## FANDOM

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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)) $$\psi(\psi_I(0)+\varepsilon_{\Omega+2})$$