User blog comment:Rgetar/Idea for FGH for larger transfinite ordinals/@comment-35470197-20190702232339/@comment-32213734-20190706212857

I did not express ordinals. I have a plan (I am not sure will it work):

1. Define a set of FGH expressions.

2. Define a way how to compare two FGH expressions. Two FGH expressions are equal only if they are the same expression. The set of FGH expressions is not well-founded.

3. Define cofinality function cof(α) of expression α.

4. Define fundamental sequence function α[n] of expression α, n < cof(α).

5. Prove that α < α[n] and of n > m then α[n] > α[m].

6. Prove that if α > β then exists n such as α[n] > β[m] for any m.

7. Define a subset of the set of FGH expressions with elements α[n0][n1][n2]...[nk] such as α[n0][n1][n2]...[nk] > α[n0][n1][n2]...[mi] for any mi < ni, i < k.

8. Prove that this subset is well-founded.

9. Define FGH for transfinite ordinals.

10. Prove that this subset corresponds to this FGH.