## FANDOM

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This blog post contains the FGH of Hierarchical Nested Subscript Array Notation. I have also added comparisons to HAN and Dollar Function.

## Up to $$\vartheta(\Omega_{\Omega})$$

FGH HNSAN HAN Dollar Function

$$\vartheta(\Omega_{\omega})$$

$$[1/[2]2]$$ $$[[_{[1]}1(1)2]]$$ $$[0 \rightarrow_{\{0\rightarrow_{\{[0],1\}}1\}} 1]$$

$$\vartheta(\Omega_{\omega2})$$

$$[1/[3]2]$$ $$[[_{[1,2]}1(1)2]]$$ $$[0 \rightarrow_{\{0\rightarrow_{\{[1],1\}}1\}} 1]$$

$$\vartheta(\Omega_{\omega^2})$$

$$[1/[1,2]2]$$

$$[[_{[1,1,2]}1(1)2]]$$ $$[0 \rightarrow_{\{0\rightarrow_{\{[[1]],1\}}1\}} 1]$$
$$\vartheta(\Omega_{\omega^{\omega}})$$ $$[1/[1[2]2]2]$$ $$[[_{[1,1,1,2]}1(1)2]]$$ $$[0 \rightarrow_{\{0\rightarrow_{\{[[[0]]],1\}}1\}} 1]$$
$$\vartheta(\Omega_{\varepsilon_0})$$ $$[1/[1[1/2]2]2]$$ $$[[_{[1,1,1,1,2]}1(1)2]]$$ $$[0 \rightarrow_{\{0\rightarrow_{\{[0]_2,1\}}1\}} 1]$$
$$\vartheta(\Omega_{\Gamma_0})$$ $$[1/[1[1[1/1\sim2]2]2]$$ $$[[_{[1(1)2]}1(1)2]]$$ $$[0 \rightarrow_{\{0\rightarrow_{\{[0,1],1\}}1\}} 1]$$
$$\vartheta(\Omega_{\vartheta(\Omega_{\varepsilon_0})})$$ $$[1/[1[1/[1[1/2]2]2]2]2]$$ $$[[_{[[_{[1,1,1,1,2]}1(1)2]]}1(1)2]]$$ $$[0 \rightarrow_{\{0\rightarrow_{\{[0 \rightarrow_{\{0\rightarrow_{\{[0]_2,1\}}1\}} 1],1\}}1\}} 1]$$
$$\vartheta(\Omega_{\Omega})$$ $$[1/[1/2]2]$$

$$[[_{[_21(1)2]}1(1)2]]$$

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

## Up to $$\psi_{I}(0)$$

FGH HNSAN HAN Dollars Function
$$\vartheta(\Omega_{\Omega})$$ $$[1/[1/2]2]$$ $$[[_{[_21]}1(1)2]]$$ $$[0 \rightarrow_{\{0\rightarrow_{\{\{0\}_2,1\}}1\}} 1]$$
$$\vartheta(\Omega_{\Omega2})$$ $$[1/[1/3]2]$$ $$[[_{[_21,2]}1(1)2]]$$ $$[0 \rightarrow_{\{0\rightarrow_{\{\{0\}_2\{0\}_2,1\}}1\}} 1]$$
$$\vartheta(\Omega_{\Omega3})$$ $$[1/[1/4]2]$$ $$[[_{[_21,3]}1(1)2]]$$ $$[0 \rightarrow_{\{0\rightarrow_{\{\{0\}_2\{0\}_2\{0\}_2,1\}}1\}} 1]$$
$$\vartheta(\Omega_{\Omega*\omega})$$ $$[1/[1/1,2]2]$$ $$[[_{[_21,[1]]}1(1)2]]$$ $$[0 \rightarrow_{\{0\rightarrow_{\{\{\{0\}_2\},1\}}1\}} 1]$$
$$\vartheta(\Omega_{\Omega*\omega^\omega})$$ $$[1/[1/1[2]2]2]$$ $$[[_{[_21,[1,1,2]]}1(1)2]]$$ $$[0 \rightarrow_{\{0\rightarrow_{\{\{\{\{0\}_2\}\},1\}}1\}} 1]$$
$$\vartheta(\Omega_{\Omega*\vartheta(\Omega_{\Omega})})$$ $$[1/[1/1[1/[1/2]2]2]2]$$ $$[[_{[_21,[[_{[_21(1)2]}1(1)2]]]}1(1)2]]$$ $$[0 \rightarrow_{\{0\rightarrow_{\{\{\{\{0\}_2\{0\}_2\}\},1\}}1\}} 1]$$
$$\vartheta(\Omega_{\Omega^2})$$ $$[1/[1/1/2]2]$$ $$[[_{[_21,1,2]}1(1)2]]$$ $$[0 \rightarrow_{\{0\rightarrow_{\{\{\{\{\{0\}_2\}\}\},1\}}1\}} 1]$$
$$\vartheta(\Omega_{\Omega^3})$$ $$[1/[1/1/1/2]2]$$ $$[[_{[_21,1,3]}1(1)2]]$$ $$[0 \rightarrow_{\{0\rightarrow_{\{\{\{\{\{0\}_2\{0\}_2\}\}\},1\}}1\}} 1]$$
$$\vartheta(\Omega_{\Omega^{\Omega}})$$ $$[1/[1[1/2\sim2]2]2]$$ $$[[_{[_21,1,1,2]}1(1)2]]$$ $$[0 \rightarrow_{\{0\rightarrow_{\{\{\{\{\{\{0\}_2\}\}\},1\}}1\}} 1]$$
$$\vartheta(\Omega_{\varepsilon_{\Omega+1}})$$ $$[1/[1[1\sim3]2]2]$$ $$[[_{[_21,1,1,1,2]}1(1)2]]$$ $$[0 \rightarrow_{\{0\rightarrow_{\{\{\{\{\{\{\{0\}_2\}\}\}\},1\}}1\}} 1]$$
$$\vartheta(\Omega_{\varphi(\omega,\Omega+1)})$$ $$[1/[1[2/_32]2]2]$$ $$[[_{[_21,1,1,1,2]w/[1]}1(1)2]]$$ $$[0 \rightarrow_{\{0\rightarrow_{\{\{\{1\}_2,1\}}1\}} 1]$$
$$\vartheta(\Omega_{\Omega_2})$$ $$[1/[1[1\sim 1/_32]2]2]$$ $$[[_{[_31(1)2]}1(1)2]]$$ $$[0 \rightarrow_{\{0\rightarrow_{\{0\rightarrow_{\{0,1\}}1\}}1\}} 1]$$
$$\vartheta(\Omega_{\Omega_{\omega}})$$ $$[1/[1/[2]2]2]$$ $$[[_{[_{[1]}1(1)2]}1(1)2]]$$ $$[0 \rightarrow_{\{0\rightarrow_{\{0\rightarrow_{\{[0],1\}}1\}}1\}} 1]$$
$$\vartheta(\Omega_{\Omega_{\Omega_{\omega}}})$$ $$[1/[1/[1/[2]2]2]2]$$ $$[[_{[_{[_{[1]}1(1)2]}1(1)2]}1(1)2]]$$ $$[0 \rightarrow_{\{0\rightarrow_{\{0\rightarrow_{\{0\rightarrow_{\{[0],1\}}1\}}1\}}1\}} 1]$$

$$\psi_{I}(0)$$

$$[1[1/_{1,2}3]2]$$

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

$$[0 \rightarrow_{\{0\}\text{&}1}1]$$

## Up to the limit of HNSAN

I'm not sure, but I think it reach the limit of Mahlo, the limit of this ( Deedlit11's blog post ), like my & operator.

FGH HNSAN Dollar Function HAN
$$\psi_I(1)$$ $$[1[1/_{1,2}4]2]$$ $$[0 \rightarrow_{\{1\}\text{&}1}1]$$ $$[[_{<1,3>}1]]$$
$$\psi_I(I)$$ $$[1[1/_{1,2}1/2]2]$$ $$[0 \rightarrow_{\{0\}_2\text{&}1}1]$$ $$[[_{<1,1,2>}1]]$$
$$\psi_I(I*\omega)$$ $$[1[1/_{1,2}1/1,2]2]$$ $$[0 \rightarrow_{\{\{0\}_2\}\text{&}1}1]$$ $$[[_{<1,1,[1]>}1]]$$
$$\psi_I(I^2)$$ $$[1[1/_{1,2}1/1/2]2]$$ $$[0 \rightarrow_{\{\{\{0\}_2\}\}\text{&}1}1]$$ $$[[_{<1,1,1,2>}1]]$$
$$\psi_I(I^I)$$ $$[1[1/_{1,2}1/1/1/2]2]$$ $$[0 \rightarrow_{\{\{\{\{0\}_2\}\}\}\text{&}1}1]$$ $$[[_{<1,1,1,1,2>}1]]$$
$$\psi_I(\varepsilon_{I+1})$$ $$[1[1/_{1,2}1[2\sim2]2]2]$$ $$[0 \rightarrow_{\{0\}_3\text{&}1}1]$$ $$[[_{<1,1,1,1,1,2>}1]]$$
$$\psi_I(\zeta_{I+1})$$

$$[1[1/_{1,2}1[1\sim1\sim2]2]2]$$

$$[0 \rightarrow_{\{0\}_4\text{&}1}1]$$ $$[[_{<1,1,1,1,1,1,2>}1]]$$
$$\psi_I(\Gamma_{I+1})$$ $$[1[1/_{1,2}1[1[1\sim1/_32]2]2]2]$$ $$[0 \rightarrow_{\{0,1\}\text{&}1}1]$$ $$[[_{<1(1)2>}1]]$$
$$\psi_{I_2}(0)$$ $$[1[1/_{1,2}1/_{1,2}2]2]$$ $$[0 \rightarrow_{\{0\}\text{&}2}1]$$ $$[[_{<_21(1)2>}1]]$$
$$\psi_{I(1)}(0)$$ $$[1[1[1/_{1,3}2]2]2]$$ $$[0 \rightarrow_{\{0\}\text{&}_21}1]$$

$$[[_{<1(1)2>⁅2⁆}1]]$$

$$\psi_{I(2)}(0)$$ $$[1[1[1[1/_{1,4}3]2]2]2]$$ $$[0 \rightarrow_{\{0\}\text{&}_31}1]$$ $$[[_{<1(1)2>⁅3⁆}1]]$$
$$\psi_{I(1,0)}(0)$$ $$[1[1/_{1,1,2}3]2]$$ $$[0 \rightarrow_{\{0\}\text{&}_{\{0\}_2}1}1]$$ $$[[_{<1(1)2>⁅1,2⁆}1]]$$
$$\psi_{I(2,0)}(0)$$ $$[1[1[1/_{1,1,3}3]2]2]$$ $$[0 \rightarrow_{\{0\}\text{&}_{\{0\}_3}1}1]$$ $$[[_{<1(1)2>⁅1,3⁆}1]]$$
$$\chi(\varepsilon_{M+1})$$ $$[1[1/_{1[1/2]2}3]2]$$ $$[0 \rightarrow_{\{0\}\text{&}_{\{0\rightarrow_21\}}1}1]$$

$$\chi(\Gamma_{M+1})$$

$$[1[1/_{1[1[1\sim2]2]2}3]2]$$ $$[0 \rightarrow_{\{0\}\text{&}_{\{0\rightarrow_{0,1}1\}}1}1]$$
$$\chi(M_2)$$

$$[1[1/_{1[1[1/_{1,1,1,2}3]2]2}2]2]$$

$$[0 \rightarrow_{\{0\}\text{&}_{\{0\}\text{&}_{\{0,1\}}1}1}1]$$
$$\chi(M_{M_2})$$ $$[1[1/_{1[1[1/_{1[1[1/_{1,1,1,2}3]2]2}2]2]2}2]2]$$ $$[0 \rightarrow_{\{0\}\text{&}_{\{0\}\text{&}_{\{0\}\text{&}_{\{0,1\}}1}1}1}1]$$
limit of normal Mahlo the variant of S(n) based on HNSAN $$[0 \rightarrow_{\{0\}\text{|}0\text{|}1} 1]$$

## Extension of HNSAN

Extension of Googleaarex to HNSAN. I hope Bird will define this soon.

FGH HNSAN Dollar Function
$$\psi_{M(3)}(0)$$ $$[1[1/_{1/2}3]2]$$ $$[0 \rightarrow_{\{0\}\text{|}1\text{|}1} 1]$$
$$\psi_{M(4)}(0)$$ $$[1[1/_{1/2}4]2]$$ $$[0 \rightarrow_{\{0\}\text{|}2\text{|}1} 1]$$
$$\psi_{M(1,0)}(0)$$ $$[1[1/_{1/2}1/2]2]$$ $$[0 \rightarrow_{\{0\}\text{|}\{0\}_2\text{|}1} 1]$$
$$\psi_{\Xi(3,0)}(2,\Xi(3,0)^\omega)$$ $$[1[1/_{1/2}1[1/1,2 \sim 2]2]2]$$ $$[0 \rightarrow_{\{0\}\text{|}\{0,0,\{0,0,1\}_{\{0\}}\}\text{|}1} 1]$$
$$\psi_{\Xi(3,0)}(3,0)$$ $$[1[1/_{1/2}1/_{1/2}2]2]$$

$$[0 \rightarrow_{\{0\}\text{|}0,1\text{|}1} 1]$$

$$\psi_{\Xi(4,0)}(0)$$ $$[1[1[1/_{1/3}2]2]2]$$ $$[0 \rightarrow_{\{0\}\text{|}2\text{|}_21} 1]$$
$$\psi_{\Xi(\omega,0)}(0)$$ $$[1[1[1/_{1/1,2}2]2]2]$$ $$[0 \rightarrow_{\{0\}\text{|}\{0\}\text{|}_21} 1]$$
$$\psi_{\Xi(1,0,0)}(0)$$ $$[1[1[1/_{1/1/2}2]2]2]$$ $$[0 \rightarrow_{\{0\}\text{|}\{0\}_2\text{|}_21} 1]$$
$$\psi_{\Xi(\varepsilon_{K+1})}(0)$$ $$[1[1[1/_{1\sim1\sim...1\sim1\sim 2}2]2]2]$$ $$[0 \rightarrow_{\{0\}\text{|}\{0\rightarrow1\}\text{|}_21} 1]$$
$$\psi_{\Xi(K_2)}(0)$$ $$[1[1[1/_{1/_{1/2}2}2]2]2]$$ $$[0 \rightarrow_{\{0\}\text{|}\{0\}\text{|}_31} 1]$$
$$\psi_{\Xi(K_{K_2})}(0)$$ $$[1[1[1/_{1/_{1/_{1/2}2}2}2]2]2]$$ $$[0 \rightarrow_{\{0\}\text{|}\{0\}\text{|}_{\{0\}\text{|}\{0\}\text{|}1}1} 1]$$
limit of compact ordinal limit of xHNSAN

current treasure function, limit of Dollar Function for now