FANDOM


Up to \(\varphi(\omega,0)\)

I think everyone agrees with the facts that linear HAN reach \(\varphi(\omega,0)\) ( see comparisons of Cloudy176 here )

Up to \(\Gamma_0\)

These comparisons are a few small steps for showing that the w/ operator iterates over entries

FGH HAN Dollar
\(f_{\varphi(\omega,0)}(a)\) \((a+3)![1]w/[1]\) \(a{$}1[0]_{[0]}\)
\(f_{\varphi(\omega,0)}(a+1)\) \((a+4)![1]w/[1]\) \(a{$}2[0]_{[0]}\)
\(f_{\varphi(\omega,0)}(f_{\varphi(\omega,0)}(a))\)

\(((a+3)![1]w/[1])![1]w/[1]\)

\(a{$}[0]_{[0]}[0]_{[0]}\)
\(f_{\varphi(\omega,0)+1}(a)\)

\(a![2]w/[1]\)

\(a{$}[[0]_{[0]}]\)
\(f_{\varphi(\omega,0)+2}(a)\) \(a![3]w/[1]\) \(a{$}[1[0]_{[0]}]\)

More:

Ordinal HAN Dollar
\(\varphi(\omega,0)+1\) \([2]w/[1]\) \([[0]_{[0]}]\)
\(\varphi(\omega,0)+2\) \([3]w/[1]\)

\([1[0]_{[0]}]\)

\(\varphi(\omega,0)+\omega\) \([[1]]w/[1]\) \([[0][0]_{[0]}]\)
\(\varphi(\omega,0)+\omega2\) \([[1,2]]w/[1]\) \([[0][0][0]_{[0]}]\)
\(\varphi(\omega,0)+\omega3\) \([[1,3]]w/[1]\) \([[0][0][0][0]_{[0]}]\)
\(\varphi(\omega,0)+\omega^2\) \([[1,1,2]]w/[1]\) \([[1][0]_{[0]}]\)
\(\varphi(\omega,0)+\omega^\omega\) \([[1,1,1,2]]w/[1]\) \([[[0]][0]_{[0]}]\)
\(\varphi(\omega,0)+\varepsilon_0\) \([[1,1,1,1,2]]w/[1]\) \([[0]_2[0]_{[0]}]\)
\(\varphi(\omega,0)2\)

\([[1]w/[1]]w/[1]=\)

\([1,2]w/[1]\)

\([[0]_{[0]}[0]_{[0]}]\)
\(\varphi(\omega,0)3\) \([1,3]w/[1]\) \([[0]_{[0]}[0]_{[0]}[0]_{[0]}]\)
\(\varphi(\omega,0)\omega\) \([1,[1]]w/[1]\) \([[[0]_{[0]}]]\)
\(\varphi(\omega,0)\omega2\) \([1,[1,2]]w/[1]\) \([[[0]_{[0]}][[0]_{[0]}]]\)
\(\varphi(\omega,0)\omega^2\) \([1,[1,1,2]]w/[1]\) \([[1[0]_{[0]}]]\)
\(\omega^{\omega^{\varphi(\omega,0)}}\) \([1]w/[3]\) \([[[[0]_{[0]}]]]\)
\(\varepsilon_{\varphi(\omega,0)+1}\) \([1]w/[4]\) \([[0]_{[0]}]_2\)

\(\zeta_{\varphi(\omega,0)+1}\)

\([1]w/[5]\)

\([[0]_{[0]}]_3\)
\(\varphi(\omega,1)\) \([1]w/[[1,2]]\) \([1]_{[0]}\)
\(\varphi(\omega,\omega)\) \([1]w/[[1]]\) \([[0]]_{[0]}\)
\(\varphi(\omega,\varphi(\omega,0))\) \([1]w/[1,2]\) \([[0]_{[0]}]_{[0]}\)
\(\varphi(\omega+1,0)\) \([1]w/[1,1,2]\) \([0]_{[0]1}\)
\(\varphi(\omega+2,0)\) \([1]w/[1,1,3]\) \([0]_{[0]2}\)
\(\varphi(\omega2,0)\) \([1]w/[1,1,1,2]\) \([0]_{[0][0]}\)
\(\varphi(\omega^2,0)\) \([1]w/[1,1,1,1,2]\) \([0]_{[1]}\)
\(\varphi(\omega^\omega,0)\) \([1]w/[1]w/5\) \([0]_{[[0]]}\)
\(\varphi(\varepsilon_0,0)\) \([1]w/[1]w/6\) \([0]_{[0]_2}\)
\(\varphi(\varphi(\omega,0),0)\) \([1]w/[1]w/[1]\) \([0]_{[0]_{[0]}}\)
\(\varphi(\varphi(\omega+1,0),0)\) \([1]w/[1]w/[1,1,2]\) \([0]_{[0]_{[0]1}}\)
\(\varphi(\varphi(\varphi(\omega,0),0),0)\) \([1]w/[1]w/[1]w/[1]\) \([0]_{[0]_{[0]_{[0]}}}\)
\(\varphi(\varphi(\varphi(\varphi(\omega,0),0),0),0)\) \([1]w/[1]w/[1]w/[1]w/[1]\) \([0]_{[0]_{[0]_{[0]_{[0]}}}}\)
\(\Gamma_0\) \([1(1)2]\) \([0,1]\)

HAN dimensional arrays

Dollar Function in the next table to avoid double comparisons

Ordinal HAN
\(\Gamma_0\) \([1(1)2]\)
\(\Gamma_0+1\) \([2(1)2]\)
\(\Gamma_0+\omega\) \([[1](1)2]\)
\(\Gamma_0+\omega2\) \([[1,2](1)2]\)
\(\Gamma_0+\omega^2\) \([[1,1,2](1)2]\)
\(\Gamma_0+\omega^\omega\) \([[1,1,1,2](1)2]\)
\(\Gamma_0+\varphi(\omega,0)\) \([[1]w/[1](1)2]\)
\(\Gamma_02\) \([1,2(1)2]\)
\(\Gamma_03\) \([1,3(1)2]\)
\(\Gamma_0\omega\) \([1,[1](1)2]\)
\(\Gamma_0\omega2\) \([1,[1,2](1)2]\)
\(\Gamma_0^2\) \([1,1,2(1)2]\)
\(\Gamma_0^{\Gamma_0}\) \([1,1,1,2(1)2]\)

\(\varepsilon_{\Gamma_0+1}\)

\([1,1,1,1,2(1)2]\)
\(\varphi(\omega,\Gamma_0+1)\) \([1(1)2]w/[1]\)
\(\Gamma_1\) \([1(1)3]\)
\(\Gamma_2\) \([1(1)4]\)
\(\Gamma_{\omega}\) \([1(1)1,2]=[1(1)[1]]\)
\(\Gamma_{\omega2}\) \([1(1)1,3]=[1(1)[1],2]\)
\(\Gamma_{\omega^2}\)

\([1(1)1,1,2]=[1(1)1,[1]]\)

\(\Gamma_{\Gamma_{\omega^2}}\) \([1(1)1,2,2]=[1(1)[1(1)1,1,2],1,2]\)
\(\varphi(1,1,0)\)

\([1(1)1,1,2,2]=[1(1)1,[1(1)1,1,1,2],1,2]\)

\(\varphi(1,2,0)\) \([1(1)1,1,1,2,2]=[1(1)1,1,[1(1)1,1,1,2],1,2]\)
\(\varphi(1,\omega,0)\) \([1(1)1]w/[1]\)
\(\varphi(1,\omega2,0)\) \([1(1)1]w/[1,2]\)
\(\varphi(1,\omega^2,0)\) \([1(1)1]w/[1,1,2]\)
\(\varphi(1,\omega^\omega,0)\) \([1(1)1]w/[1,1,1,2]\)
\(\varphi(1,\varphi(1,0,0),0)\) \([1(1)1]w/[1(1)2]\)
\(\varphi(2,0,0)\) \([1(1)1(1)2]\)
\(\Gamma_{\varphi(2,0,0)+1}\) \([1(1)2(1)2]\)
\(\varphi(2,0,1)\) \([1(1)1(1)3]\)
\(\varphi(2,0,\varphi(2,0,0))\) \([1(1)1(1)1,2,2]\)
\(\varphi(2,1,0)\) \([1(1)1(1)1,1,2,2]\)
\(\varphi(2,2,0)\) \([1(1)1(1)1,1,1,2,2]\)
\(\varphi(2,\omega,0)\) \([1(1)1(1)1]w/[1]\)
\(\varphi(3,0,0)\) \([1(1)1(1)1(1)2]\)
\(\varphi(4,0,0)\) \([1]w(1)/4\)
\(\varphi(\omega,0,0)\) \([1]w(1)/[1]\)
\(\varphi(\varphi(\omega,0,0),0,0)\) \([1]w(1)/[1]w(1)/[1]\)
\(\varphi(1,0,0,0)\)

\([1(2)2]\)

\(\varphi(1,0,0,1)\) \([1(2)3]\)
\(\varphi(1,0,1,0)\) \([1(2)1,1,2,2]\)
\(\varphi(1,1,0,0)\) \([1(2)1(1)2]\)
\(\varphi(2,0,0,0)\) \([1(2)1(2)2]\)
\(\varphi(1,0,0,0,0)\) \([1(3)2]\)
\(\vartheta(\Omega^\omega)\) \([1([1])2]\)
\(\vartheta(\Omega^{\omega2})\) \([1([1,2])2]\)
\(\vartheta(\Omega^{\omega^2})\) \([1([1,1,2])2]\)
\(\vartheta(\Omega^{\varphi(1,0,0)})\) \([1([1(1)2])2]\)
\(\vartheta(\Omega^{\vartheta(\Omega^\omega)})\) \([1([1([1])2])2]\)
\(\vartheta(\Omega^\Omega)\) \([1([1([1(...)2])2])2]\)

Bracket Types up to the Bachmann-Howard Ordinal

FGH HAN Dollar Function
\(\Gamma_0\) \([1(1)2]\) \([0,1]\)
\(\Gamma_0+1\) \([2(1)2]\) \([1,1]\)
\(\Gamma_02\) \([1,2(1)2]\) \([[0,1],1]\)
\(\Gamma_0^2\) \([1,1,2(1)2]\) \([[[0,1],1],1]\)
\(\Gamma_0^{\Gamma_0}\) \([1,1,1,2(1)2]\) \([[[[0,1],1],1],1]\)
\(\varepsilon_{\Gamma_0+1}\) \([1,1,1,1,2(1)2]\) \([[0,1]_2,1]\)
\(\zeta_{\Gamma_0+1}\) \([1,1,1,1,1,2(1)2]\) \([[0,1]_3,1]\)
\(\varphi(\omega,\Gamma_0+1)\) \([1(1)2]w/[1]\) \([[0,1]_{[0]},1]\)
\(\Gamma_1\) \([1(1)3]\) \([0,2]\)
\(\Gamma_2\) \([1(1)4]\) \([0,3]\)
\(\Gamma_\omega\) \([1(1)[1]]\) \([0,[0]]\)
\(\Gamma_{\Gamma_0}\) \([1(1)[1(1)2]]\) \([0,[0,1]]\)

\(\psi(\Omega^{\Omega+1})\)

\([1(1)[_21]]\) \([0,[0,1]_2]\)
\(\psi(\Omega^{\Omega+1}+\Omega^\Omega)\) \([1(1)[_22]]\) \([0,1[0,1]_2]\)
\(\psi(\Omega^{\Omega+1}+\Omega^\Omega2)\) \([1(1)[_23]]\) \([0,2[0,1]_2]\)
\(\psi(\Omega^{\Omega+1}+\Omega^\Omega\omega)\) \([1(1)[_2[1]]]\) \([0,[0][0,1]_2]\)
\(\psi(\Omega^{\Omega+1}+\Omega^\Omega\omega2)\) \([1(1)[_2[1,2]]]\) \([0,[0][0][0,1]_2]\)
\(\psi(\Omega^{\Omega+1}2)\) \([1(1)[_21,2]]\) \([0,[0,1]_2[0,1]_2]\)
\(\psi(\Omega^{\Omega+1}3)\) \([1(1)[_21,3]]\)

\([0,[0,1]_2[0,1]_2[0,1]_2]\)

\(\psi(\Omega^{\Omega+2})\) \([1(1)[_21,1,2]]\) \([0,[1,1]_2]\)
\(\psi(\Omega^{\Omega+3})\) \([1(1)[_21,1,3]]\)

\([0,[2,1]_2]\)

\(\psi(\Omega^{\Omega+\omega})\) \([1(1)[_21,1,[1]]]\) \([0,[[0],1]_2]\)
\(\psi(\Omega^{\Omega2})\) \([1(1)[_21,1,1,2]]\) \([0,[0,2]_2]\)
\(\psi(\Omega^{\Omega3})\) \([1(1)[_21,[_21,1,2],1,2]]\) \([0,[0,3]_2]\)
\(\psi(\Omega^{\Omega\omega})\) \([1(1)[_21,[_21,[1],2],1,2]]\) \([0,[0,[0]]_2]\)
\(\psi(\Omega^{\Omega^2})\) \([1(1)[_21,[_21,1,3],1,2]]\) \([0,[0,1]_3]\)
\(\psi(\Omega^{\Omega^22})\) \([1(1)[_21,[_21,2,3],1,2]]\) \([0,[0,2]_3]\)
\(\psi(\Omega^{\Omega^3})\) \([1(1)[_21,[_21,1,4],1,2]]\) \([0,[0,1]_4]\)
\(\psi(\Omega^{\Omega^4})\) \([1(1)[_21,[_21,1,5],1,2]]\) \([0,[0,1]_5]\)
\(\psi(\Omega^{\Omega^\omega})\) \([1(1)[_21,[_21,1,[1]],1,2]]\) \([0,[0,1]_{[0]}]\)
\(\psi(\Omega^{\Omega^{\psi(\Omega^{\Omega^\omega})}})\) \([1(1)[_21,[_21,1,[1(1)[_21,[_21,1,[1]],1,2]]],1,2]]\) \([0,[0,1]_{[0,[0,1]_{[0]}]}]\)
\(\psi(\Omega^{\Omega^\Omega})\)

\([1(1)[_21,1,2,2]]\)

\([0,0,1]\)
\(\psi(\Omega^{\Omega^\Omega}+\Omega^\Omega)\) \([1(1)[_22,1,2,2]]\) \([0,1,1]\)
\(\psi(\Omega^{\Omega^\Omega}+\Omega^{\Omega+1})\) \([1(1)[_2[_21],1,2,2]]\) \([0,[0,0,1]_2,1]\)
\(\psi(\Omega^{\Omega^\Omega}+\Omega^{\Omega+2})\) \([1(1)[_2[_21,1,2],1,2,2]]\) \([0,[1,0,1]_2,1]\)
\(\psi(\Omega^{\Omega^\Omega}+\Omega^{\Omega2})\) \([1(1)[_2[_21,1,1,2],1,2,2]]\) \([0,[0,1,1]_2,1]\)
\(\psi(\Omega^{\Omega^\Omega}+\Omega^{\Omega^2})\)

\([1(1)[_2[_21,[_21,1,3],1,2],1,2,2]]\)

\([0,[0,0,2]_2,1]\)
\(\psi(\Omega^{\Omega^\Omega}2)\) \([1(1)[_21,2,2,2]]\) \([0,[0,0,[0,0,1]_2]_2,1]\)
\(\psi(\Omega^{\Omega^\Omega+1})\) \([1(1)[_21,[_21],2,2]]\) \([0,[0,0,1]_3,1]\)
\(\psi(\Omega^{\Omega^\Omega+\Omega})\) \([1(1)[_21,[_21,2],2,2]]\) \([0,[0,0,2]_3,1]\)
\(\psi(\Omega^{\Omega^\Omega+\Omega2})\) \([1(1)[_21,[_21,3],2,2]]\) \([0,[0,0,3]_3,1]\)
\(\psi(\Omega^{\Omega^\Omega+\Omega^2})\) \([1(1)[_21,[_21,1,2],2,2]]\) \([0,[0,0,1]_4,1]\)
\(\psi(\Omega^{\Omega^\Omega2})\) \([1(1)[_21,1,3,2]]\) \([0,0,2]\)
\(\psi(\Omega^{\Omega^{\Omega+1}})\) \([1(1)[_21,1,1,3]]\) \([0,0,[0,0,1]_2]\)
\(\psi(\Omega^{\Omega^{\Omega2}})\)

\([1(1)[_21,[_21,[_21,1,2],1,2],1,3]]\)

\([0,0,[0,0,2]_2]\)
\(\psi(\Omega^{\Omega^{\Omega^2}})\) \([1(1)[_21,[_21,[_21,1,3],1,2],1,3]]\) \([0,0,[0,0,1]_3]\)
\(\psi(\Omega^{\Omega^{\Omega^\Omega}})\) \([1(1)[_21,[_21,1,2,2],1,3]]\) \([0,0,0,1]\)
\(\psi(\Omega^{\Omega^{\Omega^{\Omega^\Omega}}})\) \([1(1)[_21,[_21,[_21,1,2,2],1,3],1,4]]\) \([0,0,0,0,1]\)
\(\psi(\varepsilon_{\Omega+1})\)

\([1(1)[_21,1,1,[1]]]\)

\([0 \rightarrow 1]\)

The BHO comparison

Why is the corresponding HAN for the BHO \([1(1)[_21,1,1,[1]]]\) and not \([1(1)[_21,1,1,[_21]]]\)? That's because \(\varepsilon_{\Omega+1}=\Omega\uparrow\uparrow\omega\) and not \(\Omega\uparrow\uparrow\Omega\)

Bracket Types: Level 2

FGH HAN Dollar
\(\psi(\varepsilon_{\Omega+1})\) \([1(1)[_21,1,1,[1]]]\) \([0 \rightarrow 1]\)
\(\psi(\varepsilon_{\Omega+1}2)\)

\([1(1)[_21,2,1,[1]]]\)

\([0 \rightarrow 2]\)

\(\psi(\varepsilon_{\Omega+1}^2)\) \([1(1)[_21,1,2,[1]]]\) \([0 \rightarrow 0,1]\)
\(\psi(\varepsilon_{\Omega+1}^{\varepsilon_{\Omega+1}})\) \([1(1)[_21,1,1,[2]]]\) \([0 \rightarrow 0 \rightarrow 1]\)
\(\psi(\varepsilon_{\Omega+2})\) \([1(1)[_21,1,1,[1,2]]]\) \([0 \rightarrow_2 1]\)
\(\psi(\varepsilon_{\Omega+3})\) \([1(1)[_21,1,1,[1,3]]]\) \([0 \rightarrow_3 1]\)
\(\psi(\varepsilon_{\Omega2})\) \([1(1)[_21,1,1,1,2]]\) \([0 \rightarrow_{\{0\}_2} 1]\)
\(\psi(\zeta_{\Omega2})\)

\([1(1)[_21,1,1,1,1,2]]\)

\([0 \rightarrow_{\{0\}_3} 1]\)

\(\psi(\varphi(\omega,\Omega+1))\)

\([1(1)[_21]w/[1]]\) \([0 \rightarrow_{\{0\}_{[0]}} 1]\)
\(\psi(\varphi(\Omega,1))\) \([1(1)[_21]w/[_21]]\) \([0 \rightarrow_{\{0\}_{\{0\}_2}} 1]\)

Type 3 brackets

FGH HAN Dollar
\(\psi(\Omega_2)\) \([1(1)[_21(1)2]]\) \([0 \rightarrow_{\{0,1\}}1]\)
\(\psi(\Omega_2+\Omega^\Omega)\)

\([1(1)[_22(1)2]]\)

\([0,1 \rightarrow_{\{0,1\}}1]\)
\(\psi(\Omega_22)\) \([1(1)[_21(1)3]]\) \([0 \rightarrow_{\{0,1\}}2]\)
\(\psi(\Omega_2\Omega)\) \([1(1)[_21(1)[_21]]]\) \([0 \rightarrow_{\{0,1\}}[0 \rightarrow_{\{0,1\}}1]_2]\)
\(\psi(\Omega_2^2)\) \([1(1)[_2(1)[_31]]]\) \([0 \rightarrow_{\{0,1\}}0,1]\)
\(\psi(\Omega_2^{\Omega_2})\) \([1(1)[_2(1)[_31,1,1,2]]]\) \([0 \rightarrow_{\{0,1\}}0\rightarrow_{\{0,1\}}1]\)
\(\psi(\varepsilon_{\Omega_2+1})\) \([1(1)[_2(1)[_31,1,1,[1]]]]\) \([0 \rightarrow_{1\{0,1\}}1]\)
\(\psi(\varepsilon_{\Omega_22})\) \([1(1)[_2(1)[_31,1,1,1,2]]]\) \([0 \rightarrow_{\{0,1\}\{0,1\}}1]\)
\(\psi(\zeta_{\Omega_22})\) \([1(1)[_2(1)[_31,1,1,1,1,2]]]\) \([0 \rightarrow_{\{\{0,1\}_3,1\}}1]\)
\(\psi(\Omega_3)\) \([1(1)[_2(1)[_31(1)2]]]\) \([0 \rightarrow_{\{0,2\}}1]\)

Up to the TBF

FGH HAN Dollar
\(\psi(\Omega_3)\) \([1(1)[_2(1)[_31(1)2]]]\) \([0 \rightarrow_{\{0,2\}}1]\)
\(\psi(\Omega_3^2)\) \([1(1)[_2(1)[_31(1)[_41]]]]\) \([0 \rightarrow_{\{0,2\}}0,1]\)
\(\psi(\Omega_4)\) \([1(1)[_2(1)[_31(1)[_41(1)2]]]]\) \([0 \rightarrow_{\{0,3\}}1]\)
\(\psi(\Omega_4^2)\) \([1(1)[_2(1)[_31(1)[_41(1)[_51]]]]]\) \([0 \rightarrow_{\{0,3\}}0,1]\)
\(\psi(\Omega_5)\) \([1(1)[_2(1)[_31(1)[_41(1)[_51(1)2]]]]]\) \([0 \rightarrow_{\{0,4\}}1]\)
\(\psi(\Omega_5^2)\) \([1(1)[_2(1)[_31(1)[_41(1)[_51(1)[_61]]]]]]\) \([0 \rightarrow_{\{0,4\}}0,1]\)
\(\psi(\Omega_\omega)\) \([1(1)[_{[1]}1(1)2]]\) \([0 \rightarrow_{\{0,[0]\}}1]\)
\(\psi(\varepsilon_{\Omega_\omega+1})\) \([1(1)[_{[2]}1,1,1,[1]]]\) \([0 \rightarrow_{1\{0,[0]\}}1]\)

Array Type Brackets

FGH HAN Dollar
\(\vartheta(\Omega_\omega)\) \([1(1)[_{[1]}1(1)2]]\) \([0 \rightarrow_{1\{0,[0]\}}1]\)
\(\vartheta(\varepsilon_{\Omega_\omega+1})\) \([1(1)[_{[2]}1,1,1,[1]]]\) \([0 \rightarrow_{1\{0,[0]\}}1]\)
\(\vartheta(\Omega_{\omega+1})\) \([1(1)[_{[2]}1(1)2]]\) \([0 \rightarrow_{1\{0,[0]1\}}1]\)
\(\vartheta(\Omega_{\omega2})\) \([1(1)[_{[1,2]}1(1)2]]\) \([0 \rightarrow_{1\{0,[0][0]\}}1]\)
\(\vartheta(\Omega_{\omega^2})\) \([1(1)[_{[1,1,2]}1(1)2]]\) \([0 \rightarrow_{1\{0,[1]\}}1]\)
\(\vartheta(\Omega_{\omega^\omega})\) \([1(1)[_{[1,1,1,2]}1(1)2]]\) \([0 \rightarrow_{1\{0,[[0]]\}}1]\)
\(\vartheta(\Omega_{\varepsilon_0})\) \([1(1)[_{[1,1,1,1,2]}1(1)2]]\) \([0 \rightarrow_{1\{0,[0]_2\}}1]\)
\(\vartheta(\Omega_{\Gamma_0})\) \([1(1)[_{[1(1)2]}1(1)2]]\) \([0 \rightarrow_{1\{0,[0,1]\}}1]\)
\(\vartheta(\Omega_{\vartheta(\varepsilon_{\Omega+1})})\) \([1(1)[_{[1(1)[_21,1,1,[1]]]}1(1)2]]\) \([0 \rightarrow_{1\{0,[0\rightarrow1]\}}1]\)
\(\vartheta(\Omega_{\vartheta(\varepsilon_{\Omega_2+1})})\) \([1(1)[_{[1(1)[_21(1)[_31,1,1,[1]]]]}1(1)2]]\) \([0 \rightarrow_{1\{0,[0\rightarrow_{1\{0,1\}}1]\}}1]\)
\(\vartheta(\Omega_\Omega)\) \([1(1)[_{[_21]}1(1)2]]\) \([0 \rightarrow_{\{0,\{0\}_2\}}1]\)
\(\vartheta(\Omega_{\Omega_2})\) \([1(1)[_{[_31]}1(1)2]]\) \([0 \rightarrow_{\{0,\{0,1\}\}}1]\)
\(\vartheta(\Omega_{\Omega_\Omega})\) \([1(1)[_{[_{[_21]}1]}1(1)2]]\) \([0 \rightarrow_{\{0,\{0,\{0\}_2\}\}}1]\)
\(\vartheta(\Omega_{\Omega_{\Omega_\Omega}})\) \([1(1)[_{[_{[_{[_21]}1]}1]}1(1)2]]\) \([0 \rightarrow_{\{0,\{0,\{0,\{0\}_2\}\}\}}1]\)
\(\psi_I(0)\) \([1(1)[_{<1,2>}1]]\) \([0 \rightarrow_{\{0,\{0,1\}_2\}}1]\)

Beyond \(\psi_I(0)\)

FGH HAN Dollar
\(\psi_I(0)\) \([1(1)[_{<1,2>}1]]\) \([0 \rightarrow_{\{0,\{0,1\}_2\}}1]\)
\(\psi_I(1)\) \([1(1)[_{<1,3>}1]]\) \([0 \rightarrow_{\{0,\{1,1\}_2\}}1]\)
\(\psi_I(I)\) \([1(1)[_{<1,1,2>}1]]\)

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

\(\psi_I(I2)\) \([1(1)[_{<1,1,3>}1]]\) \([0 \rightarrow_{\{0,\{\{0\}_2\{0\}_2,1\}_2\}}1]\)
\(\psi_I(I\omega)\) \([1(1)[_{<1,1,[1]>}1]]\) \([0 \rightarrow_{\{0,\{\{\{0\}_2\},1\}_2\}}1]\)
\(\psi_I(I^2)\) \([1(1)[_{<1,1,1,2>}1]]\) \([0 \rightarrow_{\{0,\{\{1\}_2,1\}_2\}}1]\)
\(\psi_I(I^I)\) \([1(1)[_{<1,1,1,1,2>}1]]\) \([0 \rightarrow_{\{0,\{\{\{0\}_2\}_2,1\}_2\}}1]\)
\(\psi_I(\varepsilon_{I+1})\) \([1(1)[_{<1,1,1,1,1,2>}1]]\) \([0 \rightarrow_{\{0,\{\{0\}_3,1\}_2\}}1]\)
\(\psi_I(\zeta_{I+1})\) \([1(1)[_{<1,1,1,1,1,1,2>}1]]\) \([0 \rightarrow_{\{0,\{\{0\}_4,1\}_2\}}1]\)
\(\psi_I(\Gamma_{I+1})\) \([1(1)[_{<1(1)2>}1]]\) \([0 \rightarrow_{\{0,\{0,2\}_2\}}1]\)
\(\psi_I(\Omega_{I+1})\) \([1(1)[_{<1(1)<_21>>}1]]\) \([0 \rightarrow_{\{0,\{0,\{0\}_2\}_2\}}1]\)
\(\psi_I(\Omega_{I+2})\) \([1(1)[_{<1(1)<_31>>}1]]\) \([0 \rightarrow_{\{0,\{0,\{0,1\}\}_2\}}1]\)
\(\psi_I(I_2)\)

\([1(1)[_{<1>⁅2⁆}1]]\)

\([0 \rightarrow_{\{0,\{0,\{0,1\}_2\}_2\}}1]\)
\(\psi_I(I_I)\) \([1(1)[_{<1>⁅1,2⁆}1]]\) \([0 \rightarrow_{\{0,\{0,\{0,\{0\}_2\}_2\}_2\}}1]\)
\(\psi_{I(1)}(0)\) \([1(1)[_{<1>⁅1⁆w/[1]}1]]\)

\([0 \rightarrow_{\{0,\{0,1\}_3\}}1]\)

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