User blog comment:King2218/FGH Things/@comment-25418284-20140305080757

To formalize your work, we can define transfinite iteration as \(f^0 = I\) (identity function), \(f^{\alpha + 1} = f^\alpha \ring f\), and \(f^{\alpha} = \sup\{\beta < \alpha : f^\beta\}\).

We also have to define transfinite fundamental sequences, but I don't see how we can do this. It's simple enough to say that \(f_{\alpha}(\omega) = \sup\{n < \omega | f_{\alpha[n]}(\omega)\}\), but what if we tried to plug in \(\omega + 1\)? What does \(\alpha[\omega + 1]\) mean?