Mainly using BOCF with \(M\), the recursive Mahlo ordinal. For ordinals less than \(\mathrm{BHO} = \psi(\Omega_2)\), normal notation with extended Veblen’s function is also used.
Contents:
- part 1, to \(\mathrm{SCO}\)
- part 2, to \(\mathrm{FSO}\)
- part 3, to \(\mathrm{LVO}\)
- part 4, to \(\mathrm{BHO}\)
- part 5, to \(\mathrm{BO}\)
finite ordinals (\(0\) ~ \(\mathrm{FTO} = \omega\))
| \(\emptyset\) |
\(0\) |
\(0\) |
| \(()\) |
\(\psi(0)\) |
\(1\) |
| \(()()\) |
\(\psi(0)2\) |
\(2\) |
| \(()()()\) |
\(\psi(0)3\) |
\(3\) |
| \(()(1)\) |
\(\psi(\psi(0))\) |
\(\mathrm{FTO} = \omega\) |
\(\omega\) ~ \(\omega^2\)
| \(()(1)\) |
\(\psi(\psi(0))\) |
\(\omega\) |
| \(()(1)()\) |
\(\psi(\psi(0))+\psi(0)\) |
\(\omega+1\) |
| \(()(1)()()\) |
\(\psi(\psi(0))+\psi(0)2\) |
\(\omega+2\) |
| \(()(1)()(1)\) |
\(\psi(\psi(0))2\) |
\(\omega 2\) |
| \(()(1)()(1)()\) |
\(\psi(\psi(0))2+\psi(0)\) |
\(\omega 2+1\) |
| \(()(1)()(1)()(1)\) |
\(\psi(\psi(0))3\) |
\(\omega 3\) |
| \(()(1)(1)\) |
\(\psi(\psi(0)2)\) |
\(\omega^2\) |
\(\omega^2\) ~ \(\omega^\omega\)
| \(()(1)(1)\) |
\(\psi(\psi(0)2)\) |
\(\omega^2\) |
| \(()(1)(1)()\) |
\(\psi(\psi(0)2)+\psi(0)\) |
\(\omega^2+1\) |
| \(()(1)(1)()(1)\) |
\(\psi(\psi(0)2)+\psi(\psi(0))\) |
\(\omega^2+\omega\) |
| \(()(1)(1)()(1)()(1)\) |
\(\psi(\psi(0)2)+\psi(\psi(0))2\) |
\(\omega^2+\omega 2\) |
| \(()(1)(1)()(1)(1)\) |
\(\psi(\psi(0)2)2\) |
\(\omega^2 2\) |
| \(()(1)(1)()(1)(1)()(1)(1)\) |
\(\psi(\psi(0)2)3\) |
\(\omega^2 3\) |
| \(()(1)(1)(1)\) |
\(\psi(\psi(0)3)\) |
\(\omega^3\) |
| \(()(1)(1)(1)()\) |
\(\psi(\psi(0)3)+\psi(0)\) |
\(\omega^3+1\) |
| \(()(1)(1)(1)()(1)\) |
\(\psi(\psi(0)3)+\psi(\psi(0))\) |
\(\omega^3+\omega\) |
| \(()(1)(1)(1)()(1)(1)\) |
\(\psi(\psi(0)3)+\psi(\psi(0)2)\) |
\(\omega^3+\omega^2\) |
| \(()(1)(1)(1)()(1)(1)(1)\) |
\(\psi(\psi(0)3)2\) |
\(\omega^3 2\) |
| \(()(1)(1)(1)(1)\) |
\(\psi(\psi(0)4)\) |
\(\omega^4\) |
| \(()(1)(2)\) |
\(\psi(\psi(\psi(0)))\) |
\(\omega^\omega\) |
\(\omega^\omega\) ~ \(\omega^{\omega^2}\)
Using \(1 = \psi(0)\) in BOCF.
| \(()(1)(2)\) |
\(\psi(\psi(1))\) |
\(\omega^\omega\) |
| \(()(1)(2)()\) |
\(\psi(\psi(1))+1\) |
\(\omega^\omega+1\) |
| \(()(1)(2)()(1)\) |
\(\psi(\psi(1))+\psi(1)\) |
\(\omega^\omega+\omega\) |
| \(()(1)(2)()(1)(1)\) |
\(\psi(\psi(1))+\psi(2)\) |
\(\omega^\omega+\omega^2\) |
| \(()(1)(2)()(1)(2)\) |
\(\psi(\psi(1))2\) |
\(\omega^\omega 2\) |
| \(()(1)(2)(1)\) |
\(\psi(\psi(1)+1)\) |
\(\omega^{\omega+1}\) |
| \(()(1)(2)(1)()(1)(2)\) |
\(\psi(\psi(1)+1)+\psi(\psi(1))\) |
\(\omega^{\omega+1}+\omega^\omega\) |
| \(()(1)(2)(1)()(1)(2)(1)\) |
\(\psi(\psi(1)+1)2\) |
\(\omega^{\omega+1} 2\) |
| \(()(1)(2)(1)(1)\) |
\(\psi(\psi(1)+2)\) |
\(\omega^{\omega+2}\) |
| \(()(1)(2)(1)(2)\) |
\(\psi(\psi(1)2)\) |
\(\omega^{\omega 2}\) |
| \(()(1)(2)(1)(2)()(1)(2)(1)(2)\) |
\(\psi(\psi(1)2)2\) |
\(\omega^{\omega 2} 2\) |
| \(()(1)(2)(1)(2)(1)\) |
\(\psi(\psi(1)2+1)\) |
\(\omega^{\omega 2+1}\) |
| \(()(1)(2)(1)(2)(1)(1)\) |
\(\psi(\psi(1)2+2)\) |
\(\omega^{\omega 2+2}\) |
| \(()(1)(2)(1)(2)(1)(2)\) |
\(\psi(\psi(1)3)\) |
\(\omega^{\omega 3}\) |
| \(()(1)(2)(2)\) |
\(\psi(\psi(2))\) |
\(\omega^{\omega^2}\) |
\(\omega^{\omega^2}\) ~ \(\mathrm{SCO} = \varepsilon_0\)
Using \(\omega= \psi(1)\) in BOCF.
| \(()(1)(2)(2)\) |
\(\psi(\psi(2))\) |
\(\omega^{\omega^2}\) |
| \(()(1)(2)(2)(1)\) |
\(\psi(\psi(2)+1)\) |
\(\omega^{\omega^2+1}\) |
| \(()(1)(2)(2)(1)(2)\) |
\(\psi(\psi(2)+\omega)\) |
\(\omega^{\omega^2+\omega}\) |
| \(()(1)(2)(2)(1)(2)(1)(2)\) |
\(\psi(\psi(2)+\omega 2)\) |
\(\omega^{\omega^2+\omega 2}\) |
| \(()(1)(2)(2)(1)(2)(2)\) |
\(\psi(\psi(2)+\psi(2))\) |
\(\omega^{\omega^2 2}\) |
| \(()(1)(2)(2)(2)\) |
\(\psi(\psi(3))\) |
\(\omega^{\omega^3}\) |
| \(()(1)(2)(3)\) |
\(\psi(\psi(\omega))\) |
\(\omega^{\omega^\omega}\) |
| \(()(1)(2)(3)(1)\) |
\(\psi(\psi(\omega)+1)\) |
\(\omega^{\omega^\omega+1}\) |
| \(()(1)(2)(3)(1)(2)\) |
\(\psi(\psi(\omega)+\omega)\) |
\(\omega^{\omega^\omega+\omega}\) |
| \(()(1)(2)(3)(1)(2)(3)\) |
\(\psi(\psi(\omega)2)\) |
\(\omega^{\omega^{\omega 2}}\) |
| \(()(1)(2)(3)(4)\) |
\(\psi(\psi(\psi(\omega)))\) |
\(\omega^{\omega^{\omega^\omega}}\) |
| \(()(1,1)\) |
\(\psi(\psi_1(0))\) |
\(\mathrm{SCO} = \varepsilon_0\) |
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