So define ͳ(n) as f@(n) in the fast-growing hierarchy where @ is the biggest ordinal definable using Madore's ordinal trees using no more than n layers of branches, each branch having no more than n branches.
ͳ(x,y) = f@(n) where @ is the biggest ordinal definable using Madore's ordinal trees using no more than x layers of branches, each branch having no more than y branches.
Here are my approximations.
Text-Visualisation of Madore Trees[]
Example:
(0,(0,(1,1,1),(0,(1,1,1,0,2),(0,1,1),0),0,1,1,(0,1,1),2))
(x) = x
(0(0,0)) = ω
(1(0,0)) = ω+1
(x(0,0)) = ω+x
(0,(0,1)) = ω2
(1,(0,1)) = ω2+1
(x,(0,1)) = ω2+x
(y,(0,x)) = ωx+y
(0,(0,(0,0))) = ω2
(0,(0,(0,(0,0)))) = ω3
(0,(1,0)) = ωω
Basically follows nesting rules in left-to-right order, bottom-to-top, grouped by branch.
(0LxBy) = the biggest ordinal definable using Madore's ordinal trees using no more than x layers of branches, each branch having no more than y branches.
(0[6]) = (0,(0,0,0,0,0,0)
(1[6]) = (0,(1,1,1,1,1,1))
(1[3,6]) = (0,((1,1,1),(1,1,1),(1,1,1),(1,1,1),(1,1,1),(1,1,1))
(1[2,3,6]) = (0,(((1,1),(1,1),(1,1)),((1,1),(1,1),(1,1)),((1,1),(1,1),(1,1)),((1,1),(1,1),(1,1)),((1,1),(1,1),(1,1)),((1,1),(1,1),(1,1))))
(2{6}) = (2[2,2,2,2,2,2])
The real thing[]
ͳ(1) = f1(1) = 1
ͳ(2) = fω(2) = 8
ͳ(3) = f(0,(0,(0,0,0),(0,0,0),(0,0,0))(3)
ͳ(4) = f(0,(0,(0,(0,0,0,0),(0,0,0,0),(0,0,0,0),(0,0,0,0)),(0,(0,0,0,0),(0,0,0,0),(0,0,0,0),(0,0,0,0)),(0,(0,0,0,0),(0,0,0,0),(0,0,0,0),(0,0,0,0)),(0,(0,0,0,0),(0,0,0,0),(0,0,0,0),(0,0,0,0)))(4)
Some names of numbers[]
ͳ(10) = Madodec
ͳ(100) = Madohect
ͳ(10,100) = Googomad
ͳ(ͳ(10)) = Dumadohect
ͳ(ͳ(10),ͳ(100)) Googodumad
ͳ(ͳ(ͳ(10),ͳ(10)),(ͳ(100),ͳ(100)) = Googotrimad
ͳ10(100) = Decamadohect
(NOTE: This notation is copied off of a page from the GFE Wiki. I just thought it was a cool demonstration of Madore Trees and OCFs. If you think I am related to or a sockpuppet of Luckyluxiuz, a known banned user, I am not. Please don't judge.
Thanks: Kaplikua Teisinutakentun)