User blog comment:Wythagoras/Dollar function: final version/@comment-5029411-20131125204130

S(1) = W

S(2) = I

S(n) = Xi(n-1,0)

S(T) = K

S(T+n) = Xi_2(n,0)

S(T*n) = K*n

S(T*n+m) = Xi_{n+1}(m,0)

S(T^2) = K_K_... (limit of compact ordinal notation) = K(2)

K_n(2) = K_K_...K_K_{K_{n-1}(2)+1}

S(T^2)[w*2] = K_2(2)