User blog comment:Rgetar/Idea for FGH for larger transfinite ordinals/@comment-11227630-20190704063145/@comment-32213734-20190705025626

card(x)
card(x) is cardinality of x:
 * card(x) = x for finite x
 * card(x) = ω for countable x
 * card(x) = Ωα for Ωα ≤ x < Ωα + 1

Cases
I think there may be 8 cases for fα(x).

Case 1
α = 0:
 * f0(x) = x + 1

Case 2
Successor:
 * fα + 1(x) = fαx(x)

Here
 * fα0(x) = x


 * fαβ + 1(x) = fα(fαβ(x))


 * fαβ(x) = sup(fαγ < β(x)) for limit β

Case 3
cof(α) ≤ card(x):
 * fα(x) = sup(fβ < α(x))

Case 4
Case 4 - 8 are for cof(α) > card(x).

There are sequences of cardinals such as
 * ω, Ω, Ω2, Ω3, Ω4, Ω5, ...
 * I, I2, I3, I4, I5, ...
 * I(2, 0), I(2, 1), I(2, 2), I(2, 3), I(2, 4), I(2, 5), ...
 * I(1, 0, 0), I(1, 0, 1), I(1, 0, 2), I(1, 0, 3), I(1, 0, 4), I(1, 0, 5), ...
 * M, M2, M3, M4, M5, ...
 * K, K2, K3, K4, K5, ...
 * ω, L, L2, L3, L4, L5, ...

Generally,
 * A0, A1, A2, A3, A4, A5, ...

Let for infinite x An + 1 is least cardinal of "A" sequence larger than x.

Case 4 is for Ln + 1 < Lm + 1 ≤ cof(α) < Lm + 2:
 * fα(x) = fα[f α(Lm)] (x)

Case 5
An + 2 ≤ Am + 1 = cof(α) < Ln + 1:
 * fα(x) = fα[f α(Am)] (x)

Case 6
Ln + 1 ≤ cof(α) < Ln + 2:
 * fα(x) = An + 1

But which "A"? Currently I have no strictly formulated rule, but general pattern is the larger α the larger "A":
 * fL n + 1 (x) = Ωn + 1
 * fL n + 12 (x) = In + 1
 * fL n + 13 (x) = I(2, n + 1)
 * fL n + 12 (x) = I(1, 0, n + 1)
 * fΩ L n + 1 + 1 (x) = Mn + 1

Maybe it can be defined as follows:
 * fα(x) is least cardinal β such as fα[β](x) = β for cof(α) = Ln + 1
 * fα(x) is least cardinal β such as fα[f α(Ln + 1)[β]] (x) = β for Ln + 1 < cof(α) < Ln + 2

Case 7
cof(α) = fβ(x) < Ln + 1, cof(β) = Ln + 1:
 * fα(x) = fα[f β[x](x)] (x)

Example: let
 * x = ω
 * α = I = fL2(ω)
 * β = L2
 * I[n] = n
 * L2[n] = L + n

then
 * fI(ω) = fI[f L2[ω](ω)] (ω) = ff L + ω(ω) (ω) = sup(ff L(ω) (ω), ff L + 1(ω) (ω), ff L + 2(ω) (ω), ...) = sup(fΩ(ω), fΩ ω (ω), fΦ(1, 0)(ω), fΦ(1, 0, 0)(ω), fΦ(1, 0, 0, 0)(ω), ...)

Case 8
cof(α) = fβ(x) < Ln + 1, Ln + 1 < cof(β) < Ln + 2:
 * fα(x) = fα[f β[f β(Ln + 1)[x]] (x)] (x)

Example: let
 * x = ω
 * α = M = fΩ L + 1 (ω)
 * β = ΩL + 1
 * M[n] = n
 * ΩL + 1[n] = n
 * L[n] = n

then
 * Ln + 1 = L
 * fM(ω) = fM[f Ω L + 1[fΩ L + 1 (L)[ω]] (ω)] (ω) = ff f L(L)[ω] (ω) (ω) = ff f ω(L) (ω) (ω) = sup(ff f 0(L) (ω) (ω), ff f 1(L) (ω) (ω), ff f 2(L) (ω) (ω), ff f 3(L) (ω) (ω), ...) = sup(ff L + 1(ω) (ω), ff L2(ω) (ω), ff L2(ω) (ω), ff ε L2 (ω) (ω), ff ζ L2 (ω) (ω), ...) = sup(fΩ ω (ω), fI(ω), fI(1, 0, 0)(ω), ...)