User blog comment:Rgetar/Idea for FGH for larger transfinite ordinals/@comment-35470197-20190702232339/@comment-35470197-20190708060541

> then choose of them strings with minimal...

It is a circular logic. You need the well-foundedness in order to ensure the existence of the least element, you need a well-founded subset of FGH-expression for this purpose, you need the uniqueness argument in order to define such a set, and you need the least element.

> \(S_{j < i}\) are subsets of \(S_i\).

I do not think so, \(S_1\) does not contain a natural number, while \(S_0\) contains all natural numbers.

> \(S_i\) is well-founded

As I wrote above, the definition of the comparison is not complete. Please define the nibary relation \(f_{\alpha_k}^{\beta_k}(\cdots f_{\alpha_1}^{\beta_1}(x)\cdots) < f_{\gamma_m}^{\delta_m}(\cdots f_{\gamma_1}^{\delta_1}(y)\cdots)\) in \(S_i\).

I agree that there are many problems, but one of the biggest problem is that the resulting system never enumerates all ordinal below \(\Omega_2\). It will just enumerate specific ordinals below a fixed limit strictly smaller than \(\Omega_2\).