User blog comment:Rgetar/Idea for FGH for larger transfinite ordinals/@comment-25337554-20190705090411/@comment-32213734-20190811124034

No, I have only "general idea". For example, for fα(x) for special case x < L there is sequence of α with cofinalities ≥ L:
 * L, L2, L3, ..., L2, ..., L2 + L, ..., L22, ..., L3, ..., LL, ..., ΩL + 1, ..., ΩL + 1 + L, ..., ΩL + 12, ..., ΩL + 2, ..., IL + 1, ...

And there is sequence of "hierarchies of cardinals":
 * Ω, Ω2, Ω3, ...
 * I, I2, I3, ...
 * I(2, 0), I(2, 1), I(2, 2), ...
 * I(1, 0, 0), I(1, 0, 1), I(1, 0, 2), ...
 * I(1, 1, 0), I(1, 1, 1), I(1, 1, 2), ...
 * I(2, 0, 0), I(2, 0, 1), I(2, 0, 2), ...
 * I(1, 0, 0, 0), I(1, 0, 0, 1), I(1, 0, 0, 2), ...
 * M, M2, M3, ...
 * I(1, 0, 0, 0), I(1, 0, 0, 1), I(1, 0, 0, 2), ...
 * M, M2, M3, ...
 * M, M2, M3, ...
 * M, M2, M3, ...

And elements of the sequence of α correspond to elements of the sequence of "hierarchies of cardinals".

I think that the next "hierarchy of cardinals" should be "large enough" compared to previous "hierarchies of cardinals", but it is not so important which sequence of "hierarchies of cardinals" is used (similarly to "large cardinals" in OCFs). So, I'll just define that fα(x) for cof(α) ≥ L is regular:
 * cof(fα(x)) = fα(x), if cof(α) ≥ L

and
 * if α2 > α1 then fα 2 (x) > fα 1 (x)

but I think that it is not so important which specific cardinal fα(x) is.

For cof(α) ≥ L fα(x) is the least element of "hierarchy of cardinals", corresponding to α, larger than x.

fL32 + L2 + L3(ω) may be I(2, 1, 3, 0)