) \(\alpha\$[n-1]\)
\(\omega^nm\) \(\omega^nm\) \([n]...[n]\) (m [n]'s) \(E\alpha\text{#}...\text{#}(m+1)\) (n+1 \(\text{#}\)) \(\alpha\$[n-1]...[n-1]\) (m [n-1]'s)
\(\omega^\omega\) \(\omega^\omega\) \([ [1] ]\) \(E\alpha\text{#}\text{^}\text{#}\alpha\) \(\alpha\$[ [0] ]\)
\(\omega^{\omega 2}\) \(\omega^{\omega 2}\) \([ [1][1] ]\) \(\alpha\$[ [0][0] ]\)
\(\omega^{\omega n}\) \(\omega^{\omega n}\) \([ [1]...[1] ]\) (n [1]'s) \(\alpha\$[ [0]...[0] ]\) (n [0]'s)
\(\omega^{\omega^2}\) \(\omega^{\omega^2}\) \([ [2] ]\) \(\alpha\$[ [1] ]\)
\(\omega^{\omega^n}\) \(\omega^{\omega^n}\) \([ [n] ]\) \(\alpha\$[ [n-1] ]\)
\(\omega^{\omega^\omega}\) \(\omega^{\omega^\omega}\) \([ [ [1] ] ]\) \(\alpha\$[ [ [0] ] ]\)
\(\omega^{\omega^{\omega^\omega}}\) \(\omega^{\omega^{\omega^\omega}}\) \([ [ [ [1] ] ] ]\) \(\alpha\$[ [ [ [0] ] ] ]\)
\(^n\omega\) \(^n\omega\) \([ [...[ [1] ]...] ]\) (n nested) \(\alpha\$[ [...[ [0] ]...] ]\) (n nested)

Array-Type Notation

Order Type Approx Order Type using \(\omega\) FGH Notation using second entry Dollar Function
\(\varepsilon_0\) \(^\omega\omega\) \([_10]\) \(\alpha\$[ [0]_2 ]\)
\(\omega^{\varepsilon_0+1}\) \(\omega^{^\omega\omega +1}\) \([1 [_10] ]\) \(\alpha\$[ 1[0]_2 ]\)
\(\omega^{\varepsilon_0 2}\) \(\omega^{^\omega\omega 2}\) \([ [_10][_10] ]\) \(\alpha\$[ [ [0]_2 ][0]_2 ]\)
\(\omega^{\omega^{\varepsilon_0+1}}\) \(\omega^{\omega^{^\omega\omega +1}}\) \([ [1 [_10] ] ]\) \(\alpha\$[ [ 1[0]_2 ][0]_2 ]\)
\(\omega^{\omega^{\varepsilon_0 2}}\) \(^{\omega +1}\omega\) \([ [ [_10][_10] ] ]\) \(\alpha\$[ [ [ [0]_2 ][0]_2 ][0]_2 ]\)
\(\omega^{\omega^{\omega^{\varepsilon_0+1}}}\) \(\omega^{\omega^{\omega^{^\omega\omega +1}}}\) \([ [ [1 [_10] ] ] ]\) \(\alpha\$[ [ [ 1[0]_2 ][0]_2 ][0]_2 ]\)
\(\omega^{\omega^{\omega^{\omega^{\varepsilon_0+1}}}}\) \(\omega^{\omega^{\omega^{\omega^{^\omega\omega +1}}}}\) \([ [ [ [1 [_10] ] ] ] ]\) \(\alpha\$[ [ [ [ 1[0]_2 ][0]_2 ][0]_2 ][0]_2 ]\)
\(\varepsilon_1\) \(^{\omega 2}\omega\) \([_11]\) \(\alpha\$[ [0]_2[0]_2 ]\)
\(\omega^{\varepsilon_1+1}\) \(\omega^{^{\omega 2}\omega +1}\) \([1 [_11] ]\)
\(\omega^{\omega^{\varepsilon_1+1}}\) \(\omega^{\omega^{^{\omega 2}\omega +1}}\) \([ [1 [_11] ] ]\)
\(\varepsilon_n\) \(^{\omega (n+1)}\omega\) \([_1n]\)
\(\varepsilon_\omega\) \(^{\omega^2}\omega\) \([_1 [1] ]\)
\(\varepsilon_{\omega + 1}\) \(^{\omega^2+\omega}\omega\) \([_1 1[1] ]\)
\(\varepsilon_{\omega + n}\) \(^{\omega^2+\omega n}\omega\) \([_1 n[1] ]\)
\(\varepsilon_{\omega 2}\) \(^{\omega^22}\omega\) \([_1 [1][1] ]\)
\(\varepsilon_{\omega n}\) \(^{\omega^2n}\omega\) \([_1 [1]...[1] ]\) (n [1]'s)
\(\varepsilon_{\omega^2}\) \(^{\omega^3}\omega\) \([_1 [2] ]\)
\(\varepsilon_{\omega^n}\) \(^{\omega^{n+1}}\omega\) \([_1 [n] ]\)
\(\varepsilon_{\omega^\omega}\) \(^{\omega^\omega}\omega\) \([_1 [ [1] ] ]\)
\(\varepsilon_{\omega^{\omega^n}}\) \(^{\omega^{\omega^n}}\omega\) \([_1 [ [n] ] ]\)
\(\varepsilon_{\omega^{\omega^\omega}}\) \(^{\omega^{\omega^\omega}}\omega\) \([_1 [ [ [1] ] ] ]\)
\(\varepsilon_{^n\omega}\) \(^{^n\omega}\omega\) \([_1 [ [...[ [1] ]...] ]\) (n nested)
\(\varepsilon_{\varepsilon_0}\) \(^{^\omega\omega}\omega\) \([_1 [_10]\)
\(\varepsilon_{\varepsilon_{\varepsilon_0}}\) \(^{^{^\omega\omega}\omega}\omega\) \([_1 [_1 [_10] ]\)
\(\varepsilon_{..._{\varepsilon_0}}\) (n nested) \(\omega \uparrow\uparrow\uparrow n+1\) \([_1 [_1 ...[_1 [_10] ]... ]\) (n nested)
\(\zeta_0\) \(\omega \uparrow\uparrow\uparrow \omega\) \([_20]\)
\(\zeta_{\zeta_0}\) \(\omega \uparrow\uparrow\uparrow \omega \uparrow\uparrow\uparrow \omega\) \([_2 [_20] ]\)
\(\zeta_{\zeta_{\zeta_0}}\) \(\omega \uparrow\uparrow\uparrow \omega \uparrow\uparrow\uparrow \omega \uparrow\uparrow\uparrow \omega\) \([_2 [_2 [_20] ] ]\)
\(\eta_0\) \(\omega \uparrow\uparrow\uparrow\uparrow \omega\) \([_30]\)
\(\varphi(n,0)\) \(\omega \uparrow^{n+1} \omega\) \([_n0]\)
\(\varphi(\omega,0)\) \(\omega \uparrow^\omega \omega\) \([_{[1]}0]\)
\(\varphi(n,\varphi(\omega,0)+1)\) \((\omega \uparrow^\omega \omega) \uparrow^{n+1} \omega\) \([_n1 [_{[1]}0] ]\)
\(\varphi(\omega,n)\) \(\omega \uparrow^\omega (\omega n+1)\) \([_{[1]}n]\)
\(\varphi(n,\varphi(\omega,m)+1)\) \((\omega \uparrow^\omega (\omega m+1)) \uparrow^{n+1} \omega\) \([_n1 [_{[1]}m] ]\)
\(\varphi(\omega,\omega)\) \(\omega \uparrow^\omega \omega^2\) \([_{[1]} [1] ]\)
\(\varphi(\omega^\omega,0)\) \(\omega \uparrow^{\omega^\omega} \omega\) \([_{[ [1] ]}0]\)
\(\varphi(\omega^{\omega^\omega},0)\) \(\omega \uparrow^{\omega^{\omega^\omega}} \omega\) \([_{[ [ [1] ] ]}0]\)
\(\varphi(\varepsilon_0,0)\) \(\omega \uparrow^{^\omega\omega} \omega\) \([_{[_10]}0]\)
\(\varphi(\zeta_0,0)\) \(\omega \uparrow^{\omega \uparrow\uparrow\uparrow \omega} \omega\) \([_{[_20]}0]\)
\(\varphi(\eta_0,0)\) \(\omega \uparrow^{\omega \uparrow\uparrow\uparrow\uparrow \omega} \omega\) \([_{[_30]}0]\)
\(\varphi(\varphi(\omega,0),0)\) \(\omega \uparrow^{\omega \uparrow^\omega \omega} \omega\) \([_{[_{[1]}0]}0]\)
\(\varphi(\varphi(\varphi(\omega,0),0),0)\) \(\omega \uparrow^{\omega \uparrow^{\omega \uparrow^\omega \omega} \omega} \omega\) \([_{[_{[_{[1]}0]}0]}0]\)
\(\varphi(\varphi(\varphi(\varphi(\omega,0),0),0),0)\) \(\omega \uparrow^{\omega \uparrow^{\omega \uparrow^{\omega \uparrow^\omega \omega} \omega} \omega} \omega\) \([_{[_{[_{[_{[1]}0]}0]}0]}0]\)

Multi-Entry Notation

Order Type Approx Order Type using \(\omega\) FGH Notation using second entry
\(\Gamma_0\) \(\omega \{\{1\}\} \omega\) \([_{0/1}0]\)
\(\varepsilon_{\Gamma_0+1}\) \((\omega \{\{1\}\} \omega) \uparrow\uparrow \omega\) \([_11 [_{0/1}0] ]\)
\(\varphi(n,\Gamma_0 + 1)\) \((\omega \{\{1\}\} \omega) \uparrow^{n+1} \omega\) \([_n1 [_{0/1}0]\)
\(\varphi(\omega,\Gamma_0 + 1)\) \((\omega \{\{1\}\} \omega) \uparrow^\omega \omega\) \([_{[1]}1 [_{0/1}0]\)
\(\varphi(\varphi(\omega,0),\Gamma_0 + 1)\) \((\omega \{\{1\}\} \omega) \uparrow^{\omega \uparrow^\omega \omega} \omega\) \([_{[_{[1]}0]}1 [_{0/1}0]\)
\(\varphi(\varphi(\varphi(\omega,0),0),\Gamma_0 + 1)\) \((\omega \{\{1\}\} \omega) \uparrow^{\omega \uparrow^{\omega \uparrow^\omega \omega} \omega} \omega\) \([_{[_{[_{[1]}0]}0]}1 [_{0/1}0]\)
\(\varphi(\Gamma_0,1)\) \(\omega \uparrow^{\omega \{\{1\}\} (\omega - 1)} (\omega 2)\) \([_{[_{0/1}0]}1]\)
\(\varphi(\Gamma_0,\Gamma_0)\) \(\omega \uparrow^{\omega \{\{1\}\} (\omega - 1)} \omega \{\{1\}\} \omega\) \([_{[_{0/1}0]} [_{0/1}0]]\)
\(\varphi(\Gamma_0 + 1,0)\) \(\omega \uparrow^{\omega \{\{1\}\} (\omega - 1) + 1} \omega\) \([_{1 [_{0/1}0]}0\)
\(\varphi(\varepsilon_{\Gamma_0+1},0)\) \(\omega \uparrow^{(\omega \{\{1\}\} (\omega - 1)) \uparrow\uparrow \omega} \omega\) \([_{[_11 [_{0/1}0] ]}0\)
\(\varphi(\varphi(\omega,\Gamma_0+1),0)\) \(\omega \uparrow^{(\omega \{\{1\}\} (\omega - 1)) \uparrow^\omega \omega} \omega\) \([_{[_{[0]}1 [_{0/1}0] ]}0\)
\(\varphi(\varphi(\Gamma,1),0)\) \(\omega \uparrow^{\omega \uparrow^{\omega \{\{1\}\} (\omega - 2)} (\omega 2)} \omega\) \([_{[_{[_{0/1}0]}1]}0]\)
\(\varphi(\varphi(\varphi(\Gamma,1),0),0)\) \(\omega \uparrow^{\omega \uparrow^{\omega \uparrow^{\omega \{\{1\}\} (\omega - 3)} (\omega 2)} \omega} \omega\) \([_{[_{[_{[_{0/1}0]}1]}0]}0]\)
\(\Gamma_1\) \(\omega \{\{1\}\} (\omega 2)\) \([_{0/1}1]\)
\(\Gamma_n\) \(\omega \{\{1\}\} (\omega n+1)\) \([_{0/1}n]\)
\(\Gamma_{\varphi(\omega,0)}\) \(\omega \{\{1\}\} \omega \uparrow^\omega \omega\) \([_{0/1} [_{[1]}0] ]\)
\(\Gamma_{\varphi(\varphi(\omega,0),0)}\) \(\omega \{\{1\}\} \omega \uparrow^{\omega \uparrow^\omega \omega} \omega\) \([_{0/1} [_{[_{[1]}0]}0] ]\)
\(\Gamma_{\Gamma_0}\) \(\omega \{\{1\}\} \omega \{\{1\}\} \omega\) \([_{0/1} [_{0/1}0] ]\)
\(\varphi(1,1,0)\) \(\omega \{\{2\}\} \omega\) \([_{1/1}0]\)
\(\Gamma_{\varphi(1,1,0)+1}\) \(\omega \{\{2\}\} (\omega + 1)\) \([_{0/1}1 [_{1/1}0] ]\)
\(\Gamma_{\Gamma_{\varphi(1,1,0)+1}}\) \(\omega \{\{2\}\} (\omega + 2)\) \([_{0/1} [_{0/1}1 [_{1/1}0] ] ]\)
\(\varphi(1,1,1)\) \(\omega \{\{2\}\} (\omega 2)\) \([_{1/1}1]\)
\(\varphi(1,2,0)\) \(\omega \{\{3\}\} \omega\) \([_{2/1}0]\)
\(\varphi(1,\omega,0)\) \(\omega \{\{\omega\}\} \omega\) \([_{[1]/1}0]\)
\(\varphi(1,\varphi(1,0,0),0)\) \(\omega \{\{\omega \{\{1\}\} \omega\}\} \omega\) \([_{[_{0/1}0]/1}0]\)
\(\varphi(2,0,0)\) \(\omega \{\{\{1\}\}\} \omega\) \([_{0/2}0]\)
\(\varphi(\omega,0,0)\) \(\{\omega,\omega,1,\omega\}\) \([_{0/[1]}0]\)
\(\varphi(\varphi(1,0,0),0,0)\) \(\{\omega,\omega,1,\omega \{\{1\}\} \omega\}\) \([_{0/[_{0/1}0]}0]\)
\(\varphi(1,0,0,0)\) \(\{\omega,\omega,1,1,2\}\) \([_{0/0/1}0]\)
\(\varphi(\omega,0,0,0)\) \(\{\omega,\omega,1,1,\omega\}\) \([_{0/0/[0]}0]\)
\(\varphi(\varphi(1,0,0,0),0,0,0)\) \(\{\omega,\omega,1,1,\{\omega,\omega,1,1,2\}\}\) \([_{0/0/[_{0/0/1}0]}0]\)
\(\varphi(1,0,0,0,0)\) \(\{\omega,\omega,1,1,1,2\}\) \([_{0/0/0/1}0]\)
\(\varphi(1,0,...,0)\) (a entries) \(\{\omega,\omega,1,...,1,2\}\) (a+1 entries) \([_{0...0/1}0]\) (a-1 entries)

Hyper-Entry Notation

Order Type Approx Order Type using \(\omega\) FGH Notation using second entry
\(\vartheta(\Omega^\omega)\) \(\{\omega,\omega + 2 (1) 2\}\) \([_{0/_11}0]\)
\(\vartheta(\Omega^\omega + \Omega^a)\) \(\{\omega,\omega,1,...,1,2 (1) 2\}\) (a+1 entries) \([_{0...0/1/_11}0]\) (a zeroes)
\(\vartheta(\Omega^\omega a)\) \(\{\omega,\omega + 2 (1) a+1\}\) \([_{0/_1a}0]\)
\(\vartheta(\Omega^\omega \Gamma_0)\) \(\{\omega,\omega + 2 (1) \{\omega,\omega,1,2\}\}\) \([_{0/_1[_{0/1}0]}0]\)
\(\vartheta(\Omega^\omega \vartheta(\Omega^\omega))\) \(\{\omega,\omega + 2 (1) \{\omega,\omega + 2 (1) 2\}\}\) \([_{0/_1[_{0/_11}0]}0]\)
\(\vartheta(\Omega^{\omega + 1})\) \(\{\omega,\omega (1) 1,2\}\) \([_{0/_10/1]}0]\)
\(\vartheta(\Omega^{\omega + a})\) \(\{\omega,\omega (1) 1...1,2\}\) \([_{0/_10/0...0/0/1]}0]\) (a 0's)
\(\vartheta(\Omega^{\omega 2})\) \(\{\omega,\omega (1)(1) 2\}\) \([_{0/_10/_11}0]\)
\(\vartheta(\Omega^{\omega 3})\) \(\{\omega,\omega (1)(1)(1) 2\}\) \([_{0/_10/_10/_11}0]\)
\(\vartheta(\Omega^{\omega n})\) \(\{\omega,\omega (1)(1)...(1)(1) 2\}\) (n seperators) \([_{0/_10...0/_10/_11}0]\) (n 0's)
\(\vartheta(\Omega^{\omega^2})\) \(\{\omega,\omega (2) 2\}\) \([_{0/_21}0]\)
\(\vartheta(\Omega^{\omega^2 2})\) \(\{\omega,\omega (2)(2) 2\}\) \([_{0/_20/_21}0]\)
\(\vartheta(\Omega^{\omega^2 n})\) \(\{\omega,\omega (2)(2)...(2)(2) 2\}\) (n seperators) \([_{0/_20...0/_20/_21}0]\) (n 0's)
\(\vartheta(\Omega^{\omega^3})\) \(\{\omega,\omega (3) 2\}\) \([_{0/_31}0]\)
\(\vartheta(\Omega^{\omega^n})\) \(\{\omega,\omega (n) 2\}\) \([_{0/_n1}0]\)
\(\vartheta(\Omega^{\omega^\omega})\) \(\{\omega,\omega [1 [2] 2] 2\}\) \([_{0/_{[1]}1}0]\)
\(\vartheta(\Omega^{\omega^{\omega^n}})\) \(\{\omega,\omega [1 [n+1] 2] 2\}\) \([_{0/_{[n]}1}0]\)
\(\vartheta(\Omega^{\omega^{\omega^\omega}})\) \(\{\omega,\omega [1 [1,2] 2] 2\}\) \([_{0/_{[ [1] ]}1}0]\)
\(\vartheta(\Omega^{\varepsilon_0})\) \(\{\omega,\omega [1 / 2] 2\}\) \([_{0/_{[_10]}1}0]\)
\(\vartheta(\Omega^{\varepsilon_{\varepsilon_0}})\) \(\{\omega,\omega [1 / 1 [1 / 2] 2] 2\}\) \([_{0/_{[_1 [_10] ]}1}0]\)

Tetrational-Omega Growth Notation

\(\vartheta(\varepsilon_{\Omega+1})\)?

Ultimate-Omega Growth Notation

\(\vartheta(\Omega_2)\)?

Omega-Type Notation

\(\vartheta(\Omega_\Omega)\)?

Impossible-Omega Notation

\(\psi_I(0)\)?

Inomegable Notation

\(\psi_{I(1,0)}(0)\)?

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