FANDOM


Rules are here.

Up to \(\psi(\psi_I(0))\)

Dollar FGH
\([0 \rightarrow 1]\) \(\vartheta(\varepsilon_{\Omega+1})\)
\([0 \rightarrow_2 1]\) \(\vartheta(\varepsilon_{\Omega+2})\)
\([0 \rightarrow_{[0]} 1]\) \(\vartheta(\varepsilon_{\Omega+\omega})\)
\([0 \rightarrow_{[0]\text{&}1} 1]\) \(\vartheta(\varepsilon_{\Omega2})\)
\([0 \rightarrow_{[0]\text{&}1[0]\text{&}1} 1]\) \(\vartheta(\varepsilon_{\Omega3})\)

\([0 \rightarrow_{[1]\text{&}1} 1]\)

\(\vartheta(\varepsilon_{\Omega\omega})\)
\([0 \rightarrow_{[[0]\text{&}1]\text{&}1} 1]\) \(\vartheta(\varepsilon_{\Omega^2})\)
\([0 \rightarrow_{[[1]\text{&}1]\text{&}1} 1]\) \(\vartheta(\varepsilon_{\Omega^\omega})\)
\([0 \rightarrow_{[[[0]\text{&}1]\text{&}1]\text{&}1} 1]\) \(\vartheta(\varepsilon_{\Omega^\Omega})\)
\([0 \rightarrow_{[0]_2\text{&}1} 1]\) \(\vartheta(\varepsilon_{\varepsilon_{\Omega+1}+1})\)
\([0 \rightarrow_{[0]_3\text{&}1} 1]\) \(\vartheta(\zeta_{\Omega+1})\)
\([0 \rightarrow_{[0]_{[0]}\text{&}1} 1]\) \(\vartheta(\varphi(\omega,\Omega+1))\)
\([0 \rightarrow_{[0]_{[0]\text{&}1}\text{&}1} 1]\) \(\vartheta(\varphi(\Omega,1))\)
\([0 \rightarrow_{[0]_{[0]_{[0]\text{&}1}\text{&}1}\text{&}1} 1]\) \(\vartheta(\varphi(\varphi(\Omega,1),1))\)
\([0 \rightarrow_{[0,1]\text{&}1} 1]\) \(\vartheta(\Omega_2)\)
\([0 \rightarrow_{[1,1]\text{&}1} 1]\) \(\vartheta(\Omega_2\omega)\)
\([0 \rightarrow_{[[0]\text{&}1,1]\text{&}1} 1]\) \(\vartheta(\Omega_2\Omega)\)
\([0 \rightarrow_{[[0,1]\text{&}1,1]\text{&}1} 1]\) \(\vartheta(\Omega_2^2)\)
\([0 \rightarrow_{[[[0,1]\text{&}1,1]\text{&}1,1]\text{&}1} 1]\) \(\vartheta(\Omega_2^{\Omega_2})\)
\([0 \rightarrow_{[[0,1]_2\text{&}1,1]\text{&}1} 1]\) \(\vartheta(\varepsilon_{\Omega_2+1})\)
\([0 \rightarrow_{[0,2]\text{&}1} 1]\) \(\vartheta(\Omega_3)\)
\([0 \rightarrow_{[0,3]\text{&}1} 1]\) \(\vartheta(\Omega_4)\)
\([0 \rightarrow_{[0,[0]]\text{&}1} 1]\) \(\vartheta(\Omega_\omega)\)
\([0 \rightarrow_{[0,[0]\text{&}1]\text{&}1} 1]\) \(\vartheta(\Omega_\Omega)\)

\([0 \rightarrow_{[0,[0,[0]\text{&}1]\text{&}1]\text{&}1} 1]\)

\(\vartheta(\Omega_{\Omega_\Omega})\)

Beyond \(\psi(\psi_I(0))\)

Dollar FGH HAN
\([0 \rightarrow_{[0,[0,1]_2\text{&}1]\text{&}1} 1]\)

\(\psi(\psi_I(0))\)

\([[_{<1,2>}1]]\)
\([0 \rightarrow_{1[0,[0,1]_2\text{&}1]\text{&}1} 1]\) \(\psi(\varepsilon_{\psi_I(0)+1})\) \([[_{<1,2>}1,1,1,1,2]]\)
\([0 \rightarrow_{[1,[0,1]_2\text{&}1]\text{&}1} 1]\) \(\psi(\varepsilon_{\psi_I(0)\omega})\)
\([0 \rightarrow_{[[0]\text{&}1,[0,1]_2\text{&}1]\text{&}1} 1]\) \(\psi(\varepsilon_{\psi_I(0)\Omega})\)
\([0 \rightarrow_{[[[0,1]_2\text{&}1,[0,1]_2\text{&}1]\text{&}1} 1]\) \(\psi(\varepsilon_{\psi_I(0)^2})\)
\([0 \rightarrow_{[[[[0,1]_2\text{&}1],[0,1]_2\text{&}1]\text{&}1} 1]\) \(\psi(\varepsilon_{\varepsilon_{\psi_I(0)+1}})\)
\([0 \rightarrow_{[[[0,1]_3\text{&}1,[0,1]_2\text{&}1]\text{&}1} 1]\) \(\psi(\zeta_{\psi_I(0)+1})\) \([[_{<1,2>}1,1,1,1,1,2]]\)
\([0 \rightarrow_{[0,1[0,1]_2\text{&}1]\text{&}1} 1]\) \(\psi(\Omega_{\psi_I(0)+1})\) \([[_{<1,2>}1(1)2]]\)
\([0 \rightarrow_{[0,[0,1]_2\text{&}1[0,1]_2\text{&}1]\text{&}1} 1]\) \(\psi(\Omega_{\psi_I(0)2})\) \([[_{<[_{<1,2>}1],2>}1]]\)
\([0 \rightarrow_{[0,[[0,1]_2\text{&}1]]\text{&}1} 1]\) \(\psi(\Omega_{\psi_I(0)\omega})\) \([[_{<[_{<1,2>}1,1,2],2>}1]]\)
\([0 \rightarrow_{[0,[1,1]_2\text{&}1]\text{&}1} 1]\) \(\psi(\Omega_{\varepsilon_{\psi_I(0)+1}})\) \([[_{<[_{<1,2>}1,1,1,1,2],2>}1]]\)
\([0 \rightarrow_{[0,[0,1]_3\text{&}1]\text{&}1} 1]\) \(\psi(\Omega_{\zeta_{\psi_I(0)+1}})\) \([[_{<[_{<1,2>}1,1,1,1,1,2],2>}1]]\)
\([0 \rightarrow_{[0,0,1]\text{&}1} 1]\) \(\psi(\Omega_{\Omega_{\psi_I(0)+1}})\) \([[_{<[_{<2,2>}1],2>}1]]\)
\([0 \rightarrow_{[0 \rightarrow 1]\text{&}1} 1]\) \(\psi(\psi_I(1))\) \([[_{<1,3>}1]]\)
\([0 \rightarrow_{[0,1 \rightarrow 1]\text{&}1} 1]\) \(\psi(\Omega_{\psi_I(1)+1})\)
\([0 \rightarrow_{[0,0,1 \rightarrow 1]\text{&}1} 1]\) \(\psi(\Omega_{\Omega_{\psi_I(1)+1}})\)
\([0 \rightarrow_{[0 \rightarrow 2]\text{&}1} 1]\) \(\psi(\psi_I(2))\) \([[_{<1,4>}1]]\)
\([0 \rightarrow_{[0 \rightarrow [0]]\text{&}1} 1]\) \(\psi(\psi_I(\omega))\) \([[_{<1,[1]>}1]]\)
\([0 \rightarrow_{[0 \rightarrow [0]\text{&}1]\text{&}1} 1]\) \(\psi(\psi_I(\Omega))\) \([[_{<1,[_21]>}1]]\)
\([0 \rightarrow_{[0 \rightarrow [0,1]\text{&}1]\text{&}1} 1]\) \(\psi(\psi_I(\Omega_2))\) \([[_{<1,[_31]>}1]]\)
\([0 \rightarrow_{[0 \rightarrow [0,1]_2\text{&}1]\text{&}1} 1]\) \(\psi(\psi_I(\psi_I(0)))\)
\([0 \rightarrow_{[0 \rightarrow [0 \rightarrow 1]\text{&}1]\text{&}1} 1]\) \(\psi(\psi_I(\psi_I(1)))\)
\([0 \rightarrow_{[0 \rightarrow [0 \rightarrow [0 \rightarrow 1]\text{&}1]\text{&}1]\text{&}1} 1]\) \(\psi(\psi_I(\psi_I(\psi_I(1))))\)
\([0 \rightarrow_{[0 \rightarrow [0 \rightarrow 1]_2\text{&}1]\text{&}1} 1]\) \(\psi(\psi_I(I))\) \([[_{<1,1,2>}1]]\)
\([0 \rightarrow_{[0 \rightarrow [0 \rightarrow 1]_2\text{&}1[0 \rightarrow 1]_2\text{&}1]\text{&}1} 1]\) \(\psi(\psi_I(I2))\) \([[_{<1,2,2>}1]]\)

\([0 \rightarrow_{[0 \rightarrow [[0 \rightarrow 1]_2\text{&}1]]\text{&}1} 1]\)

\(\psi(\psi_I(I\omega))\) \([[_{<1,[1],2>}1]]\)
\([0 \rightarrow_{[0 \rightarrow [0 \rightarrow 1]_3\text{&}1]\text{&}1} 1]\) \(\psi(\psi_I(I^2))\)

\([[_{<1,1,3>}1]]\)

\([0 \rightarrow_{[0 \rightarrow [0 \rightarrow 1]_{[0]\text{&}1}\text{&}1]\text{&}1} 1]\) \(\psi(\psi_I(I^\Omega))\) \([[_{<1,1,[_21]>}1]]\)
\([0 \rightarrow_{[0 \rightarrow [0 \rightarrow 1]_{[0,1]_2\text{&}1}\text{&}1]\text{&}1} 1]\) \(\psi(\psi_I(I^{\psi_I(0)}))\) \([[_{<1,1,[_{<1,2>}1]>}1]]\)
\([0 \rightarrow_{[0 \rightarrow [0 \rightarrow 1]_{[0\rightarrow 1]\text{&}1}\text{&}1]\text{&}1} 1]\)

\(\psi(\psi_I(I^{\psi_I(1)}))\)

\([[_{<1,1,[_{<1,3>}1]>}1]]\)
\([0 \rightarrow_{[0 \rightarrow 0,1]\text{&}1} 1]\) \(\psi(\psi_I(I^I))\) \([[_{<1,1,1,2>}1]]\)
\([0 \rightarrow_{[0 \rightarrow 0 \rightarrow 1]\text{&}1} 1]\) \(\psi(\psi_I(\varepsilon_{I+1}))\) \([[_{<1,1,1,[1]>}1]]\)
\([0 \rightarrow_{[0 \rightarrow_2 1]\text{&}1} 1]\) \(\psi(\psi_I(\varepsilon_{I+2}))\) \([[_{<1,1,1,[1,2]>}1]]\)
\([0 \rightarrow_{[0 \rightarrow_{[0]\text{&}1} 1]\text{&}1} 1]\) \(\psi(\psi_I(\varepsilon_{I+\Omega}))\) \([[_{<1,1,1,[_21]>}1]]\)
\([0 \rightarrow_{[0]\text{&}2} 1]\)

\(\psi(\psi_I(\varepsilon_{I2}))\)

\([[_{<1,1,1,1,2>}1]]\)

Level 2 brackets

Dollar FGH HAN
\([0 \rightarrow_{[0]\text{&}2} 1]\) \(\psi(\psi_I(\varepsilon_{I2}))\) \([[_{<1,1,1,1,2>}1]]\)
\([0 \rightarrow_{[0]\text{&}2[0]\text{&}2} 1]\) \(\psi(\varepsilon_{\psi_I(\varepsilon_{I2})2})\)
\([0 \rightarrow_{[1]\text{&}2} 1]\) \(\psi(\varepsilon_{\psi_I(\varepsilon_{I2})\omega})\)
\([0 \rightarrow_{[[0]\text{&}1]\text{&}2} 1]\) \(\psi(\varepsilon_{\psi_I(\varepsilon_{I2})\Omega})\)
\([0 \rightarrow_{[[0]\text{&}2]\text{&}2} 1]\) \(\psi(\varepsilon_{\psi_I(\varepsilon_{I2})^2})\)
\([0 \rightarrow_{[0]_2\text{&}2} 1]\) \(\psi(\varepsilon_{\varepsilon_{\psi_I(\varepsilon_{I2})+1}+1})\)
\([0 \rightarrow_{[0]_3\text{&}2} 1]\) \(\psi(\zeta_{\psi_I(\varepsilon_{I2})})\)
\([0 \rightarrow_{[0,1]\text{&}2} 1]\) \(\psi(\Omega_{\psi_I(\varepsilon_{I2})+1})\)
\([0 \rightarrow_{[0,2]\text{&}2} 1]\) \(\psi(\Omega_{\psi_I(\varepsilon_{I2})+2})\)
\([0 \rightarrow_{[0,[0]\text{&}2]\text{&}2} 1]\) \(\psi(\Omega_{\psi_I(\varepsilon_{I2})2})\)
\([0 \rightarrow_{[0,[0]_2\text{&}2]\text{&}2} 1]\) \(\psi(\Omega_{\varepsilon_{\psi_I(\varepsilon_{I2})}})\)
\([0 \rightarrow_{[0,[0,1]\text{&}2]\text{&}2} 1]\) \(\psi(\Omega_{\Omega_{\psi_I(\varepsilon_{I2})}})\)
\([0 \rightarrow_{[0,[0,1]_2\text{&}2]\text{&}2} 1]\) \(\psi(\psi_I(\varepsilon_{I2}+1))\)
\([0 \rightarrow_{[0 \rightarrow 1]\text{&}2} 1]\) \(\psi(\psi_I(\varepsilon_{I2}+2))\)
\([0 \rightarrow_{[0 \rightarrow 2]\text{&}2} 1]\) \(\psi(\psi_I(\varepsilon_{I2}+3))\)
\([0 \rightarrow_{[0 \rightarrow [0 \rightarrow 1]_2\text{&}2]\text{&}2} 1]\) \(\psi(\psi_I(\varepsilon_{I2}+I))\)
\([0 \rightarrow_{[0 \rightarrow [1 \rightarrow 1]_2\text{&}2]\text{&}2} 1]\) \(\psi(\psi_I(\varepsilon_{I2}+\varepsilon_{I+1}))\)
\([0 \rightarrow_{[0 \rightarrow [[0 \rightarrow 1]_2\text{&}2 \rightarrow 1]_2\text{&}2]\text{&}2} 1]\) \(\psi(\psi_I(\varepsilon_{I2}2))\)
\([0 \rightarrow_{[0 \rightarrow [0,1\rightarrow 1]_2\text{&}2]\text{&}2} 1]\) \(\psi(\psi_I(\varepsilon_{I2+1}))\)
\([0 \rightarrow_{[0 \rightarrow [0 \rightarrow 2]_2\text{&}2]\text{&}2} 1]\) \(\psi(\psi_I(\varepsilon_{I3}))\)
\([0 \rightarrow_{[0 \rightarrow [0 \rightarrow 1]_3\text{&}2]\text{&}2} 1]\) \(\psi(\psi_I(\zeta_{I+1}))\) \([[_{<1,1,1,1,[1]>}1]]\)
\([0 \rightarrow_{[0 \rightarrow 0,1]\text{&}2} 1]\) \(\psi(\psi_{\Omega_{I+1}}(\Omega_{I+1}))\) \([[_{<1(1)<_21>>}1]]\)

\([0 \rightarrow_{[0 \rightarrow 0,2]\text{&}2} 1]\)

\(\psi(\psi_{\Omega_{I+2}}(\Omega_{I+2}))\) \([[_{<1(1)<_22>>}1]]\)
\([0 \rightarrow_{[0 \rightarrow 0,[0 \rightarrow 0,1]_2]\text{&}2} 1]\) \(\psi(\psi_{I_2}(0))\) \([1(1)[_{<1>⁅2⁆}1]]\)
\([0 \rightarrow_{[0 \rightarrow 0 \rightarrow 1]\text{&}2} 1]\) \(\psi(\psi_{I_2}(1))\) \([1(1)[_{<1,2>⁅2⁆}1]]\)
\([0 \rightarrow_{[0 \rightarrow 0 \rightarrow 2]\text{&}2} 1]\) \(\psi(\psi_{I_2}(2))\) \([1(1)[_{<1,3>⁅2⁆}1]]\)
\([0 \rightarrow_{[0]\text{&}3} 1]\) \(\psi(\psi_I(\varepsilon_{I_22}))\) \([1(1)[_{<1,1,1,1,2>⁅2⁆}1]]\)

Higher level brackets

Dollar FGH HAN
\([0 \rightarrow_{[0]\text{&}3} 1]\) \(\psi(\psi_{I_2}(\varepsilon_{I_22}))\) \([1(1)[_{<1,1,1,1,2>⁅2⁆}1]]\)
\([0 \rightarrow_{[0,[0,1]_2\text{&}3]\text{&}3} 1]\) \(\psi(\psi_{I_3}(0))\) \([1(1)[_{<1>⁅3⁆}1]]\)
\([0 \rightarrow_{[0]\text{&}4} 1]\) \(\psi(\psi_{I_3}(\varepsilon_{I_32}))\) \([1(1)[_{<1,1,1,1,2>⁅3⁆}1]]\)
\([0 \rightarrow_{[0,[0,1]_2\text{&}4]\text{&}4} 1]\) \(\psi(\psi_{I_4}(0))\) \([1(1)[_{<1>⁅4⁆}1]]\)
\([0 \rightarrow_{[0]\text{&}[0]} 1]\) \(\psi(\psi_{I_\omega}(0))\) \([1(1)[_{<1>⁅[1]⁆}1]]\)
\([0 \rightarrow_{[0]\text{&}([0]\text{&}1)} 1]\) \(\psi(\psi_{I_\Omega}(0))\) \([1(1)[_{<1>⁅[_21]⁆}1]]\)
\([0 \rightarrow_{[0]\text{&}([0]\text{&}2)} 1]\) \(\psi(\psi_{I_I}(0))\) \([1(1)[_{<1>⁅1,2⁆}1]]\)
\([0 \rightarrow_{[0]\text{&}([0]\text{&}3)} 1]\) \(\psi(\psi_{I_{I_2}}(0))\)
\([0 \rightarrow_{[0]\text{&}([0]\text{&}([0]\text{&}2))} 1]\) \(\psi(\psi_{I_{I_I}}(0))\)
\([0 \rightarrow_{[0]\text{&}0\text{&}1} 1]\)

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

& arrays

Dollar FGH
\([0 \rightarrow_{[0]\text{&}0\text{&}1} 1]\) \(\psi(\psi_{I(1,0)}(0))\)
\([0 \rightarrow_{[0]\text{&}1\text{&}1} 1]\) \(\psi(\psi_{I(1,1)}(0))\)
\([0 \rightarrow_{[0]\text{&}0\text{&}2} 1]\) \(\psi(\psi_{I(2,0)}(0))\)
\([0 \rightarrow_{[0]\text{&}0\text{&}0\text{&}1} 1]\) \(\psi(\psi_{I(1,0,0)}(0))\)
\([0 \rightarrow_{[0]\text{&}\rightarrow1} 1]\) \(\psi(\chi(M^\omega))\)
\([0 \rightarrow_{[0]\text{&}\rightarrow2} 1]\) \(\psi(\chi(M^{\omega2}))\)
\([0 \rightarrow_{[0]\text{&}\rightarrow([0]]\text{&}1)} 1]\) \(\psi(\chi(M^{\Omega}))\)
\([0 \rightarrow_{[0]\text{&}\rightarrow0\text{&}1} 1]\) \(\psi(\chi(M^{M}))\)
\([0 \rightarrow_{[0]\text{&}\rightarrow0\text{&}\rightarrow1} 1]\) \(\psi(\chi(\varepsilon_{M+1}))\)
\([0 \rightarrow_{[0]\text{&}\rightarrow_21} 1]\) \(\psi(\chi(\varepsilon_{M+2}))\)
\([0 \rightarrow_{[0]\text{&}\rightarrow_{[0]\text{&}1}1} 1]\) \(\psi(\chi(\varepsilon_{M+\Omega}))\)
\([0 \rightarrow_{[0]\text{&}\rightarrow_{[0]\text{&}2}1} 1]\) \(\psi(\chi(\varepsilon_{M+I}))\)
\([0 \rightarrow_{[0]\text{&}\rightarrow_{[0]\text{&}[0]}1} 1]\) \(\psi(\chi(\varepsilon_{M+I(\omega)}))\)
\([0 \rightarrow_{[0]\text{&}\rightarrow_{[0]\text{&}0\text{&}1}1} 1]\) \(\psi(\chi(\varepsilon_{M+I(1,0)}))\)
\([0 \rightarrow_{[0]\text{&}\rightarrow_{[0]\text{&}\rightarrow1}1} 1]\) \(\psi(\chi(\varepsilon_{M+\chi(M)}))\)
\([0 \rightarrow_{[0]\text{&}\rightarrow_{[0]\text{&}\rightarrow0\text{&}\rightarrow1}1}1]\) \(\psi(\chi(\varepsilon_{M+\chi(\varepsilon_{M+1})}))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0]\text{&&}1}1}1]\)

\(\psi(\chi(\varepsilon_{M2}))\)

Up to the limit of weakly Mahlo cardinals!

Dollar FGH
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0]\text{&&}1}1}1]\) \(\psi(\chi(\varepsilon_{M2}))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{1[0]\text{&&}1}1}1]\) \(\psi(\chi(\varepsilon_{M2+1}))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0]\text{&&}1[0]\text{&&}1}1}1]\) \(\psi(\chi(\varepsilon_{M3}))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0]_2\text{&&}1}1}1]\) \(\psi(\chi(\varepsilon_{\varepsilon_{M+1}}))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0]_3\text{&&}1}1}1]\) \(\psi(\chi(\zeta_{M+1}))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0,1]\text{&&}1}1}1]\) \(\psi(\chi(\Omega_{M+1}))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0,2]\text{&&}1}1}1]\) \(\psi(\chi(\Omega_{M+2}))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0,[0]\text{&&}1]\text{&&}1}1}1]\) \(\psi(\chi(\Omega_{M2}))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0,[0,1]\text{&&}1]\text{&&}1}1}1]\) \(\psi(\chi(\Omega_{\Omega_{M+1}}))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0,[0,1]_2\text{&&}1]\text{&&}1}1}1]\) \(\psi(\chi(I_{M+1}))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0\rightarrow1]\text{&&}1}1}1]\) \(\psi(\chi(I_{M+2}))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0\rightarrow[0\rightarrow1]_2\text{&&}1]\text{&&}1}1}1]\) \(\psi(\chi(I_{M2}))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0\rightarrow[1\rightarrow1]_2\text{&&}1]\text{&&}}1}1]\) \(\psi(\chi(I_{\varepsilon_{M+1}}))\)

\([0\rightarrow_{[0]\text{&}\rightarrow_{[0\rightarrow[0\rightarrow1]_3\text{&&}1]\text{&&}1}1}1]\)

\(\psi(\chi(I_{\zeta_{M+1}}))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0\rightarrow0,1]\text{&&}1}1}1]\) \(\psi(\chi(I_{\Omega_{M+1}}))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0\rightarrow0\rightarrow1]\text{&&}1}1}1]\) \(\psi(\chi(I_{I_{M+1}}))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0\rightarrow_21]\text{&&}1}1}1]\) \(\psi(\chi(M_2))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0\rightarrow_31]\text{&&}1}1}1]\) \(\psi(\chi(M_3))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0\rightarrow_{[0]\text{&&}1}1]\text{&&}1}1}1]\)

\(\psi(\chi(M_M))\)

\([0\rightarrow_{[0]\text{&}\rightarrow_{[0]\text{&&}2}1}1]\) \(\psi(\chi(M(1,0)))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0]\text{&&}0\text{&}1}1}1]\) \(\psi(\chi(M(1,1)))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0]\text{&&}0\text{&&}1}1}1]\)

\(\psi(\chi(M(2,0)))\)

Multiple &'s

Dollar FGH
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0]\text{&&}\rightarrow1}1}1]\) \(\psi(\Psi_{\Xi(3,0)}(\Xi(3,0)^\omega))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0]\text{&&}\rightarrow0\text{&&}\rightarrow1}1}1]\)

\(\psi(\Psi_{\Xi(3,0)}(\varepsilon_{\Xi(3,0)+1}))\)

\([0\rightarrow_{[0]\text{&}\rightarrow_{[0]\text{&&}\rightarrow_21}1}1]\) \(\psi(\Psi_{\Xi(3,0)}(\varepsilon_{\Xi(3,0)+2}))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0]\text{&&}\rightarrow_{[0]\text{&&&}1}1}1}1]\) \(\psi(\Psi_{\Xi(3,0)}(\varepsilon_{\Xi(3,0)2}))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0]\text{&&}\rightarrow_{[1]\text{&&&}1}1}1}1]\) \(\psi(\Psi_{\Xi(3,0)}(\varepsilon_{\Xi(3,0)\omega}))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0]\text{&&}\rightarrow_{[0]_2\text{&&&}1}1}1}1]\) \(\psi(\Psi_{\Xi(3,0)}(\varepsilon_{\varepsilon_{\Xi(3,0)+1}+1}))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0]\text{&&}\rightarrow_{[0,1]\text{&&&}1}1}1}1]\)

\(\psi(\Psi_{\Xi(3,0)}(\Omega_{\Xi(3,0)+1}))\)

\([0\rightarrow_{[0]\text{&}\rightarrow_{[0]\text{&&}\rightarrow_{[0,[0,1]_2\text{&&&}1]\text{&&&}1}1}1}1]\) \(\psi(\Psi_{\Xi(3,0)}(I_{\Xi(3,0)+1}))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0]\text{&&}\rightarrow_{[0\rightarrow1]\text{&&&}1}1}1}1]\) \(\psi(\Psi_{\Xi(3,0)}(I_{\Xi(3,0)+2}))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0]\text{&&}\rightarrow_{[0\rightarrow0\rightarrow1]\text{&&&}1}1}1}1]\) \(\psi(\Psi_{\Xi(3,0)}(M_{\Xi(3,0)+2}))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0]\text{&&}\rightarrow_{[0]\text{&&&}2}1}1}1]\) \(\psi(\Psi_{\Xi(3,1)}(\Xi(3,1)))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0]\text{&&}\rightarrow_{[0]\text{&&&}\rightarrow_{[0]\text{&&&&}1}1}1}1}1]\)

\(\psi(\Psi_{\Xi(4,0)}(\Xi(4,0)))\)

& structures

Dollar FGH
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0][_{\text{&}}0]1}1}1]\) \(\psi(\Psi_{\Xi(\omega,0)}(\Xi(\omega,0)))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0][_{\text{&}}1]1}1}1]\) \(\psi(\Psi_{\Xi(\omega^2,0)}(\Xi(\omega^2,0)))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0][_{\text{&}}[0][_{\text{&}}0]1]1}1}1]\) \(\psi(\Psi_{\Xi(\Xi(\omega,0),0)}(\Xi(\Xi(\omega,0),0)))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0][_{\text{&}}\text{&}]1}1}1]\) \(\psi(\Psi_{\Xi(K)}(\Xi(K)))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0][_{\text{&}}\text{&&}]1}1}1]\) \(\psi(\Psi_{\Xi(K2)}(\Xi(K2)))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0][_{\text{&}}[_{\text{&}}0]]1}1}1]\) \(\psi(\Psi_{\Xi(K\omega)}(\Xi(K\omega)))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0][_{\text{&}}[_{\text{&}}\text{&}]]1}1}1]\) \(\psi(\Psi_{\Xi(K^2)}(\Xi(K^2)))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0][_{\text{&}}[_{\text{&}}[_{\text{&}}0]]]1}1}1]\) \(\psi(\Psi_{\Xi(K^K)}(\Xi(K^K)))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0][_{\text{&}}0]_21}1}1]\) \(\psi(\Psi_{\Xi(\varepsilon_{K+1})}(\Xi(\varepsilon_{K+1})))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0][_{\text{&}}0]_31}1}1]\) \(\psi(\Psi_{\Xi(\zeta_{K+1})}(\Xi(\zeta_{K+1})))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0][_{\text{&}}0,1]1}1}1]\) \(\psi(\Psi_{\Xi(\Omega_{K+1})}(\Xi(\Omega_{K+1})))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0]([_{\text{&}}0]\text{&}1)1}1}1]\) \(\psi(\Psi_{\Xi(\varepsilon_{\Omega_{K+1}+1})}(\Xi(\varepsilon_{\Omega_{K+1}+1})))\)

\([0\rightarrow_{[0]\text{&}\rightarrow_{[0]([_{\text{&}}0,1]_2\text{&}1)1}1}1]\)

\(\psi(\Psi_{\Xi(I_{K+1})}(\Xi(I_{K+1})))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0]([_{\text{&}}0]\text{&}\rightarrow_{[_{\text{&}}0]\text{&&}1}1)1}1}1]\) \(\psi(\Psi_{\Xi(M_{K+1})}(\Xi(M_{K+1})))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0][_{\text{&&}}0]1}1}1]\) \(\psi(\Psi_{\Xi(K_2)}(\Xi(K_2)))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0][_{[_{\text{&}}0]}0]1}1}1]\) \(\psi(\Psi_{\Xi(K_{\omega})}(\Xi(K_{\omega})))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0][_{[_{\text{&&}}0]}0]1}1}1]\) \(\psi(\Psi_{\Xi(K_{K})}(\Xi(K_{K})))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0][_{[_{[_{\text{&&}}0]}0]}0]1}1}1]\) \(\psi(\Psi_{\Xi(K_{K_{K}})}(\Xi(K_{K_{K}})))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0]\{0,1\}1}1}1]\)

\(\psi(\Psi_{\Xi_{\Xi_2(1,0)}}(\Xi_{\Xi_2(1,0)}(0)))\) inaccessible notation with K instead of Ω.

\([0\rightarrow_{[0]\text{&}\rightarrow_{[0]\{0,2\}1}1}1]\) \(\psi(\Psi_{\Xi_{\Xi_2(1,0)}}(\Xi_{\Xi_2(1,0)}(1)))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0]\{0,0,1\}1}1}1]\) \(\psi(\Psi_{\Xi_{\Xi_2(1,0)}}(\Xi_{\Xi_2(1,0)}(\Xi_2(1,0))))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0]\{0,0,0,1\}1}1}1]\) \(\psi(\Psi_{\Xi_{\Xi_2(1,0)}}(\Xi_{\Xi_2(1,0)}(\Xi_2(1,0)^2)))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0]\{0\rightarrow1\}1}1}1]\) \(\psi(\Psi_{\Xi_{\Xi_2(1,0)}}(\Xi_{\Xi_2(1,0)}(\varphi(\omega,\Xi_2(1,0)))))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0]\{0\rightarrow_21\}1}1}1]\) \(\psi(\Psi_{\Xi_{\Xi_2(1,0)}}(\Xi_{\Xi_2(1,0)}(\varphi(\omega^2,\Xi_2(1,0)))))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0]\{0\rightarrow_{[0]}1\}1}1}1]\) \(\psi(\Psi_{\Xi_{\Xi_2(1,0)}}(\Xi_{\Xi_2(1,0)}(\varphi(\omega^\omega,\Xi_2(1,0)))))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0]\{0\rightarrow_{[0]_2}1\}1}1}1]\) \(\psi(\Psi_{\Xi_{\Xi_2(1,0)}}(\Xi_{\Xi_2(1,0)}(\varphi(\varepsilon_0,\Xi_2(1,0)))))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0]\{0\rightarrow_{[0]\text{&}1}1\}1}1}1]\) \(\psi(\Psi_{\Xi_{\Xi_2(1,0)}}(\Xi_{\Omega_{\Xi_2(1,0)+1}}(\Omega_{\Xi_2(1,0)+1})))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0]\{0\rightarrow_{[0]\text{&}2}1\}1}1}1]\) \(\psi(\Psi_{\Xi_{\Xi_2(1,0)}}(\Xi_{I_{\Xi_2(1,0)+1}}(I_{\Xi_2(1,0)+1})))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0]\{0\rightarrow_{[0]\text{&}[0]}1\}1}1}1]\) \(\psi(\Psi_{\Xi_{\Xi_2(1,0)}}(\Xi_{I_{\Xi_2(1,0)+\omega}}(I_{\Xi_2(1,0)+\omega})))\)
\([0\rightarrow_{[0]\text{&}\rightarrow_{[0]\{0\rightarrow_{[0]\text{&}[0]\text{&}1}1\}1}1}1]\) \(\psi(\Psi_{\Xi_{\Xi_2(1,0)}}(\Xi_{I_{\Xi_2(1,0)2}}(I_{\Xi_2(1,0)2})))\)

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