10,822 Pages

## 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

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