FANDOM


Pair sequence system is 2-row Bashicu Matrix System. I am analyzing this notation fully and carefully to double check.

Analysis

Notation Level
(0,0) 0
(0,0)(0,0) 1
(0,0)(0,0)(0,0) 2
(0,0)(0,0)(0,0)... n
(0,0)(1,0) \(\omega\)
(0,0)(1,0)(0,0) \(\omega+1\)
(0,0)(1,0)(0,0)(0,0) \(\omega+2\)
(0,0)(1,0)(0,0)(0,0)(0,0)... \(\omega+n\)
(0,0)(1,0)(0,0)(1,0) \(\omega 2\)
(0,0)(1,0)(0,0)(1,0)(0,0) \(\omega 2+1\)
(0,0)(1,0)(0,0)(1,0)(0,0)(1,0) \(\omega 3\)
(0,0)(1,0)(0,0)(1,0)(0,0)(1,0)(0,0)(1,0) \(\omega 4\)
(0,0)(1,0)(0,0)(1,0)(0,0)(1,0)... \(\omega n\)
(0,0)(1,0)(1,0) \(\omega^2\)
(0,0)(1,0)(1,0)(0,0) \(\omega^2+1\)
(0,0)(1,0)(1,0)(0,0)(1,0) \(\omega^2+\omega\)
(0,0)(1,0)(1,0)(0,0)(1,0)(1,0) \(\omega^22\)
(0,0)(1,0)(1,0)(0,0)(1,0)(1,0)(0,0)(1,0)(1,0) \(\omega^23\)
(0,0)(1,0)(1,0)(0,0)(1,0)(1,0)(0,0)(1,0)(1,0)... \(\omega^2n\)
(0,0)(1,0)(1,0)(1,0) \(\omega^3\)
(0,0)(1,0)(1,0)(1,0)(0,0)(1,0)(1,0)(1,0) \(\omega^32\)
(0,0)(1,0)(1,0)(1,0)(1,0) \(\omega^4\)
(0,0)(1,0)(1,0)(1,0)(1,0)(1,0) \(\omega^5\)
(0,0)(1,0)(1,0)(1,0)... \(\omega^n\)
(0,0)(1,0)(2,0) \(\omega^{\omega}\)
(0,0)(1,0)(2,0)(0,0)(1,0) \(\omega^{\omega}+\omega\)
(0,0)(1,0)(2,0)(0,0)(1,0)(1,0) \(\omega^{\omega}+\omega 2\)
(0,0)(1,0)(2,0)(0,0)(1,0)(1,0)(1,0)... \(\omega^{\omega}+\omega^n\)
(0,0)(1,0)(2,0)(0,0)(1,0)(2,0) \(\omega^{\omega}2\)
(0,0)(1,0)(2,0)(0,0)(1,0)(2,0)(0,0)(1,0)(2,0) \(\omega^{\omega}3\)
(0,0)(1,0)(2,0)(0,0)(1,0)(2,0)(0,0)(1,0)(2,0)... \(\omega^{\omega}n\)
(0,0)(1,0)(2,0)(1,0) \(\omega^{\omega+1}\)
(0,0)(1,0)(2,0)(1,0)(0,0)(1,0)(2,0)(1,0) \(\omega^{\omega+1}2\)
(0,0)(1,0)(2,0)(1,0)(1,0) \(\omega^{\omega+2}\)
(0,0)(1,0)(2,0)(1,0)(2,0) \(\omega^{\omega 2}\)
(0,0)(1,0)(2,0)(1,0)(2,0)(1,0)(2,0) \(\omega^{\omega 3}\)
(0,0)(1,0)(2,0)(1,0)(2,0)(1,0)(2,0)... \(\omega^{\omega n}\)
(0,0)(1,0)(2,0)(2,0) \(\omega^{\omega^2}\)
(0,0)(1,0)(2,0)(2,0)(1,0)(2,0)(2,0) \(\omega^{\omega^22}\)
(0,0)(1,0)(2,0)(2,0)(2,0) \(\omega^{\omega^3}\)
(0,0)(1,0)(2,0)(3,0) \(\omega^{\omega^{\omega}}\)
(0,0)(1,0)(2,0)(3,0)(3,0) \(\omega^{\omega^{\omega^2}}\)
(0,0)(1,0)(2,0)(3,0)(4,0) \(\omega^{\omega^{\omega^{\omega}}}\)
(0,0)(1,0)(2,0)(3,0)(4,0)(5,0) \(\omega^{\omega^{\omega^{\omega^{\omega}}}}\)
(0,0)(1,0)(2,0)... \(\varepsilon_0[n]\)
(0,0)(1,1) (Limit of primitive sequence system) \(\varepsilon_0\)
(0,0)(1,1)(0,0) \(\varepsilon_0+1\)
(0,0)(1,1)(0,0)(1,0) \(\varepsilon_0+\omega\)
(0,0)(1,1)(0,0)(1,0)(2,0) \(\varepsilon_0+\omega^{\omega}\)
(0,0)(1,1)(0,0)(1,0)(2,0)... \(\varepsilon_02[n]\)
(0,0)(1,1)(0,0)(1,1) \(\varepsilon_02\)
(0,0)(1,1)(1,0) \(\varepsilon_0\omega\)
(0,0)(1,1)(1,0)(1,0) \(\varepsilon_0\omega^2\)
(0,0)(1,1)(1,0)(2,0) \(\varepsilon_0\omega^{\omega}\)
(0,0)(1,1)(1,0)(2,0)(3,0) \(\varepsilon_0\omega^{\omega^{\omega}}\)
(0,0)(1,1)(1,0)(2,0)(3,0)... \(\varepsilon_0^2[n]\)
(0,0)(1,1)(1,0)(2,1) \(\varepsilon_0^2\)
(0,0)(1,1)(1,0)(2,1)(0,0)(1,1) \(\varepsilon_0^2+\varepsilon_0\)
(0,0)(1,1)(1,0)(2,1)(0,0)(1,1)(1,0)(2,1) \(\varepsilon_0^22\)
(0,0)(1,1)(1,0)(2,1)(1,0) \(\varepsilon_0^2\omega\)
(0,0)(1,1)(1,0)(2,1)(1,0)(2,1) \(\varepsilon_0^3\)
(0,0)(1,1)(1,0)(2,1)(1,0)(2,1)(1,0)(2,1) \(\varepsilon_0^4\)
(0,0)(1,1)(1,0)(2,1)(1,0)(2,1)(1,0)(2,1)... \(\varepsilon_0^n\)
(0,0)(1,1)(1,0)(2,1)(2,0) \(\varepsilon_0^{\omega}\)
(0,0)(1,1)(1,0)(2,1)(2,0)(1,0)(2,1) \(\varepsilon_0^{\omega+1}\)
(0,0)(1,1)(1,0)(2,1)(2,0)(1,0)(2,1)(2,0) \(\varepsilon_0^{\omega 2}\)
(0,0)(1,1)(1,0)(2,1)(2,0)(2,0) \(\varepsilon_0^{\omega^2}\)
(0,0)(1,1)(1,0)(2,1)(2,0)(3,0) \(\varepsilon_0^{\omega^{\omega}}\)
(0,0)(1,1)(1,0)(2,1)(2,0)(3,0)(4,0) \(\varepsilon_0^{\omega^{\omega^{\omega}}}\)
(0,0)(1,1)(1,0)(2,1)(2,0)(3,0)(4,0)... \(\varepsilon_0^{\varepsilon_0}[n]\)
(0,0)(1,1)(1,0)(2,1)(2,0)(3,1) \(\varepsilon_0^{\varepsilon_0}\)
(0,0)(1,1)(1,0)(2,1)(2,0)(3,1)(1,0)(2,1)(2,0)(3,1) \(\varepsilon_0^{\varepsilon_02}\)
(0,0)(1,1)(1,0)(2,1)(2,0)(3,1)(2,0) \(\varepsilon_0^{\varepsilon_0\omega}\)
(0,0)(1,1)(1,0)(2,1)(2,0)(3,1)(2,0)(3,1) \(\varepsilon_0^{\varepsilon_0^2}\)
(0,0)(1,1)(1,0)(2,1)(2,0)(3,1)(3,0) \(\varepsilon_0^{\varepsilon_0^{\omega}}\)
(0,0)(1,1)(1,0)(2,1)(2,0)(3,1)(3,0)(4,1)(4,0) \(\varepsilon_0^{\varepsilon_0^{\varepsilon_0^{\omega}}}\)
(0,0)(1,1)(1,0)(2,1)(2,0)(3,1)... \(\varepsilon_1[n]\)
(0,0)(1,1)(1,1) \(\varepsilon_1\)
(0,0)(1,1)(1,1)(0,0)(1,1) \(\varepsilon_1+\varepsilon_0\)
(0,0)(1,1)(1,1)(0,0)(1,1)(1,1) \(\varepsilon_12\)
(0,0)(1,1)(1,1)(1,0) \(\varepsilon_1\omega\)
(0,0)(1,1)(1,1)(1,0)(2,1)(2,1) \(\varepsilon_1^2\)
(0,0)(1,1)(1,1)(1,0)(2,1)(2,1)(2,0) \(\varepsilon_1^{\omega}\)
(0,0)(1,1)(1,1)(1,0)(2,1)(2,1)(2,0)(3,1)(3,1)(3,0) \(\varepsilon_1^{\varepsilon_1^{\omega}}\)
(0,0)(1,1)(1,1)(1,0)(2,1)(2,1)(2,0)(3,1)(3,1)... \(\varepsilon_2[n]\)
(0,0)(1,1)(1,1)(1,1) \(\varepsilon_2\)
(0,0)(1,1)(1,1)(1,1)(1,0)(2,1)(2,1)(2,1) \(\varepsilon_2^2\)
(0,0)(1,1)(1,1)(1,1)(1,1) \(\varepsilon_3\)
(0,0)(1,1)(1,1)(1,1)(1,1)(1,1) \(\varepsilon_4\)
(0,0)(1,1)(1,1)(1,1)... \(\varepsilon_n\)
(0,0)(1,1)(2,0) \(\varepsilon_{\omega}\)
(0,0)(1,1)(2,0)(0,0)(1,1)(2,0) \(\varepsilon_{\omega}2\)
(0,0)(1,1)(2,0)(1,0)(2,1)(3,0) \(\varepsilon_{\omega}^2\)
(0,0)(1,1)(2,0)(1,1) \(\varepsilon_{\omega+1}\)
(0,0)(1,1)(2,0)(1,1)(2,0) \(\varepsilon_{\omega 2}\)
(0,0)(1,1)(2,0)(2,0) \(\varepsilon_{\omega^2}\)
(0,0)(1,1)(2,0)(3,0) \(\varepsilon_{\omega^{\omega}}\)
(0,0)(1,1)(2,0)(3,1) \(\varepsilon_{\varepsilon_0}\)
(0,0)(1,1)(2,0)(3,1)(1,1)(2,0)(3,1) \(\varepsilon_{\varepsilon_02}\)
(0,0)(1,1)(2,0)(3,1)(3,1) \(\varepsilon_{\varepsilon_1}\)
(0,0)(1,1)(2,0)(3,1)(4,0)(5,1) \(\varepsilon_{\varepsilon_{\varepsilon_0}}\)
(0,0)(1,1)(2,0)(3,1)(4,0)(5,1)(6,0)(7,1) \(\varepsilon_{\varepsilon_{\varepsilon_{\varepsilon_0}}}\)
(0,0)(1,1)(2,0)(3,1)(4,0)(5,1)... \(\zeta_0[n]\)
(0,0)(1,1)(2,1) \(\zeta_0\)
(0,0)(1,1)(2,1)(0,0)(1,1)(2,1) \(\zeta_02\)
(0,0)(1,1)(2,1)(1,0)(2,1)(3,1) \(\zeta_0^2\)
(0,0)(1,1)(2,1)(1,1) \(\varepsilon_{\zeta_0+1}\)
(0,0)(1,1)(2,1)(1,1)(2,0) \(\varepsilon_{\zeta_0+\omega}\)
(0,0)(1,1)(2,1)(1,1)(2,0)(3,0) \(\varepsilon_{\zeta_0+\omega^{\omega}}\)
(0,0)(1,1)(2,1)(1,1)(2,0)(3,1) \(\varepsilon_{\zeta_0+\varepsilon_0}\)
(0,0)(1,1)(2,1)(1,1)(2,0)(3,1)(4,0)(5,1) \(\varepsilon_{\zeta_0+\varepsilon_{\varepsilon_0}}\)
(0,0)(1,1)(2,1)(1,1)(2,0)(3,1)(4,0)(5,1)(6,0)... \(\varepsilon_{\zeta_02}[n]\)
(0,0)(1,1)(2,1)(1,1)(2,0)(3,1)(4,1) \(\varepsilon_{\zeta_02}\)
(0,0)(1,1)(2,1)(1,1)(2,0)(3,1)(4,1)(1,1)(2,0)(3,1)(4,1) \(\varepsilon_{\zeta_03}\)
(0,0)(1,1)(2,1)(1,1)(2,0)(3,1)(4,1)(2,0) \(\varepsilon_{\zeta_0\omega}\)
(0,0)(1,1)(2,1)(1,1)(2,0)(3,1)(4,1)(2,0)(3,1)(4,1) \(\varepsilon_{\zeta_0^2}\)
(0,0)(1,1)(2,1)(1,1)(2,0)(3,1)(4,1)(3,0) \(\varepsilon_{\zeta_0^{\omega}}\)
(0,0)(1,1)(2,1)(1,1)(2,0)(3,1)(4,1)(3,0)(3,0) \(\varepsilon_{\zeta_0^{\omega^2}}\)
(0,0)(1,1)(2,1)(1,1)(2,0)(3,1)(4,1)(3,0)(4,0) \(\varepsilon_{\zeta_0^{\omega^{\omega}}}\)
(0,0)(1,1)(2,1)(1,1)(2,0)(3,1)(4,1)(3,0)(4,1) \(\varepsilon_{\zeta_0^{\varepsilon_0}}\)
(0,0)(1,1)(2,1)(1,1)(2,0)(3,1)(4,1)(3,0)(4,1)(5,1) \(\varepsilon_{\zeta_0^{\zeta_0}}\)
(0,0)(1,1)(2,1)(1,1)(2,0)(3,1)(4,1)(3,1) \(\varepsilon_{\varepsilon_{\zeta_0+1}}\)
(0,0)(1,1)(2,1)(1,1)(2,0)(3,1)(4,1)(3,1)(4,0)(5,1)(6,1)(5,1) \(\varepsilon_{\varepsilon_{\varepsilon_{\zeta_0+1}}}\)
(0,0)(1,1)(2,1)(1,1)(2,0)(3,1)(4,1)(3,1)(4,0)(5,1)(6,1)(5,1)... \(\zeta_1[n]\)
(0,0)(1,1)(2,1)(1,1)(2,1) \(\zeta_1\)
(0,0)(1,1)(2,1)(1,1)(2,1)(1,1) \(\varepsilon_{\zeta_1+1}\)
(0,0)(1,1)(2,1)(1,1)(2,1)(1,1)(2,0)(3,1)(4,1)(3,1)(4,1)(2,1) \(\varepsilon_{\varepsilon_{\zeta_1+1}}\)
(0,0)(1,1)(2,1)(1,1)(2,1)(1,1)(2,1) \(\zeta_2\)
(0,0)(1,1)(2,1)(1,1)(2,1)(1,1)(2,1)(1,1)(2,1) \(\zeta_3\)
(0,0)(1,1)(2,1)(1,1)(2,1)(1,1)(2,1)... \(\zeta_n\)
(0,0)(1,1)(2,1)(2,0) \(\zeta_{\omega}\)
(0,0)(1,1)(2,1)(2,0)(1,1) \(\varepsilon_{\zeta_{\omega}+1}\)
(0,0)(1,1)(2,1)(2,0)(1,1)(2,0)(3,1)(4,1)(4,0)(3,1) \(\varepsilon_{\varepsilon_{\zeta_{\omega}+1}}\)
(0,0)(1,1)(2,1)(2,0)(1,1)(2,1) \(\zeta_{\omega+1}\)
(0,0)(1,1)(2,1)(2,0)(1,1)(2,1)(2,0) \(\zeta_{\omega 2}\)
(0,0)(1,1)(2,1)(2,0)(2,0) \(\zeta_{\omega^2}\)
(0,0)(1,1)(2,1)(2,0)(3,1)(4,1) \(\zeta_{\zeta_0}\)
(0,0)(1,1)(2,1)(2,0)(3,1)(4,1)(4,0)(5,1)(6,1) \(\zeta_{\zeta_{\zeta_0}}\)
(0,0)(1,1)(2,1)(2,0)(3,1)(4,1)(4,0)(5,1)(6,1)... \(\eta_0[n]\)
(0,0)(1,1)(2,1)(2,1) \(\eta_0\)
(0,0)(1,1)(2,1)(2,1)(1,1) \(\varepsilon_{\eta_0+1}\)
(0,0)(1,1)(2,1)(2,1)(1,1)(2,1) \(\zeta_{\eta_0+1}\)
(0,0)(1,1)(2,1)(2,1)(1,1)(2,1)(2,0)(3,1)(4,1)(4,1)(3,1)(4,1) \(\zeta_{\zeta_{\eta_0+1}}\)
(0,0)(1,1)(2,1)(2,1)(1,1)(2,1)(2,1) \(\eta_1\)
(0,0)(1,1)(2,1)(2,1)(1,1)(2,1)(2,1)(1,1)(2,1)(2,1) \(\eta_2\)
(0,0)(1,1)(2,1)(2,1)(1,1)(2,1)(2,1)(1,1)(2,1)(2,1)... \(\eta_n\)
(0,0)(1,1)(2,1)(2,1)(2,0) \(\eta_{\omega}\)
(0,0)(1,1)(2,1)(2,1)(2,0)(3,1)(4,1)(4,1) \(\eta_{\eta_0}\)
(0,0)(1,1)(2,1)(2,1)(2,0)(3,1)(4,1)(4,1)(4,0)(5,1)(6,1)(6,1) \(\eta_{\eta_{\eta_0}}\)
(0,0)(1,1)(2,1)(2,1)(2,0)(3,1)(4,1)(4,1)(4,0)(5,1)(6,1)(6,1)... \(\varphi(4,0)[n]\)
(0,0)(1,1)(2,1)(2,1)(2,1) \(\varphi(4,0)\)
(0,0)(1,1)(2,1)(2,1)(2,1)(2,0)(3,1)(4,1)(4,1)(4,1) \(\varphi(4,\varphi(4,0))\)
(0,0)(1,1)(2,1)(2,1)(2,1)(2,1) \(\varphi(5,0)\)
(0,0)(1,1)(2,1)(2,1)(2,1)(2,1)(2,1) \(\varphi(6,0)\)
(0,0)(1,1)(2,1)(2,1)(2,1)... \(\varphi(n,0)\)
(0,0)(1,1)(2,1)(3,0) \(\varphi(\omega,0)\)
(0,0)(1,1)(2,1)(3,0)(4,1)(5,1)(6,0) \(\varphi(\varphi(\omega,0),0)\)
(0,0)(1,1)(2,1)(3,0)(4,1)(5,1)(6,0)(7,1)(8,1)... \(\varphi(1,0,0)[n]\)
(0,0)(1,1)(2,1)(3,1) \(\varphi(1,0,0)\)
(0,0)(1,1)(2,1)(3,1)(1,1) \(\varepsilon_{\varphi(1,0,0)+1}\)
(0,0)(1,1)(2,1)(3,1)(1,1)(2,1) \(\zeta_{\varphi(1,0,0)+1}\)
(0,0)(1,1)(2,1)(3,1)(1,1)(2,1)(3,0) \(\varphi(\omega,\varphi(1,0,0)+1)\)
(0,0)(1,1)(2,1)(3,1)(1,1)(2,1)(3,0)(4,1)(5,1)(6,1) \(\varphi(\varphi(1,0,0),1)\)
(0,0)(1,1)(2,1)(3,1)(1,1)(2,1)(3,0)(4,1)(5,1)(6,1)(4,1)(5,1)(6,0)(7,1)(8,1)(9,1) \(\varphi(\varphi(\varphi(1,0,0),1),0)\)
(0,0)(1,1)(2,1)(3,1)(1,1)(2,1)(3,1) \(\varphi(1,0,1)\)
(0,0)(1,1)(2,1)(3,1)(2,0) \(\varphi(1,0,\omega)\)
(0,0)(1,1)(2,1)(3,1)(2,0)(3,1)(4,1)(5,1) \(\varphi(1,0,\varphi(1,0,0))\)
(0,0)(1,1)(2,1)(3,1)(2,1) \(\varphi(1,1,0)\)
(0,0)(1,1)(2,1)(3,1)(2,1)(3,0) \(\varphi(1,\omega,0)\)
(0,0)(1,1)(2,1)(3,1)(2,1)(3,0)(4,1)(5,1)(6,1)(5,1)(6,0) \(\varphi(1,\varphi(1,\omega,0),0)\)
(0,0)(1,1)(2,1)(3,1)(2,1)(3,1) \(\varphi(2,0,0)\)
(0,0)(1,1)(2,1)(3,1)(3,0) \(\varphi(\omega,0,0)\)
(0,0)(1,1)(2,1)(3,1)(3,0)(4,1)(5,1)(6,1)(6,0) \(\varphi(\varphi(\omega,0,0),0,0)\)
(0,0)(1,1)(2,1)(3,1)(3,1) \(\varphi(1,0,0,0)\)
(0,0)(1,1)(2,1)(3,1)(3,1)(3,0) \(\varphi(2,0,0,0)\)
(0,0)(1,1)(2,1)(3,1)(3,1)(3,1) \(\varphi(1,0,0,0,0)\)
(0,0)(1,1)(2,1)(3,1)(3,1)(3,1)(3,1) \(\varphi(1,0,0,0,0,0)\)
(0,0)(1,1)(2,1)(3,1)(3,1)(3,1)... \(\psi(\Omega^{\Omega^n})\)
(0,0)(1,1)(2,1)(3,1)(4,0) \(\psi(\Omega^{\Omega^{\omega}})\)
(0,0)(1,1)(2,1)(3,1)(4,0)(5,1)(6,1)(7,1)(8,0) \(\psi(\Omega^{\Omega^{\psi(\Omega^{\Omega^{\omega}})}})\)
(0,0)(1,1)(2,1)(3,1)(4,1) \(\psi(\Omega^{\Omega^{\Omega}})\)
(0,0)(1,1)(2,1)(3,1)(4,1)(1,1) \(\psi(\Omega^{\Omega^{\Omega}}+1)\)
(0,0)(1,1)(2,1)(3,1)(4,1)(1,1)(2,0)(3,1)(4,1)(5,1)(6,1) \(\psi(\Omega^{\Omega^{\Omega}}+\psi(\Omega^{\Omega^{\Omega}}))\)
(0,0)(1,1)(2,1)(3,1)(4,1)(2,0)(3,1)(4,1)(5,1)(6,1) \(\psi(\Omega^{\Omega^{\Omega}}\psi(\Omega^{\Omega^{\Omega}}))\)
(0,0)(1,1)(2,1)(3,1)(4,1)(2,1) \(\psi(\Omega^{\Omega^{\Omega}+1})\)
(0,0)(1,1)(2,1)(3,1)(4,1)(2,1)(3,0)(4,1)(5,1)(6,1)(7,1) \(\psi(\Omega^{\Omega^{\Omega}+\psi(\Omega^{\Omega^{\Omega}})})\)
(0,0)(1,1)(2,1)(3,1)(4,1)(3,0)(4,1)(5,1)(6,1)(7,1) \(\psi(\Omega^{\Omega^{\Omega}\psi(\Omega^{\Omega^{\Omega}})})\)
(0,0)(1,1)(2,1)(3,1)(4,1)(3,1) \(\psi(\Omega^{\Omega^{\Omega+1}})\)
(0,0)(1,1)(2,1)(3,1)(4,1)(3,1)(4,0)(5,1)(6,1)(7,1)(8,1) \(\psi(\Omega^{\Omega^{\Omega+\psi(\Omega^{\Omega^{\Omega}})}})\)
(0,0)(1,1)(2,1)(3,1)(4,1)(4,0)(5,1)(6,1)(7,1)(8,1) \(\psi(\Omega^{\Omega^{\Omega\psi(\Omega^{\Omega^{\Omega}})}})\)
(0,0)(1,1)(2,1)(3,1)(4,1)(4,1) \(\psi(\Omega^{\Omega^{\Omega^2}})\)
(0,0)(1,1)(2,1)(3,1)(4,1)(5,1) \(\psi(\Omega^{\Omega^{\Omega^{\Omega}}})\)
(0,0)(1,1)(2,1)(3,1)(4,1)(5,1)(6,1) \(\psi(\Omega^{\Omega^{\Omega^{\Omega^{\Omega}}}})\)
(0,0)(1,1)(2,1)(3,1)... \(\psi(\varepsilon_{\Omega+1})[n]\)
(0,0)(1,1)(2,2) \(\psi(\varepsilon_{\Omega+1})\)
(0,0)(1,1)(2,2)(1,1) \(\psi(\varepsilon_{\Omega+1}+1)\)
(0,0)(1,1)(2,2)(1,1)(2,1) \(\psi(\varepsilon_{\Omega+1}+\Omega)\)
(0,0)(1,1)(2,2)(1,1)(2,1)(3,1) \(\psi(\varepsilon_{\Omega+1}+\Omega^{\Omega})\)
(0,0)(1,1)(2,2)(1,1)(2,2) \(\psi(\varepsilon_{\Omega+1}2)\)
(0,0)(1,1)(2,2)(2,0) \(\psi(\varepsilon_{\Omega+1}\omega)\)
(0,0)(1,1)(2,2)(2,0)(3,0) \(\psi(\varepsilon_{\Omega+1}\omega^{\omega})\)
(0,0)(1,1)(2,2)(2,0)(3,1) \(\psi(\varepsilon_{\Omega+1}\varepsilon_0)\)
(0,0)(1,1)(2,2)(2,0)(3,1)(4,2) \(\psi(\varepsilon_{\Omega+1}\psi(\varepsilon_{\Omega+1}))\)
(0,0)(1,1)(2,2)(2,1) \(\psi(\varepsilon_{\Omega+1}\Omega)\)
(0,0)(1,1)(2,2)(2,1)(2,1) \(\psi(\varepsilon_{\Omega+1}\Omega^2)\)
(0,0)(1,1)(2,2)(2,1)(3,0) \(\psi(\varepsilon_{\Omega+1}\Omega^{\omega})\)
(0,0)(1,1)(2,2)(2,1)(3,1) \(\psi(\varepsilon_{\Omega+1}\Omega^{\Omega})\)
(0,0)(1,1)(2,2)(2,1)(3,2) \(\psi(\varepsilon_{\Omega+1}^2)\)
(0,0)(1,1)(2,2)(2,1)(3,2)(3,1)(4,2) \(\psi(\varepsilon_{\Omega+1}^{\varepsilon_{\Omega+1}})\)
(0,0)(1,1)(2,2)(2,1)(3,2)(3,1)(4,2)(4,1)(5,2) \(\psi(\varepsilon_{\Omega+1}^{\varepsilon_{\Omega+1}^{\varepsilon_{\Omega+1}}})\)
(0,0)(1,1)(2,2)(2,1)(3,2)(3,1)(4,2)... \(\psi(\varepsilon_{\Omega+2})[n]\)
(0,0)(1,1)(2,2)(2,2) \(\psi(\varepsilon_{\Omega+2})\)
(0,0)(1,1)(2,2)(2,2)(2,1)(3,2)(3,2) \(\psi(\varepsilon_{\Omega+2}^2)\)
(0,0)(1,1)(2,2)(2,2)(2,1)(3,2)(3,2)(3,1)(4,2)(4,2) \(\psi(\varepsilon_{\Omega+2}^{\varepsilon_{\Omega+2}})\)
(0,0)(1,1)(2,2)(2,2)(2,2) \(\psi(\varepsilon_{\Omega+3})\)
(0,0)(1,1)(2,2)(2,2)(2,2)(2,2) \(\psi(\varepsilon_{\Omega+4})\)
(0,0)(1,1)(2,2)(2,2)(2,2)... \(\psi(\varepsilon_{\Omega+b})\)
(0,0)(1,1)(2,2)(3,0) \(\psi(\varepsilon_{\Omega+\omega})\)
(0,0)(1,1)(2,2)(3,1) \(\psi(\varepsilon_{\Omega 2})\)
(0,0)(1,1)(2,2)(3,1)(2,2)(3,1) \(\psi(\varepsilon_{\Omega 3})\)
(0,0)(1,1)(2,2)(3,1)(4,2) \(\psi(\varepsilon_{\varepsilon_{\Omega+1}})\)
(0,0)(1,1)(2,2)(3,1)(4,2)(5,1)(6,2) \(\psi(\varepsilon_{\varepsilon_{\varepsilon_{\Omega+1}}})\)
(0,0)(1,1)(2,2)(3,1)(4,2)(5,1)(6,2)... \(\psi(\Omega_2)[n]\)
(0,0)(1,1)(2,2)(3,2) \(\psi(\Omega_2)\)
(0,0)(1,1)(2,2)(3,2)(1,1)(2,2)(3,2) \(\psi(\Omega_2+\psi_1(\Omega_2))\)
(0,0)(1,1)(2,2)(3,2)(2,1)(3,2)(4,2) \(\psi(\Omega_2+\psi_1(\Omega_2)^2)\)
(0,0)(1,1)(2,2)(3,2)(2,1)(3,2)(4,2)(3,1)(4,2)(5,2) \(\psi(\Omega_2+\psi_1(\Omega_2)^{\psi_1(\Omega_2)})\)
(0,0)(1,1)(2,2)(3,2)(2,2) \(\psi(\Omega_2+\psi_1(\Omega_2+1))\)
(0,0)(1,1)(2,2)(3,2)(2,2)(3,1) \(\psi(\Omega_2+\psi_1(\Omega_2+\Omega))\)
(0,0)(1,1)(2,2)(3,2)(2,2)(3,1)(4,2)(5,2) \(\psi(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2)))\)
(0,0)(1,1)(2,2)(3,2)(2,2)(3,2) \(\psi(\Omega_2 2)\)
(0,0)(1,1)(2,2)(3,2)(3,2) \(\psi(\Omega_2^2)\)
(0,0)(1,1)(2,2)(3,2)(4,2) \(\psi(\Omega_2^{\Omega_2})\)
(0,0)(1,1)(2,2)(3,2)(4,2)(5,2) \(\psi(\Omega_2^{\Omega_2^{\Omega_2}})\)
(0,0)(1,1)(2,2)(3,2)(4,2)... \(\psi(\varepsilon_{\Omega_2+1})[n]\)
(0,0)(1,1)(2,2)(3,3) \(\psi(\varepsilon_{\Omega_2+1})\)
(0,0)(1,1)(2,2)(3,3)(2,2)(3,3) \(\psi(\varepsilon_{\Omega_2+1}2)\)
(0,0)(1,1)(2,2)(3,3)(3,2)(4,3) \(\psi(\varepsilon_{\Omega_2+1}^2)\)
(0,0)(1,1)(2,2)(3,3)(3,3) \(\psi(\varepsilon_{\Omega_2+2})\)
(0,0)(1,1)(2,2)(3,3)(4,3) \(\psi(\Omega_3)\)
(0,0)(1,1)(2,2)(3,3)(4,3)(5,3) \(\psi(\Omega_3^{\Omega_3})\)
(0,0)(1,1)(2,2)(3,3)(4,3)(5,3)... \(\psi(\varepsilon_{\Omega_3+1})[n]\)
(0,0)(1,1)(2,2)(3,3)(4,4) \(\psi(\varepsilon_{\Omega_3+1})\)
(0,0)(1,1)(2,2)(3,3)(4,4)(5,4) \(\psi(\Omega_4)\)
(0,0)(1,1)(2,2)(3,3)(4,4)(5,5) \(\psi(\varepsilon_{\Omega_4+1})\)
(0,0)(1,1)(2,2)(3,3)(4,4)(5,5)(6,6) \(\psi(\varepsilon_{\Omega_5+1})\)
(0,0)(1,1)(2,2)... \(\psi(\varepsilon_{\Omega_n+1})\)

The limit level of Pair sequence system is \(\psi(\Omega_{\omega})\).

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