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## 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
$$\psi_I(0)$$ $$[1(1)[_{<1,2>}1]]$$ $$[0 \rightarrow_{\{0,\{0,1\}_2\}}1]$$
$$\psi_I(1)$$ $$[1(1)[_{<1,3>}1]]$$ $$[0 \rightarrow_{\{0,\{1,1\}_2\}}1]$$
$$\psi_I(I)$$ $$[1(1)[_{<1,1,2>}1]]$$

$$[0 \rightarrow_{\{0,\{\{0\}_2,1\}_2\}}1]$$

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

$$[1(1)[_{<1>⁅2⁆}1]]$$

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

$$[0 \rightarrow_{\{0,\{0,1\}_3\}}1]$$