finite ordinals (\(0\) ~ \(\mathrm{FTO} = \omega\))
| BMS | BOCF | normal notation, NN |
|---|---|---|
| \(\emptyset\) | \(0\) | \(0\) |
| \(()\) | \(\psi(0)\) | \(1\) |
| \(()()\) | \(\psi(0)2\) | \(2\) |
| \(()()()\) | \(\psi(0)3\) | \(3\) |
| \(()(1)\) | \(\psi(\psi(0))\) | \(\mathrm{FTO} = \omega\) |
one-column BMS (\(\omega\) ~ \(\mathrm{SCO} = \varepsilon_0\))
\(\omega\) ~ \(\omega^2\)
| BMS | BOCF | NN |
|---|---|---|
| \(()(1)\) | \(\psi(\psi(0))\) | \(\omega\) |
| \(()(1)()\) | \(\psi(\psi(0))+\psi(0)\) | \(\omega+1\) |
| \(()(1)()()\) | \(\psi(\psi(0))+\psi(0)2\) | \(\omega+2\) |
| \(()(1)()(1)\) | \(\psi(\psi(0))2\) | \(\omega 2\) |
| \(()(1)()(1)()\) | \(\psi(\psi(0))2+\psi(0)\) | \(\omega 2+1\) |
| \(()(1)()(1)()(1)\) | \(\psi(\psi(0))3\) | \(\omega 3\) |
| \(()(1)(1)\) | \(\psi(\psi(0)2)\) | \(\omega^2\) |
\(\omega^2\) ~ \(\omega^\omega\)
| BMS | BOCF | NN |
|---|---|---|
| \(()(1)(1)\) | \(\psi(\psi(0)2)\) | \(\omega^2\) |
| \(()(1)(1)()\) | \(\psi(\psi(0)2)+\psi(0)\) | \(\omega^2+1\) |
| \(()(1)(1)()(1)\) | \(\psi(\psi(0)2)+\psi(\psi(0))\) | \(\omega^2+\omega\) |
| \(()(1)(1)()(1)()(1)\) | \(\psi(\psi(0)2)+\psi(\psi(0))2\) | \(\omega^2+\omega 2\) |
| \(()(1)(1)()(1)(1)\) | \(\psi(\psi(0)2)2\) | \(\omega^2 2\) |
| \(()(1)(1)()(1)(1)()(1)(1)\) | \(\psi(\psi(0)2)3\) | \(\omega^2 3\) |
| \(()(1)(1)(1)\) | \(\psi(\psi(0)3)\) | \(\omega^3\) |
| \(()(1)(1)(1)()\) | \(\psi(\psi(0)3)+\psi(0)\) | \(\omega^3+1\) |
| \(()(1)(1)(1)()(1)\) | \(\psi(\psi(0)3)+\psi(\psi(0))\) | \(\omega^3+\omega\) |
| \(()(1)(1)(1)()(1)(1)\) | \(\psi(\psi(0)3)+\psi(\psi(0)2)\) | \(\omega^3+\omega^2\) |
| \(()(1)(1)(1)()(1)(1)(1)\) | \(\psi(\psi(0)3)2\) | \(\omega^3 2\) |
| \(()(1)(1)(1)(1)\) | \(\psi(\psi(0)4)\) | \(\omega^4\) |
| \(()(1)(2)\) | \(\psi(\psi(\psi(0)))\) | \(\omega^\omega\) |
\(\omega^\omega\) ~ \(\omega^{\omega^2}\)
Using \(1 = \psi(0)\) in BOCF.
| BMS | BOCF | NN |
|---|---|---|
| \(()(1)(2)\) | \(\psi(\psi(1))\) | \(\omega^\omega\) |
| \(()(1)(2)()\) | \(\psi(\psi(1))+1\) | \(\omega^\omega+1\) |
| \(()(1)(2)()(1)\) | \(\psi(\psi(1))+\psi(1)\) | \(\omega^\omega+\omega\) |
| \(()(1)(2)()(1)(1)\) | \(\psi(\psi(1))+\psi(2)\) | \(\omega^\omega+\omega^2\) |
| \(()(1)(2)()(1)(2)\) | \(\psi(\psi(1))2\) | \(\omega^\omega 2\) |
| \(()(1)(2)(1)\) | \(\psi(\psi(1)+1)\) | \(\omega^{\omega+1}\) |
| \(()(1)(2)(1)()(1)(2)\) | \(\psi(\psi(1)+1)+\psi(\psi(1))\) | \(\omega^{\omega+1}+\omega^\omega\) |
| \(()(1)(2)(1)()(1)(2)(1)\) | \(\psi(\psi(1)+1)2\) | \(\omega^{\omega+1} 2\) |
| \(()(1)(2)(1)(1)\) | \(\psi(\psi(1)+2)\) | \(\omega^{\omega+2}\) |
| \(()(1)(2)(1)(2)\) | \(\psi(\psi(1)2)\) | \(\omega^{\omega 2}\) |
| \(()(1)(2)(1)(2)()(1)(2)(1)(2)\) | \(\psi(\psi(1)2)2\) | \(\omega^{\omega 2} 2\) |
| \(()(1)(2)(1)(2)(1)\) | \(\psi(\psi(1)2+1)\) | \(\omega^{\omega 2+1}\) |
| \(()(1)(2)(1)(2)(1)(1)\) | \(\psi(\psi(1)2+2)\) | \(\omega^{\omega 2+2}\) |
| \(()(1)(2)(1)(2)(1)(2)\) | \(\psi(\psi(1)3)\) | \(\omega^{\omega 3}\) |
| \(()(1)(2)(2)\) | \(\psi(\psi(2))\) | \(\omega^{\omega^2}\) |
\(\omega^{\omega^2}\) ~ \(\mathrm{SCO} = \varepsilon_0\)
Using \(\omega= \psi(1)\) in BOCF.
| BMS | BOCF | NN |
|---|---|---|
| \(()(1)(2)(2)\) | \(\psi(\psi(2))\) | \(\omega^{\omega^2}\) |
| \(()(1)(2)(2)(1)\) | \(\psi(\psi(2)+1)\) | \(\omega^{\omega^2+1}\) |
| \(()(1)(2)(2)(1)(2)\) | \(\psi(\psi(2)+\omega)\) | \(\omega^{\omega^2+\omega}\) |
| \(()(1)(2)(2)(1)(2)(1)(2)\) | \(\psi(\psi(2)+\omega 2)\) | \(\omega^{\omega^2+\omega 2}\) |
| \(()(1)(2)(2)(1)(2)(2)\) | \(\psi(\psi(2)+\psi(2))\) | \(\omega^{\omega^2 2}\) |
| \(()(1)(2)(2)(2)\) | \(\psi(\psi(3))\) | \(\omega^{\omega^3}\) |
| \(()(1)(2)(3)\) | \(\psi(\psi(\omega))\) | \(\omega^{\omega^\omega}\) |
| \(()(1)(2)(3)(1)\) | \(\psi(\psi(\omega)+1)\) | \(\omega^{\omega^\omega+1}\) |
| \(()(1)(2)(3)(1)(2)\) | \(\psi(\psi(\omega)+\omega)\) | \(\omega^{\omega^\omega+\omega}\) |
| \(()(1)(2)(3)(1)(2)(3)\) | \(\psi(\psi(\omega)2)\) | \(\omega^{\omega^{\omega 2}}\) |
| \(()(1)(2)(3)(4)\) | \(\psi(\psi(\psi(\omega)))\) | \(\omega^{\omega^{\omega^\omega}}\) |
| \(()(1,1)\) | \(\psi(\psi_1(0))\) | \(\mathrm{SCO} = \varepsilon_0\) |
two-column BMS (\(\varepsilon_0\) ~ \(\mathrm{BO} = \psi(\Omega_\omega)\))
\(\varepsilon_0\) ~ \(\mathrm{CO} = \zeta_0\)
\(\varepsilon_0\) ~ \(\varepsilon_1\)
Using \(\forall \alpha< \varepsilon_0, \omega^\alpha= \psi(\alpha)\) in BOCF.
| BMS | BOCF | NN |
|---|---|---|
| \(()(1,1)\) | \(\psi(\psi_1(0))\) | \(\varepsilon_0\) |
| \(()(1,1)()(1,1)\) | \(\psi(\psi_1(0))2\) | \(\varepsilon_0 2\) |
| \(()(1,1)(1)\) | \(\psi(\psi_1(0)+1)\) | \(\omega^{\varepsilon_0+1}\) |
| \(()(1,1)(1)(1)\) | \(\psi(\psi_1(0)+2)\) | \(\omega^{\varepsilon_0+2}\) |
| \(()(1,1)(1)(2)\) | \(\psi(\psi_1(0)+\omega)\) | \(\omega^{\varepsilon_0+\omega}\) |
| \(()(1,1)(1)(2)(2)\) | \(\psi(\psi_1(0)+\omega^2)\) | \(\omega^{\varepsilon_0+\omega^2}\) |
| \(()(1,1)(1)(2)(3)\) | \(\psi(\psi_1(0)+\omega^3)\) | \(\omega^{\varepsilon_0+\omega^3}\) |
| \(()(1,1)(1)(2,1)\) | \(\psi(\psi_1(0)+\psi(\psi_1(0)))\) | \(\omega^{\varepsilon_0 2}\) |
| \(()(1,1)(1)(2,1)(1)(2,1)\) | \(\psi(\psi_1(0)+\psi(\psi_1(0))2)\) | \(\omega^{\varepsilon_0 3}\) |
| \(()(1,1)(1)(2,1)(2)\) | \(\psi(\psi_1(0)+\psi(\psi_1(0)+1))\) | \(\omega^{\omega^{\varepsilon_0+1}}\) |
| \(()(1,1)(1)(2,1)(2)(1)(2,1)\) | \(\psi(\psi_1(0)+\psi(\psi_1(0)+1)+\psi(\psi_1(0)))\) | \(\omega^{\omega^{\varepsilon_0+1}+\varepsilon_0}\) |
| \(()(1,1)(1)(2,1)(2)(1)(2,1)(1)(2,1)\) | \(\psi(\psi_1(0)+\psi(\psi_1(0)+1)+\psi(\psi_1(0))2)\) | \(\omega^{\omega^{\varepsilon_0+1}+\varepsilon_0 2}\) |
| \(()(1,1)(1)(2,1)(2)(1)(2,1)(2)\) | \(\psi(\psi_1(0)+\psi(\psi_1(0)+1)2)\) | \(\omega^{\omega^{\varepsilon_0+1} 2}\) |
| \(()(1,1)(1)(2,1)(2)(2)\) | \(\psi(\psi_1(0)+\psi(\psi_1(0)+2))\) | \(\omega^{\omega^{\varepsilon_0+2}}\) |
| \(()(1,1)(1)(2,1)(2)(3)\) | \(\psi(\psi_1(0)+\psi(\psi_1(0)+\omega))\) | \(\omega^{\omega^{\varepsilon_0+\omega}}\) |
| \(()(1,1)(1)(2,1)(2)(3,1)\) | \(\psi(\psi_1(0)+\psi(\psi_1(0)+\psi(\psi_1(0))))\) | \(\omega^{\omega^{\varepsilon_0 2}}\) |
| \(()(1,1)(1,1)\) | \(\psi(\psi_1(0)2)\) | \(\varepsilon_1\) |
\(\varepsilon_1\) ~ \(\varepsilon_{\varepsilon_0}\)
| BMS | BOCF | NN |
|---|---|---|
| \(()(1,1)(1,1)\) | \(\psi(\psi_1(0)2)\) | \(\varepsilon_1\) |
| \(()(1,1)(1,1)(1)\) | \(\psi(\psi_1(0)2+1)\) | \(\omega^{\varepsilon_1+1}\) |
| \(()(1,1)(1,1)(1)(2,1)\) | \(\psi(\psi_1(0)2+\psi(\psi_1(0)))\) | \(\omega^{\varepsilon_1+\varepsilon_0}\) |
| \(()(1,1)(1,1)(1)(2,1)(2,1)\) | \(\psi(\psi_1(0)2+\psi(\psi_1(0)2))\) | \(\omega^{\varepsilon_1 2}\) |
| \(()(1,1)(1,1)(1)(2,1)(2,1)(2)(3,1)(3,1)\) | \(\psi(\psi_1(0)2+\psi(\psi_1(0)2+\psi(\psi_1(0)2)))\) | \(\omega^{\omega^{\varepsilon_1 2}}\) |
| \(()(1,1)(1,1)(1,1)\) | \(\psi(\psi_1(0)3)\) | \(\varepsilon_2\) |
| \(()(1,1)(2)\) | \(\psi(\psi_1(1))\) | \(\varepsilon_\omega\) |
| \(()(1,1)(2)(1)(2,1)(3)\) | \(\psi(\psi_1(1)+\psi(\psi_1(1)))\) | \(\omega^{\varepsilon_\omega 2}\) |
| \(()(1,1)(2)(1,1)\) | \(\psi(\psi_1(1)+\psi_1(0))\) | \(\varepsilon_{\omega+1}\) |
| \(()(1,1)(2)(1,1)(1,1)\) | \(\psi(\psi_1(1)+\psi_1(0)2)\) | \(\varepsilon_{\omega+2}\) |
| \(()(1,1)(2)(1,1)(2)\) | \(\psi(\psi_1(1)2)\) | \(\varepsilon_{\omega 2}\) |
| \(()(1,1)(2)(2)\) | \(\psi(\psi_1(2))\) | \(\varepsilon_{\omega^2}\) |
| \(()(1,1)(2)(3)\) | \(\psi(\psi_1(\omega))\) | \(\varepsilon_{\omega^\omega}\) |
| \(()(1,1)(2)(3)(4)\) | \(\psi(\psi_1(\omega^\omega))\) | \(\varepsilon_{\omega^{\omega^\omega}}\) |
| \(()(1,1)(2)(3,1)\) | \(\psi(\psi_1(\psi(\psi_1(0))))\) | \(\varepsilon_{\varepsilon_0}\) |
\(\varepsilon_{\varepsilon_0}\) ~ \(\mathrm{CO} = \zeta_0\)
| BMS | BOCF | NN |
|---|---|---|
| \(()(1,1)(2)(3,1)\) | \(\psi(\psi_1(\psi(\psi_1(0))))\) | \(\varepsilon_{\varepsilon_0}\) |
| \(()(1,1)(2)(1)(2,1)(3)\) | \(\psi(\psi_1(1)+\psi(\psi_1(1)))\) | \(\omega^{\varepsilon_\omega 2}\) |
| \(()(1,1)(2)(1,1)\) | \(\psi(\psi_1(1)+\psi_1(0))\) | \(\varepsilon_{\omega+1}\) |
| \(()(1,1)(2)(1,1)(1,1)\) | \(\psi(\psi_1(1)+\psi_1(0)2)\) | \(\varepsilon_{\omega+2}\) |
| \(()(1,1)(2)(1,1)(2)\) | \(\psi(\psi_1(1)2)\) | \(\varepsilon_{\omega 2}\) |
| \(()(1,1)(2)(2)\) | \(\psi(\psi_1(2))\) | \(\varepsilon_{\omega^2}\) |
| \(()(1,1)(2)(3)\) | \(\psi(\psi_1(\omega))\) | \(\varepsilon_{\omega^\omega}\) |
| \(()(1,1)(2)(3)(4)\) | \(\psi(\psi_1(\omega^\omega))\) | \(\varepsilon_{\omega^{\omega^\omega}}\) |
| \(()(1,1)(2)(3,1)\) | \(\psi(\psi_1(\psi(\psi_1(0))))\) | \(\varepsilon_{\varepsilon_0}\) |
| \(()(1,1)(2)(3,1)(1,1)\) | \(\psi(\psi_1(\psi(\psi_1(0)))+\psi_1(0))\) | \(\varepsilon_{\varepsilon_0+1}\) |
| \(()(1,1)(2)(3,1)(1,1)(2)(3,1)\) | \(\psi(\psi_1(\psi(\psi_1(0)))2)\) | \(\varepsilon_{\varepsilon_0 2}\) |
| \(()(1,1)(2)(3,1)(2)\) | \(\psi(\psi_1(\psi(\psi_1(0))+1))\) | \(\varepsilon_{\omega^{\varepsilon_0+1}}\) |
| \(()(1,1)(2)(3,1)(2)(1,1)(2)(3,1)(2)\) | \(\psi(\psi_1(\psi(\psi_1(0))+1)2)\) | \(\varepsilon_{\omega^{\varepsilon_0+1}2}\) |
| \(()(1,1)(2)(3,1)(2)(2)\) | \(\psi(\psi_1(\psi(\psi_1(0))+2))\) | \(\varepsilon_{\omega^{\varepsilon_0+2}}\) |
| \(()(1,1)(2)(3,1)(2)(3)\) | \(\psi(\psi_1(\psi(\psi_1(0))+\omega))\) | \(\varepsilon_{\omega^{\varepsilon_0+\omega}}\) |
| \(()(1,1)(2)(3,1)(2)(3,1)\) | \(\psi(\psi_1(\psi(\psi_1(0))2)\) | \(\varepsilon_{\omega^{\varepsilon_0 2}}\) |
| \(()(1,1)(2)(3,1)(3)\) | \(\psi(\psi_1(\psi(\psi_1(0)+1))\) | \(\varepsilon_{\omega^{\omega^{\varepsilon_0+1}}}\) |
| \(()(1,1)(2)(3,1)(3)(4)\) | \(\psi(\psi_1(\psi(\psi_1(0)+\omega))\) | \(\varepsilon_{\omega^{\omega^{\varepsilon_0+\omega}}}\) |
| \(()(1,1)(2)(3,1)(3)(4,1)\) | \(\psi(\psi_1(\psi(\psi_1(0)+\psi(\psi_1(0))))\) | \(\varepsilon_{\omega^{\omega^{\varepsilon_0 2}}}\) |
| \(()(1,1)(2)(3,1)(3,1)\) | \(\psi(\psi_1(\psi(\psi_1(0)2))\) | \(\varepsilon_{\varepsilon_1}\) |
| \(()(1,1)(2)(3,1)(4)\) | \(\psi(\psi_1(\psi(\psi_1(1)))\) | \(\varepsilon_{\varepsilon_\omega}\) |
| \(()(1,1)(2)(3,1)(4)(5,1)\) | \(\psi(\psi_1(\psi(\psi_1(\psi(\psi_1(0)))))\) | \(\varepsilon_{\varepsilon_{\varepsilon_0}}\) |
| \(()(1,1)(2,1)\) | \(\psi(\psi_1(\psi_1(0)))\) | \(\mathrm{CO} = \zeta_0\) |
\(\zeta_0\) ~ \(\mathrm{FSO} = \Gamma_0 = \varphi(1,0,0)\)
\(\zeta_0\) ~ \(\mathrm{HCO} = \varphi(\omega,0)\)
Using \(\forall \alpha< \psi_1(\psi_2(0)) = \varepsilon_{\Omega+1}, \omega^{\Omega+\alpha} = \psi_1(\alpha)\) in BOCF.
| BMS | BOCF | NN |
|---|---|---|
| \(()(1,1)(2,1)\) | \(\psi(\omega^{\Omega 2})\) | \(\zeta_0\) |
| \(()(1,1)(2,1)(1)\) | \(\psi(\omega^{\Omega 2}+1)\) | \(\omega^{\zeta_0+1}\) |
| \(()(1,1)(2,1)(1)(2,1)\) | \(\psi(\omega^{\Omega 2}+\psi(\Omega))\) | \(\omega^{\zeta_0+\varepsilon_0}\) |
| \(()(1,1)(2,1)(1)(2,1)(3,1)\) | \(\psi(\omega^{\Omega 2}+\psi(\omega^{\Omega 2}))\) | \(\omega^{\zeta_0 2}\) |
| \(()(1,1)(2,1)(1,1)\) | \(\psi(\omega^{\Omega 2}+\Omega)\) | \(\varepsilon_{\zeta_0+1}\) |
| \(()(1,1)(2,1)(1,1)(2)\) | \(\psi(\omega^{\Omega 2}+\omega^{\Omega+1}))\) | \(\varepsilon_{\zeta_0+\omega}\) |
| \(()(1,1)(2,1)(1,1)(2)(3,1)(4,1)\) | \(\psi(\omega^{\Omega 2}+\omega^{\Omega+\psi(\omega^{\Omega 2})})\) | \(\varepsilon_{\zeta_0 2}\) |
| \(()(1,1)(2,1)(1,1)(2)(3,1)(4,1)(3,1)\) | \(\psi(\omega^{\Omega 2}+\omega^{\Omega+\psi(\omega^{\Omega 2}+\Omega)})\) | \(\varepsilon_{\varepsilon_{\zeta_0+1}}\) |
| \(()(1,1)(2,1)(1,1)(2,1)\) | \(\psi(\omega^{\Omega 2}2)\) | \(\zeta_1\) |
| \(()(1,1)(2,1)(1,1)(2,1)(1,1)\\(2)(3,1)(4,1)(3,1)(4,1)\) | \(\psi(\omega^{\Omega 2}2+\psi(\omega^{\Omega 2}))\) | \(\varepsilon_{\zeta_1 2}\) |
| \(()(1,1)(2,1)(1,1)(2,1)(1,1)(2,1)\) | \(\psi(\omega^{\Omega 2}3)\) | \(\zeta_2\) |
| \(()(1,1)(2,1)(2)\) | \(\psi(\omega^{\Omega 2+1})\) | \(\zeta_\omega\) |
| \(()(1,1)(2,1)(2)(1,1)(2,1)\) | \(\psi(\omega^{\Omega 2+1}+\omega^{\Omega 2})\) | \(\zeta_{\omega+1}\) |
| \(()(1,1)(2,1)(2)(1,1)(2,1)(2)\) | \(\psi(\omega^{\Omega 2+1}2)\) | \(\zeta_{\omega 2}\) |
| \(()(1,1)(2,1)(2)(2)\) | \(\psi(\omega^{\Omega 2+2})\) | \(\zeta_{\omega^2}\) |
| \(()(1,1)(2,1)(2)(3)\) | \(\psi(\omega^{\Omega 2+\omega})\) | \(\zeta_{\omega^\omega}\) |
| \(()(1,1)(2,1)(2)(3,1)\) | \(\psi(\omega^{\Omega 2+\psi(\Omega)})\) | \(\zeta_{\varepsilon_0}\) |
| \(()(1,1)(2,1)(2)(3,1)(4,1)\) | \(\psi(\omega^{\Omega 2+\psi(\omega^{\Omega 2})})\) | \(\zeta_{\zeta_0}\) |
| \(()(1,1)(2,1)(2,1)\) | \(\psi(\omega^{\Omega 3})\) | \(\eta_0\) |
| \(()(1,1)(2,1)(2,1)(1,1)\) | \(\psi(\omega^{\Omega 3}+\Omega)\) | \(\varepsilon_{\eta_0+1}\) |
| \(()(1,1)(2,1)(2,1)(1,1)(2,1)\) | \(\psi(\omega^{\Omega 3}+\omega^{\Omega 2})\) | \(\zeta_{\eta_0+1}\) |
| \(()(1,1)(2,1)(2,1)(1,1)(2,1)(2,1)\) | \(\psi(\omega^{\Omega 3}2)\) | \(\eta_1\) |
| \(()(1,1)(2,1)(2,1)(2)\) | \(\psi(\omega^{\Omega 3+1})\) | \(\eta_\omega\) |
| \(()(1,1)(2,1)(2,1)(2)(3,1)(4,1)(4,1)\) | \(\psi(\omega^{\Omega 3+\psi(\omega^{\Omega 3})})\) | \(\eta_{\eta_0}\) |
| \(()(1,1)(2,1)(2,1)(2,1)\) | \(\psi(\omega^{\Omega 4})\) | \(\varphi(4,0)\) |
| \(()(1,1)(2,1)(3)\) | \(\psi(\omega^{\omega^{\Omega+1}})\) | \(\mathrm{HCO} = \varphi(\omega,0)\) |
\(\varphi(\omega,0)\) ~ \(\mathrm{FSO} = \Gamma_0 = \varphi(1,0,0)\)
| BMS | BOCF | NN |
|---|---|---|
| \(()(1,1)(2,1)(3)\) | \(\psi(\omega^{\omega^{\Omega+1}})\) | \(\varphi(\omega,0)\) |
| \(()(1,1)(2,1)(3)(1,1)\) | \(\psi(\omega^{\omega^{\Omega+1}}+\Omega)\) | \(\varepsilon_{\varphi(\omega,0)+1}\) |
| \(()(1,1)(2,1)(3)(1,1)(2,1)\) | \(\psi(\omega^{\omega^{\Omega+1}}+\omega^{\Omega 2})\) | \(\zeta_{\varphi(\omega,0)+1}\) |
| \(()(1,1)(2,1)(3)(1,1)(2,1)(2,1)\) | \(\psi(\omega^{\omega^{\Omega+1}}+\omega^{\Omega 3})\) | \(\eta_{\varphi(\omega,0)+1}\) |
| \(()(1,1)(2,1)(3)(1,1)(2,1)(3)\) | \(\psi(\omega^{\omega^{\Omega+1}}2)\) | \(\varphi(\omega,1)\) |
| \(()(1,1)(2,1)(3)(2)\) | \(\psi(\omega^{\omega^{\Omega+1}+1})\) | \(\varphi(\omega,\omega)\) |
| \(()(1,1)(2,1)(3)(2)(3,1)\) | \(\psi(\omega^{\omega^{\Omega+1}+\psi(\Omega)})\) | \(\varphi(\omega,\varepsilon_0)\) |
| \(()(1,1)(2,1)(3)(2)(3,1)(4,1)\) | \(\psi(\omega^{\omega^{\Omega+1}+\psi(\omega^{\Omega 2})})\) | \(\varphi(\omega,\zeta_0)\) |
| \(()(1,1)(2,1)(3)(2)(3,1)(4,1)(5)\) | \(\psi(\omega^{\omega^{\Omega+1}+\psi(\omega^{\omega^{\Omega+1}})})\) | \(\varphi(\omega,\varphi(\omega,0))\) |
| \(()(1,1)(2,1)(3)(2,1)\) | \(\psi(\omega^{\omega^{\Omega+1}+\Omega})\) | \(\varphi(\omega+1,0)\) |
| \(()(1,1)(2,1)(3)(2,1)(1,1)(2,1)(3)(2,1)\) | \(\psi(\omega^{\omega^{\Omega+1}+\Omega}2)\) | \(\varphi(\omega+1,1)\) |
| \(()(1,1)(2,1)(3)(2,1)(2)\) | \(\psi(\omega^{\omega^{\Omega+1}+\Omega+1})\) | \(\varphi(\omega+1,\omega)\) |
| \(()(1,1)(2,1)(3)(2,1)(2)(3,1)(4,1)(5)(4,1)\) | \(\psi(\omega^{\omega^{\Omega+1}+\Omega+\psi(\omega^{\omega^{\Omega+1}+\Omega})})\) | \(\varphi(\omega+1,\varphi(\omega+1,0))\) |
| \(()(1,1)(2,1)(3)(2,1)(2,1)\) | \(\psi(\omega^{\omega^{\Omega+1}+\Omega 2})\) | \(\varphi(\omega+2,0)\) |
| \(()(1,1)(2,1)(3)(2,1)(3)\) | \(\psi(\omega^{\omega^{\Omega+1}2})\) | \(\varphi(\omega 2,0)\) |
| \(()(1,1)(2,1)(3)(3)\) | \(\psi(\omega^{\omega^{\Omega+2}})\) | \(\varphi(\omega^2,0)\) |
| \(()(1,1)(2,1)(3)(4)\) | \(\psi(\omega^{\omega^{\Omega+\omega}})\) | \(\varphi(\omega^\omega,0)\) |
| \(()(1,1)(2,1)(3)(4,1)\) | \(\psi(\omega^{\omega^{\Omega+\psi(\Omega)}})\) | \(\varphi(\varepsilon_0,0)\) |
| \(()(1,1)(2,1)(3)(4,1)(5,1)\) | \(\psi(\omega^{\omega^{\Omega+\psi(\omega^{\Omega 2})}})\) | \(\varphi(\zeta_0,0)\) |
| \(()(1,1)(2,1)(3)(4,1)(5,1)(6)\) | \(\psi(\omega^{\omega^{\Omega+\psi(\omega^{\omega^{\Omega+1}})}})\) | \(\varphi(\varphi(\omega,0),0)\) |
| \(()(1,1)(2,1)(3)(4,1)(5,1)(6)(7,1)(8,1)\) | \(\psi(\omega^{\omega^{\Omega+\psi(\omega^{\omega^{\Omega+\psi(\omega^{\Omega 2})}})}})\) | \(\varphi(\varphi(\zeta_0,0),0)\) |
| \(()(1,1)(2,1)(3,1)\) | \(\psi(\omega^{\omega^{\Omega 2}})\) | \(\mathrm{FSO} = \Gamma_0\) |
\(\Gamma_0\) ~ \(\mathrm{SVO} = \varphi(1@\omega)\)
\(\varphi(1@\omega)\) ~ \(\mathrm{LVO} = \varphi(1@(1,0))\)
\(\varphi(1@(1,0))\) ~ \(\mathrm{BHO} = \psi(\Omega_2)\)
\(\psi(\Omega_2)\) ~ \(\mathrm{BO} = \psi(\Omega_\omega)\)
To be continued …