Let's start from where we left at, from φ(...,0,0,0) with an ordinal amount of arguments!
Now there are multiple notations: Lets start with the most popular choice: Buchholz' ψ.
ψ(Ω) = ε0
ψ(Ω+1) = ε0ω
ψ(Ω+2) = ε0*(ω^2)
...
ψ(Ω+ω) = ε0*(ω^ω)
...
ψ(Ω2) = ε1
...
ψ(Ω^2) = ζ0
ψ(Ω^3) = η0
ψ(Ω^ω) = φ(ω,0)
ψ(Ω^ψ(Ω)) = φ(ε0,0)
... and when we reach the limit of raising Ω to countable ordinals we simply use Ω as the exponent itself ...
ψ(Ω^Ω) = φ(1,0,0) = Γ0
ψ(Ω^(Ω2)) = φ(2,0,0)
ψ(Ω^(Ω3)) = φ(3,0,0)
...
ψ(Ω^(Ωω)) = φ(ω,0,0)
... just the same as before until we reach ...
ψ(Ω^Ω^2) = φ(1,0,0,0)
ψ(Ω^Ω^3) = φ(1,0,0,0,0)
ψ(Ω^Ω^4) = φ(1,0,0,0,0,0)
ψ(Ω^Ω^5) = φ(1,0,0,0,0,0,0)
ψ(Ω^Ω^6) = φ(1,0,0,0,0,0,0,0)
...
Now we reach the limit of the Veblen's φ function.
We usually call this ordinal the "Small Veblen Ordinal", or the SVO for short.
...
ψ(Ω^Ω^ω) = SVO
If you know beaf up here I will give some examples
ψ(Ω^Ω^Ω) = LVO
In the FGH this ordinal in beaf would be about {X,X,2(1)2} & X
Why? Why did we only reach ONE LEVEL OF RECURSION?
Remember that there are multiple levels of recursion recursions!
Let's clarify this:
ψ(Ω^Ω^ω)[ψ(Ω^Ω^ω)] < ψ(Ω^Ω^ω) + 1 < ψ(Ω^Ω^ω+1) < ψ(Ω^Ω^Ω)
and it is probably trivial to continue.
Next impasse: ψ(Ω^^ω) for beginners, though that is not the correct way to write.
We call this the "Bachmann-Howard ordinal", or the BHO for short.
BHO = ψ(ε_(Ω+1))
epsilon-capital omega plus one because of the fixed point problem ε_Ω -> Ω.
Ω is bigger than the first epsilon fixed point zeta-0 (basically it IS an epsilon fixed point), thus a +1.
Don't understand yet? remember when we had ε0^ε0^ε0^ε0^ε0^ε0^..., Then we added one to the epsilon number.
remember Γ0^Γ0^Γ0^Γ0^Γ0^Γ0^... was ε(Γ0+1) ? same for all the other big ordinals. now we can apply the same technique with the Ω symbol.
just like when we were with Γ0, we can have ψ(ε_(Ω+2)), ψ(ε_(Ω2)), ψ(ε_(Ω^2)), ψ(ε_(Ω^Ω)), ψ(ε_ε_(Ω+1)), ψ(ε_ε_ε_(Ω+1)), ..., and with zetas, etas, binary phis, multiargument phis, and so on...
examples (assuming base 10 for the fgh)
ψ(ε_(Ω+1)) ~ 10^^10 & 10 & 10
ψ(ε_(Ω+1)+1) ~ (10^^10)+1 & 10 & 10
ψ(ε_(Ω+2)) ~ 10^^10|2 & 10 & 10
ψ(ε_(Ω+ω)) ~ 10^^10|10 & 10 & 10 ~ 10^^(10+1) & 10 & 10
ψ(ε_(Ω2)) ~ 10^^(10*2) & 10 & 10
ψ(ε_(Ωω)) ~ X_2^^(X_2*X) & 10 & 10
ψ(ε_(Ω^2)) ~ X_2^^(X_2^2) & 10 & 10
ψ(ε_(Ω^Ω)) ~ X_2^^(X_2^X_2) & 10 & 10
...
ψ(ε_ε_(Ω+1)) ~ X_2^^(X_2^^X) & 10 & 10 ~ 10^^10^^10 & 10 & 10
ψ(ε_ε_ε_(Ω+1))~ 10^^^4 & 10 & 10
...
and then we have
ψ(ζ_Ω+1) ~ X_2^^^X & 10 & 10
ψ(η_Ω+1) ~ X_2^^^^X & 10 & 10
...
ψ(φ(ω,Ω+1)) ~ {X_2,X_2,X} & X & 10 ~ 3 & X_2 & X & 10
And the ternary phi's...
ψ(Γ_Ω+1) ~ {X_2,X_2,X_2} & X & 10
...
we can do the same with more multiargument phi's until...
{X_2,X(1)2} & X & 10 ~ 10 & 10 & 10 & 10
Oh-oh... We lost the notation! Now ψ(ψ(Ω^Ω^ω)_Ω+1) makes absolutely no sense.
Let's go back to when we were at ψ(ε_(Ω+1)) -- let's see... what about ψ(Ω_2) much simpler!
if we have ψ(Γ_(Ω+1)) we can have ψ(Ω_2^Ω_2) -- get it? Γ0 = ψ(Ω^Ω).
So now we can have ψ(Ω_2^Ω_2^ω) for X & X & X & X -- Now we have a second level of Small Veblen Ordinal!
In the same manner we can reach ψ(Ω_3^Ω_3^ω) for {10,5/2}, ψ(Ω_4^Ω_4^ω) for {10,6/2}, ψ(Ω_5^Ω_5^ω) for {10,7/2}, and so on... until ...?
EBOCF can continue with ordinals as the level.
ψ(Ω_ω) ~ {a,a/2} level.
Congratulations!!! We reached the first point where fgh_ord(a) -> sgh_ord(a) !
but a common mistake is to think that now FGH = SGH beyond this level, that is not true. because
SGH(ψ(Ω_ω)+1) ~ {a,a/2}+1 level and FGH(ψ(Ω_ω)+1) ~ {a,a,{a,a/2}+1} (or mathematically precisely {a,a,hyperlog_a({a,a/2})+1}) (oh no)
EBOCF please save this mess!!!
wait... EBOCF does save this mess. it lets us continue with {X,X/2} + 1 & X structures and similar and even
{X,X^X^2+X+1*3(X^X^2+X+1*3(X^X^2+X+1*3,X^X^2+X+1*3)X^X^2+X+1*3)X^X^2+X+1*3} & X_X+1 | ... | & X & 10 structures.
... @_@;; ...