User blog comment:Eners49/A whole new superclass of infinities?/@comment-35470197-20180722021627

First of all, the existence of the largest ordinal number is unprovable under \(\textrm{ZFC}\), the usual set theory. Her, the "largest" means the greatest with respect to \(\in\).

Therefore in order to extend FGH in such a way, you need to define another binary relation. It is very easy.

For example, let \(\overline{\textrm{ON}}\) denote the proper class consisting of all finite sequences of ordinals such that any entry other than the last one is \(\Omega\). Then \(\textrm{ON}\) is naturally regarded as a subclass of \(\overline{\textrm{ON}}\), and the function \(\overline{\textrm{ON}} \to \overline{\textrm{ON}}\) sending \(\alpha\) to the concatenation of \(\Omega\) and \(\alpha) is well-defined. I denote the image of \(\alpha\) by \(\Omega(\alpha)\), which is your function.

I denote by \(\alpha \overline{\in} \beta\) the binary relation on \(\overline{\textrm{ON}}) defined in the following recursive way: Then \((\overline{\textrm{ON}},\overline{\in})\) admits an order-preserving bijective function to \(\omega \times \textrm{ON}\) equipped with the lexicographic order, and hence well-founded.
 * 1) If \(\alpha, \beta \in \textrm{ON}\), then \(\alpha \overline{\in} \beta\) \stackrel{\textrm{def}}{\Leftrightarrow} \alpha \in \beta\).
 * 2) If \(\alpha \in \textrm{ON}\) and \(\beta \notin \textrm{ON}\), then \(\alpha \overline{\in} \beta\) is true.
 * 3) If \(\alpha \notin \textrm{ON}\) and \(\beta \in \textrm{ON}\), then \(\alpha \overline{\in} \beta\) is false.
 * 4) If \(\alpha = \Omega(\alpha')\) and \(\beta = \Omega(\beta')\), then \(\alpha \overline{\in} \beta \stackrel{\textrm{def}}{\Leftrightarrow} \alpha' \overline{\in} \beta'\).

In particular, FGH can be extended in the following recursive way: It is precisely what you defined. I note that the third condition is very weak compared with the usual FGH. So you need to replace it by another strong recursive condition if you want to make use of your "hyperordinal" in definitions of new large numbers.
 * 1) \(f_{\alpha}(x)\) is the usual FGH if \(\alpha \in \textrm{ON}\)
 * 2) \(f_{\Omega^{n+1}(0)}(x) = \sup_{\alpha \in \Omega^{n+1}(0)} f_{\alpha}(x)\)
 * 3) \(f_{\Omega^{n+1}(\alpha+1)}(x) = \Omega^{n+1}(\alpha)\)

In order to do so, it is better to use additional functions symbols other than \(\Omega\) such as \(+\), \(\varphi_{\alpha}^{\beta}\), \(\psi_{\alpha}(\beta)\), and so on. Then you may find that the resulting construction is just the same as what googologists always do when they construct ordinal collapsing functions and ordinal notations.