急数列システム[]
\begin{array}{ll} SS(0,1,2)&\approx&\Gamma_0\\ SS(0,1,2,0,1)&\approx&\Gamma_0\cdot \omega\\ SS(0,1,2,0,1,0,1,2)&\approx&\Gamma_1\\ SS(0,1,2,0,1,1,0,1,2)&\approx&\phi(1,1,0)\\ SS(0,1,2,0,1,1,2)&\approx&\psi_\Omega(\psi_{\Omega_2}(0))\\ SS(0,1,2,0,1,1,2,2)&\approx&\psi_\Omega(\psi_{\Omega_2}(\Omega_2))\\ SS(0,1,2,0,1,1,2,2,3)&\approx&\psi_\Omega(\psi_{\Omega_2}(\Omega_2^\omega))\\ SS(0,1,2,0,1,2)&\approx&\psi_\Omega(\psi_{\Omega_2}(\Omega_2^\Omega))\\ SS(0,1,2,0,1,2,0,1,1,2)&\approx&\psi_\Omega(\psi_{\Omega_2}(\Omega_2^{\psi_{\Omega_2}(0)}))\\ SS(0,1,2,1,0,1,2,0,1,1,2)&\approx&\psi_\Omega(\psi_{\Omega_2}(\Omega_2^{\Omega_2}))\\ SS(0,1,2,1,1,2)&\approx&\psi_\Omega(\psi_{\Omega_2}(\psi_{\Omega_3}(0)))\\ SS(0,1,2,1,1,2,2,3)&\approx&\psi_\Omega(\psi_{\Omega_2}(\psi_{\Omega_3}(\Omega_3^\omega)))\\ SS(0,1,2,1,2)&\approx&\psi_\Omega(\psi_{\Omega_2}(\psi_{\Omega_3}(\Omega_3^\Omega)))\\ SS(0,1,2,1,2,0,1,1,2)&\approx&\psi_\Omega(\psi_{\Omega_2}(\psi_{\Omega_3}(\Omega_3^{\psi_{\Omega_2}(0)})))\\ SS(0,1,2,1,2,2,3)&\approx&\psi_\Omega(\psi_{\Omega_2}(\psi_{\Omega_3}(\Omega_3^{\Omega_3})))\\ SS(0,1,2,1,2,3)&\approx&\psi_\Omega(\psi_{\Omega_2}(\psi_{\Omega_3}(\psi_{\Omega_4}(0))))\\ SS(0,1,2,2)&\approx&\psi_\Omega(\Omega_\omega)\\ SS(0,1,2,2,3)&\approx&(0,0,0,0)(1,1,1,1)\\ SS(0,1,2,3,3,4)&\approx&(0,0,0,0,0)(1,1,1,1,1)\\ SSS&\approx&BMS\\ \end{array}
BTMS vs BHMS[]
\begin{array}{ll} BTM(0)(1)(2,1)(3,2,1)(4,3)&\approx&\\ BHM(0)(1,1,1)(2,1)(3,2,1)(3,2)\\ \\ BTM(0)(1)(2,1)(3,2,1)(4,3,1)&\approx&\\ BHM(0)(1,1,1)(2,1,1)\\ \\ BTM(0)(1)(2,1)(3,2,1)(4,3,1)(5,4,0)&\approx&\\ BHM(0)(1,1,1)(2,1,1)(3,1)(4,2,1)(4,2)\\ \\ BTM(0)(1)(2,1)(3,2,1)(4,3,2)&\approx&\\ BHM(0)(1,1,1)(2,1,1)(3,1,1)\\ \\ BTM(0)(1)(2,1)(3,2,1)(4,3,2,1)&\approx&\\ BHM(0)(1,1,1)(2,1,1)(3,1,1)(4,1)(3,1,1)\\ \\ BTM(0)(1)(2,1)(3,2,1)(4,3,2,1)(5,3,2)&\approx&\\ BHM(0)(1,1,1)(2,1,1)(3,1,1)(4,1)(5,2,1)(5,2)\\ \\ BTM(0)(1)(2,1)(3,2,1)(4,3,2,1)(5,4)&\approx&\\ BHM(0)(1,1,1)(2,1,1)(3,1,1)(4,1)(5,2,1)(5,2)(6,3)\\ \\ BTM(0)(1)(2,1)(3,2,1)(4,3,2,1)(5,4,2)&\approx&\\ BHM(0)(1,1,1)(2,1,1)(3,1,1)(4,1)(5,2,1)(6,2,1)(6,2)(7,3)\\ \\ BTM(0)(1)(2,1)(3,2,1)(4,3,2,1)(5,4,3)&\approx&\\ BHM(0)(1,1,1)(2,1,1)(3,1,1)(4,1)(5,2,1)(6,2,1)(7,2,1)(7,2)\\ \\ BTM(0)(1)(2,1)(3,2,1)(4,3,2,1)(5,4,3,2)&\approx&\\ BHM(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)\\ \\ BTM(0)(1)(2,1)(3,2,1)(4,3,2,1)(5,4,3,2,1)(6,5,4,3,2)&\approx&\\ BHM(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(5,1,1)\\ \\ BTM&\approx&\\ BHM(0)(1,1,1)(2,2)\\ \end{array}
Ysequence vs BTMS[]
\begin{array}{ll} Y(1,3)&\approx&\\ BM&\approx&\\ (0)(1)(2,1)(3,2,1)(4,2)(4)\\ \\ Y(1,3,3)&\approx&\\ (0)(1)(2,1)(3,2,1)(4,2)(4)(3,2)(4,3,1)(5,2)(5)\\ \\ Y(1,3,4)&\approx&\\ (0)(1)(2,1)(3,2,1)(4,2)(4)(3,2)(4,3,1)(5,2)(5)(4)\\ \\ Y(1,3,5)&\approx&\\ (0)(1)(2,1)(3,2,1)(4,2)(4)(3,2)(4,3,1)(5,2)(5)(4,2)\\ \\ Y(1,3,6)(2,3)&\approx&\\ (0)(1)(2,1)(3,2,1)(4,2)(4)(3,2)(4,3,1)(5,2)(5)(4,2)(5,3,1)\\ \\ Y(1,3,7)(2,4)&\approx&\\ (0)(1)(2,1)(3,2,1)(4,2)(4)(3,2)(4,3,1)(5,2)(5)(4,2)(5,3,1)(6,2)(6)\\ \\ Y(1,3,8)(2,5)(3)&\approx&\\ (0)(1)(2,1)(3,2,1)(4,2)(4)(3,2)(4,3,1)(5,2)(5)(4,3)(5,4,1)(6,2)(6)\\ \\ Y(1,3,9)(2,6)(4)&\approx&\\ (0)(1)(2,1)(3,2,1)(4,2)(4)(3,2)(4,3,1)(5,2)(5)(4,3,1)\\ \\ Y(1,3,9,21)(2,6,12)(4,6)(2)&\approx&\\ (0)(1)(2,1)(3,2,1)(4,2)(4)(3,2)(4,3,1)(5,2)(5)(4,3,1)(5,2)(5)\\ \\ Y(1,3,9,24)(2,6,15)(4,9)(5)&\approx&\\ (0)(1)(2,1)(3,2,1)(4,2)(4)(3,2)(4,3,1)(5,2)(5)(5)\\ \\ Y(1,3,9,25)(2,6,16)(4,10)(6)&\approx&\\ (0)(1)(2,1)(3,2,1)(4,2)(4)(3,2)(4,3,1)(5,3)\\ \\ Y(1,3,9,26)(2,6,17)(4,11)(7)&\approx&\\ (0)(1)(2,1)(3,2,1)(4,2)(4)(3,2)(4,3,1)(5,3)(4,3,1)(5,2)(5)\\ \\ Y(1,3,9,27)(2,6,18)(4,12)(8)&\approx&\\ (0)(1)(2,1)(3,2,1)(4,2)(4)(3,2)(4,3,1)(5,3)(4,3,1)(5,3)\\ \\ Y(1,4)&\approx&\\ (0)(1)(2,1)(3,2,1)(4,2)(4)(3,2)(4,3,1)(5,3)(5)\\ \\ Y(1,\omega)&\approx&\\ (0)(1)(2,1)(3,2,1)(4,2)(4)(3,2,1)\\ \\ \end{array}
超順序数崩壊関数バージョン3 vs BHMS[]
\begin{array}{ll} (0)(1,1,1)(1)(2)(1,1)(2,2)&\approx&\\ \Psi_\Omega(\Psi_{\Phi_1(\omega+1)}(0))\\ \\ (0)(1,1,1)(1)(2)(2)(1,1)(2,2)&\approx&\\ \Psi_\Omega(\Psi_{\Phi_1(\omega^2+1)}(0))\\ \\ (0)(1,1,1)(1,1)&\approx&\\ \Psi_\Omega(\Phi_1(\Omega))\\ \\ (0)(1,1,1)(1,1)(1,1,1)&\approx&\\ \Psi_\Omega(\Phi_1(\Phi_1(\omega)))\\ \\ (0)(1,1,1)(1,1)(2)&\approx&\\ \Psi_\Omega(\Psi_{\Omega_2}(0))\\ \\ (0)(1,1,1)(1,1)(2)&\approx&\\ \Psi_\Omega(\Psi_{\Omega_2}(0))\\ \\ (0)(1,1,1)(2,1)(3,2)&\approx&\\ \Psi_\Omega(\Psi_{\Omega_2}(\Omega_2))\\ \\ (0)(1,1,1)(2,1)(3,2,1)(4,2)(5,3)&\approx&\\ \Psi_\Omega(\Psi_{\Omega_2}(\Psi_{\Omega_3}(\Omega_3)))\\ \\ (0)(1,1,1)(2,1,1)&\approx&\\ \Psi_\Omega(\Omega_\omega)\\ \\ (0)(1,1,1)(2,1,1)(2)&\approx&\\ \Psi_\Omega(\Phi_\omega(0))\\ \\ (0)(1,1,1)(2,1,1)(2,1,1)&\approx&\\ \Psi_\Omega(\Omega_{\omega^2})\\ \\ (0)(1,1,1)(2,1,1)(3)&\approx&\\ \Psi_\Omega(\Omega_{\omega^\omega})\\ \\ (0)(1,1,1)(2,1,1)(3,1)(2)&\approx&\\ OFP\\ \\ \end{array}