User blog:Hyp cos/Analysis - BEAF, FGH and SGH (part 3)

Part 2 is here.

Now an ordinal function for SGH-catching-FGH-points comes in. -

Catching function
I use C(α) to stand for this function. It's defined as follows:

Condition α=0: C(0) is first ordinal β such that g_β(n) is comparable to f_β(n).

Condition α=S: C(α+1) is the next ordinal β such that g_β(n) is comparable to f_β(n) after C(α).

Condition α=L: C(α)[n]=C(α[n]).

Also, C(α) is the smallest ordinal β such that g_β(n) is comparable to f_β(n) but it's larger than C(γ) for all γ<α.

For comparisons in part 2, we know C(0)=ψ(Ω_ω), C(1)=ψ(Ω_Ω_ω) and C(ω)=ψ(ψ_I(0)).