User blog comment:Syst3ms/A formal definition for UNOCF/@comment-35470197-20180728080043/@comment-35470197-20180728103206

If you do not construct the associated ordinal notation simultaneously, then a problem is that the definition of \(\psi_{\kappa}(\ddots \kapps)\) heavily depends on the expression of \(\ddots \kappa\) rather than the value of \(\ddots \kappa\).

For example, if \(\alpha = \ddots_1 \kappa = \ddots_2 \kappa\) for some disctinct expressions \(\ddots_1\) and \(\ddots_2\), then \(\psi_{\kappa}(\alpha)\) is not well-defined unless you show \(\psi_{\kappa}(\ddots_1 \kappa) = \psi_{\kappa}(\ddots_2 \kappa)\). At least, you can formally define a similar rule when you work with an ordinal notation (especially with a recursive subset of standard forms).