TON (Taranovsky’s ordinal notation) will defined in here.
Up to \(\psi(\psi_I(1))\)[]
Let \(I\) will be \(\chi(0)\). TON vs further ordinals will begin here.
| TON | Ordinal |
|---|---|
| C(C(Ω2*2,0),0) | \(\psi(\psi_I(0))\) |
| C(0,C(C(Ω2*2,0),0)) | \(\psi(\psi_I(0))+1\) |
| C(1,C(C(Ω2*2,0),0)) | \(\psi(\psi_I(0))+\omega\) |
| C(n,C(C(Ω2*2,0),0)) | \(\psi(\psi_I(0))+\omega^n\) |
| C(Ω,C(C(Ω2*2,0),0)) | \(\psi(\psi_I(0)+1)\) |
| C(Ω,C(Ω,C(C(Ω2*2,0),0)) | \(\psi(\psi_I(0)+2)\) |
| C(C(0,Ω),C(C(Ω2*2,0),0)) | \(\psi(\psi_I(0)+\omega)\) |
| C(C(0,C(0,Ω)),C(C(Ω2*2,0),0)) | \(\psi(\psi_I(0)+\omega^2)\) |
| C(C(1,Ω),C(C(Ω2*2,0),0)) | \(\psi(\psi_I(0)+\omega^{\omega})\) |
| C(C(0,C(1,Ω)),C(C(Ω2*2,0),0)) | \(\psi(\psi_I(0)+\omega^{\omega+1})\) |
| C(C(1,C(1,Ω)),C(C(Ω2*2,0),0)) | \(\psi(\psi_I(0)+\omega^{\omega2})\) |
| C(C(2,Ω),C(C(Ω2*2,0),0)) | \(\psi(\psi_I(0)+\omega^{\omega^2})\) |
| C(C(n,Ω),C(C(Ω2*2,0),0)) | \(\psi(\psi_I(0)+\omega^{\omega^n})\) |
| C(C(Ω,Ω),C(C(Ω2*2,0),0)) | \(\psi(\psi_I(0)+\Omega)\) |
| C(C(Ω,Ω),C(C(Ω,Ω),C(C(Ω2*2,0),0))) | \(\psi(\psi_I(0)+\Omega2)\) |
| C(C(Ω,C(Ω,Ω)),C(C(Ω2*2,0),0)) | \(\psi(\psi_I(0)+\Omega^2)\) |
| C(C(C(Ω,Ω),Ω),C(C(Ω2*2,0),0)) | \(\psi(\psi_I(0)+\Omega^{\Omega})\) |
| C(C(C(Ω,Ω),C(C(Ω,Ω),Ω)),C(C(Ω2*2,0),0)) | \(\psi(\psi_I(0)+\Omega^{\Omega2})\) |
| C(C(C(Ω,C(Ω,Ω)),Ω),C(C(Ω2*2,0),0)) | \(\psi(\psi_I(0)+\Omega^{\Omega^2})\) |
| C(C(C(C(Ω,Ω),Ω),Ω),C(C(Ω2*2,0),0)) | \(\psi(\psi_I(0)+\Omega^{\Omega^{\Omega}})\) |
| C(C(C(C(C(Ω,Ω),Ω),Ω),Ω),C(C(Ω2*2,0),0)) | \(\psi(\psi_I(0)+\Omega^{\Omega^{\Omega^{\Omega}}})\) |
| C(C(Ω2,Ω),C(C(Ω2*2,0),0)) | \(\psi(\psi_I(0)+\varepsilon_{\Omega+1})\) |
| C(C(Ω2,Ω),C(C(Ω2,Ω),C(C(Ω2*2,0),0))) | \(\psi(\psi_I(0)+\varepsilon_{\Omega+1}*2)\) |
| C(C(0,C(Ω2,Ω)),C(C(Ω2*2,0),0)) | \(\psi(\psi_I(0)+\varepsilon_{\Omega+1}*\omega)\) |
| C(C(Ω,C(Ω2,Ω)),C(C(Ω2*2,0),0)) | \(\psi(\psi_I(0)+\varepsilon_{\Omega+1}*\Omega)\) |
| C(C(C(Ω,Ω),C(Ω2,Ω)),C(C(Ω2*2,0),0)) | \(\psi(\psi_I(0)+\varepsilon_{\Omega+1}*\Omega^\Omega)\) |
| C(C(C(C(Ω2,Ω),Ω),C(Ω2,Ω)),C(C(Ω2*2,0),0)) | \(\psi(\psi_I(0)+\varepsilon_{\Omega+1}^2)\) |
| C(C(C(0,C(C(Ω2,Ω),Ω)),C(Ω2,Ω)),C(C(Ω2*2,0),0)) | \(\psi(\psi_I(0)+\varepsilon_{\Omega+1}^\omega)\) |
| C(C(C(Ω,C(C(Ω2,Ω),Ω)),C(Ω2,Ω)),C(C(Ω2*2,0),0)) | \(\psi(\psi_I(0)+\varepsilon_{\Omega+1}^\Omega)\) |
| C(C(C(C(Ω,Ω),C(C(Ω2,Ω),Ω)),C(Ω2,Ω)),C(C(Ω2*2,0),0)) | \(\psi(\psi_I(0)+\varepsilon_{\Omega+1}^{\Omega^\Omega})\) |
| C(C(C(C(C(Ω2,Ω),Ω),C(C(Ω2,Ω),Ω)),C(Ω2,Ω)),C(C(Ω2*2,0),0)) | \(\psi(\psi_I(0)+\varepsilon_{\Omega+1}^{\varepsilon_{\Omega+1}})\) |
| C(C(C(C(0,C(Ω2,Ω)),Ω),C(Ω2,Ω)),C(C(Ω2*2,0),0)) | \(\psi(\psi_I(0)+\varepsilon_{\Omega+1}^{\varepsilon_{\Omega+1}^\omega})\) |
| C(C(C(C(C(C(Ω2,Ω),Ω),C(Ω2,Ω)),Ω),C(Ω2,Ω)),C(C(Ω2*2,0),0)) | \(\psi(\psi_I(0)+\varepsilon_{\Omega+1}^{\varepsilon_{\Omega+1}^{\varepsilon_{\Omega+1}}})\) |
| C(C(C(Ω2,Ω),C(Ω2,Ω)),C(C(Ω2*2,0),0)) | \(\psi(\psi_I(0)+\varepsilon_{\Omega+2})\) |