User blog comment:Deedlit11/Ordinal Notations V: Up to a weakly Mahlo cardinal/@comment-24509095-20140509071457/@comment-5150073-20140605092916

But without FGH and $$\psi$$, $$\chi(\alpha)$$ would be a meaningless symbol for googology. I can imagine that process for evaluating $$f_{\psi_{\chi(\alpha)}(\chi(\alpha))}(n)$$ must go as follows:

$$f_{\psi_{\chi(\alpha)}(\chi(\alpha))}(n) = f_{\psi_{\chi(\alpha)}(\chi(\alpha))[n]}(n) = f_{\psi_{\chi(\alpha)}(\chi(\alpha)[n])}(n)$$

Then the correct step would be evaluate $$\chi(\alpha)[n]$$ to $$\psi_{\chi(\alpha)}(\psi_{\chi(\alpha)}(\psi_{\chi(\alpha)}(\cdots(\psi_{\chi(\alpha)}(0))\cdots)))$$ (with n-1 $$\psi_{\chi(\alpha)}$$'s. I don't see where we must use something like $$\chi(\alpha)[\varepsilon_1]$$.