User blog comment:Nayuta Ito/What We've Agreed on Stage Cardinal/@comment-2001:56A:71A8:5F00:C801:4C6D:105A:CD53-20180524213756

This is how I think it should work, using the C function we talked about in discord: (I am ecl1 by the way)

S(0) = C(0) = Ω S(1) = C(1) = Ω_2 S(w) = C(w) = Ω_w S(T) = C(1;0) = M S(T+1) = C(C(1;0)+1) = W_(M+1) S(T2) = C(1;1) = M_2 S(Tw) = C(1;w) = M_w S(T^2) = C(2;0) = N S(T^w) = C(w;0) = Xi(w) S(T^T) = C(1;0;0) = K S(T^T*2) = C(1;0;1) = K_2 S(T^(T+1)) = C(1;1;0) = "mahlo compact" cardinal S(T^(T+2)) = C(1;2;0) = "mahlo mahlo compact" cardinal S(T^(T2)) = C(2;0;0) = "compact compact" cardinal S(T^T^2) = C(1;0;0;0) = stage-4 cardinal S(T^T^3) = C(1;0;0;0;0) = stage-5 cardinal S(T^T^w) = stage-w cardinal