User blog comment:Deedlit11/Ordinal Notations V: Up to a weakly Mahlo cardinal/@comment-24509095-20140509071457/@comment-5529393-20140605085010

I know this is very late, I was gone for a while.

King2218: Like Ikosarakt1 said, it is best not to mix different notations; $$\chi(\alpha) = I_\alpha$$, so we don't really have a $$\chi(\alpha)_2$$ or a $$\chi(\alpha)_{\chi(\alpha)}$$

Ikosarakt1: $$\chi(\alpha)$$ is always an inaccessible cardinal, which is always regular, meaning its cofinality is itself, not $$\omega$$. So we cannot define an $$\omega$$ length fundamental sequences for such cardinals. Sometimes it is useful to define fundamental sequences for ordinals of greater cofinality than $$\omega$$; in such a case, a regular cardinal simply uses itself as a fundamental sequences, i.e. if $$\alpha$$ is a regular cardinal than $$\alpha[\beta] = \beta$$ for all $$\beta < \alpha$$.