User blog comment:Nayuta Ito/What We've Agreed on Stage Cardinal/@comment-11227630-20180522030604

Taranovsky's notation can naturally show how "stage cardinals" work. In Taranovsky's notation, Ω2 just means a large ordinal, not the 3rd infinity cardinal.


 * C(Ω2,0) works similar to S(1) - the first uncountable ordinal - Ω in normal notation.
 * C(Ω2,C(Ω2,0)) works similar to S(2) = Ω2 in normal notation.
 * C(Ω2+1,0) works similar to S(ω) = Ωω in normal notation.
 * C(Ω2+C(Ω2,0),0) works similar to S(Ω) = ΩΩ in normal notation.
 * C(Ω2+C(Ω22,0),0) works similar to S(S(T)) = ψI(0) in normal notation.
 * C(Ω2,C(Ω2+C(Ω22,0),0)) works similar to S(S(T)+1) = Ωψ I(0)+1 in normal notation.
 * C(Ω2+C(Ω22,0),C(Ω2+C(Ω22,0),0)) works similar to S(S(T)2) = Ωψ I(0)2 in normal notation.
 * C(Ω2+C(Ω22,0)+1,0) works similar to S(S(T)ω) = Ωψ I(0)ω in normal notation.
 * C(Ω2+C(Ω22,0)2,0) works similar to S(S(T)2) = Ωψ I(0)2 in normal notation.
 * C(Ω2+C(Ω22,0)2,0) works similar to S(S(T)S(T)) = Ωψ I(0)ψI(0) in normal notation.
 * C(Ω2+C(Ω22,0)C(Ω22,0),0) works similar to S(S(T)S(T) S(T) ) = Ωψ I(0)ψI(0) ψI(0) in normal notation.
 * C(Ω2+ε(C(Ω22,0)+1),0) works similar to S(ε(S(T)+1)) = Ωε(ψ I(0)+1) in normal notation.
 * C(Ω2+C(Ω2,C(Ω22,0)),0) works similar to S(T) - first weakly inaccessible ordinal - I in normal notation.

Let t = C(Ω2,C(Ω22,0)), then
 * C(Ω2,C(Ω2+t,0)) works similar to S(T+1) = ΩI+1 in normal notation.
 * C(Ω2+C(Ω2+t,0),C(Ω2+t,0)) works similar to S(T+S(T)) = ΩI2 in normal notation.
 * C(Ω2+t,C(Ω2+t,0)) works similar to S(T2) = I2 in normal notation.
 * C(Ω2+t+1,0) works similar to S(Tω) = Iω in normal notation.
 * C(Ω2+t+C(Ω2+t,0),0) works similar to S(TS(T)) = II in normal notation.
 * C(Ω2+t2,0) works similar to S(T2) - first weakly 2-inaccessible ordinal - I(2,0) in normal notation.
 * C(Ω2+t3,0) works similar to S(T3) - first weakly 3-inaccessible ordinal - I(3,0) in normal notation.
 * C(Ω2+tC(Ω2+t,0),0) works similar to S(TS(T)) = ψI(I,0)(0) in normal notation.
 * C(Ω2+t2,0) works similar to S(TT) - first weakly Mahlo ordinal - M in normal notation.
 * C(Ω2,C(Ω2+t2,0)) works similar to S(TT+1) - next uncountable ordinal after M - ΩM+1 in normal notation.
 * C(Ω2+t,C(Ω2+t2,0)) works similar to S(TT+T) - next weakly inaccessible ordinal after M - IM+1 in normal notation.
 * C(Ω2+t2,C(Ω2+t2,0)) works similar to S(TT+T2) - next weakly 2-inaccessible ordinal after M - I(2,M+1) in normal notation.
 * C(Ω2+t2,C(Ω2+t2,0)) works similar to S(TT2) - the 2nd weakly Mahlo ordinal - M2 in normal notation.
 * C(Ω2+t2+1,0) works similar to S(TTω) = Mω in normal notation.
 * C(Ω2+t2+t,0) works similar to S(TT+1) - first weakly inaccessible ordinal that is limit of Mahlo ordinals in normal notation.
 * C(Ω2+t22,0) works similar to S(TT2) - first weakly Mahlo oridnal that is limit of Mahlo ordinals in normal notation.
 * C(Ω2+t3,0) works similar to S(TT 2 ) - first weakly 2-Mahlo ordinal (i.e. first weakly Mahlo ordinal that weakly Mahlo ordinals below are stationary) in normal notation.
 * C(Ω2+t4,0) works similar to S(TT 3 ) - first weakly 3-Mahlo ordinal in normal notation.
 * C(Ω2+tt,0) works similar to S(TT T ) - first weakly compact ordinal in normal notation.
 * C(Ω2+tt,C(Ω2+tt,0)) works similar to S(TT T 2) - the 2nd weakly compact ordinal in normal notation.
 * C(Ω2+tt+t,0) works similar to S(TT T+1 ) - first weakly inaccessible ordinal that is limit of weakly compact ordinals in normal notation.
 * C(Ω2+tt2,0) works similar to S(TT T2 ) - first weakly compact ordinal that is limit of weakly compact ordinals in normal notation.
 * C(Ω2+tt+1,0) works similar to S(TT T+1 ) - first weakly Mahlo ordinal that weakly compact ordinals below are stationary in normal notation.
 * C(Ω2+tt2,0) works similar to S(TT T2 ) - first weakly compact ordinal that weakly compact ordinals below are stationary in normal notation.
 * C(Ω2+tt 2 ,0) works similar to S(TT T 2 ) - first 2-weakly compact ordinal in normal notation.
 * C(Ω2+tt t ,0) works similar to S(TT T T ) in normal notation.
 * C(Ω2+tt t t ,0) works similar to S(TT T T T   ) in normal notation.
 * C(Ω2+ε(t+1),0) works similar to S(ε(T+1)) in normal notation.