User blog comment:Wythagoras/Dollar function formal definition/@comment-5529393-20130616091711

Unfortunately, evaluating the subscripts until you reach a number won't do what you want. We do have n$[0]_[0] = n$[0]_n ~ f_phi(omega, 0) (n), but n$[0]_g(n) will equal f_phi(omega, 0) (g(n)). So n$[0]_[0]_[0] = n$[0]_[0]_n ~ f_phi(omega, 0) (f_phi(omega, 0)(n)). So n$[0, 1] = n$[0]_[0]_..._[0] with n [0]'s = f_phi(omega, 0) (f_phi(omega,0 ... (f_phi(omega, 0)(n)...)) = f_{phi(omega,0) + 1} (n), not f_Gamma_0 (n).