I'll post the guide once more for everybody who knows nothing about googology.
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ωεζηφαψΓ
0,1,2,3,4,5,6... ω
ω, ω+1, ω+2, ω+3, ω+4,... ω+ω = ω2
ω, ω2, ω3, ω4, ω5,... ω*ω = ω^2
ω^2, ω^3, ω^4, ω^5,... ω^ω
ω, ω^ω, ω^ω^ω, ω^ω^ω^ω,... ω^ω^ω^ω^ω^ω^ω^... = ε_0
0th fixed point of α = ω^α
ε_0+1, ω^(ε_0+1), ω^ω^(ε_0+1), ω^ω^ω^(ε_0+1),... ε_1
1st fixed point of α = ω^α
ε_1+1, ω^(ε_1+1), ω^ω^(ε_1+1), ω^ω^ω^(ε_1+1),... ε_2
2nd fixed point of α = ω^α
ε_0, ε_1, ε_2, ε_3, ε_4, ε_5,... ε_ω,... ε_ε_0
ε_ε_0, ε_ε_1, ε_ε_2, ε_ε_3,... ε_ε_ω
ε_0, ε_ε_0, ε_ε_ε_0, ε_ε_ε_ε_0, ε_ε_ε_ε_ε_0,... ε_ε_ε_ε_ε_ε_ε_... = ζ_0
(Cantor's ordinal) 0th fixed point of α = ε_α
ζ_0+1, ε_(ζ_0+1), ε_ε_(ζ_0+1), ε_ε_ε_(ζ_0+1),... ζ_1
1st fixed point of α = ε_α
ζ_0, ζ_1, ζ_2, ζ_3, ζ_4,... ζ_ω,... ζ_ζ_0
ζ_ζ_0, ζ_ζ_1, ζ_ζ_2,... ζ_ζ_ω
ζ_0, ζ_ζ_0, ζ_ζ_ζ_0, ζ_ζ_ζ_ζ_0,... ζ_ζ_ζ_ζ_ζ_ζ_ζ_ζ_ζ_... = η_0
Table of φ(α,β)
0) 1) 2) 3) fixed point
φ(0, ω ω^ω ω^ω^ω ω^ω^ω^ω ω^ω^ω^ω^ω^... = ε_0 α = φ(0,α)
φ(1, ε_0 ε_1 ε_2 ε_3 ε_ε_ε_ε_... = ζ_0 α = φ(1,α) = ε_α
φ(2, ζ_0 ζ_1 ζ_2 ζ_3 ζ_ζ_ζ_ζ_... = η_0 α = φ(2,α) = ζ_α
φ(3, η_0 η_1 η_2 η_3 η_η_η_η_... = φ(4,0) α = η_α
...
ω = φ(0,0), ε_0 = φ(1,0), ζ_0 = φ(2,0), η_0 = φ(3,0), φ(4,0), φ(5,0),... φ(ω,0)
φ(ω,0), φ(φ(ω,0),0), φ(φ(φ(ω,0),0),0),... φ(φ(φ(φ(... ...),0),0),0),0) = Γ_0 = φ(1,0,0)
(Feferman–Schütte ordinal) 0th fixed point of α = φ(α,0)
Γ_0+1, φ(Γ_0+1,0), φ(φ(Γ_0+1,0),0), φ(φ(φ(Γ_0+1,0),0),0)... Γ_1 = φ(1,0,1)
1st fixed point of α = φ(α,0)
Γ_0 = φ(1,0,0), Γ_1 = φ(1,0,1), Γ_2 = φ(1,0,2), Γ_3, Γ_4, Γ_5,... Γ_ω
Γ_ω, Γ_Γ_ω, Γ_Γ_Γ_ω,... Γ_Γ_Γ_Γ_Γ_Γ_... = φ(1,1,0)
0th fixed point of α = φ(1,0,α)
φ(1,1,0)+1, Γ_(φ(1,1,0)+1), Γ_Γ_(φ(1,1,0)+1), Γ_Γ_Γ_(φ(1,1,0)+1),... φ(1,1,1)
1st fixed point of α = φ(1,0,α)
φ(1,1,0), φ(1,1,1), φ(1,1,2), φ(1,1,3),... φ(1,1,φ(1,1,φ(1,1,φ(1,1,...)))) = φ(1,2,0)
0th fixed point of α = φ(1,1,α)
φ(1,0,0), φ(1,1,0), φ(1,2,0), φ(1,3,0),... φ(1,φ(1,φ(1,φ(1,...,0),0),0),0) = φ(2,0,0)
0th fixed point of α = φ(1,α,0)
φ(1,0,0), φ(2,0,0), φ(3,0,0),... φ(φ(φ(φ(...,0,0),0,0),0,0),0,0), = φ(1,0,0,0)
(Ackermann's ordinal) 0th fixed point of α = φ(α,0,0)
φ(1,0), φ(1,0,0), φ(1,0,0,0), φ(1,0,0,0,0),... φ(1,0,0,0,0,0,0,0,0,...) = SVO
(Small Veblen ordinal)
ω = φ([i]0[/i]), 0 0s (not counting the 1st and here only argument which is a 0 unfortunately)
φ(1,0,0,0,0,0,0,0,...), ω 0s = SVO
φ(1,0,0,0,0,0,0,0,...), SVO 0s
φ(1,0,0,0,0,0,0,0,...), w/ that many 0s
φ(1,0,0,0,0,0,0,0,...), w/ that many 0s
...
φ(1,0,0,0,0,0,0,0,...) (infinite layers) = LVO (Large Veblen ordinal)
0th fixed point of α = φ(1,0,0,0,0,0,0,0,...(w/ α 0s)), supremum of Veblen's hierarchy
Ordinal collapsing function (OCF) - a function used to write large ordinals using even larger uncountable ordinals
Ω = ω_1, the first uncountable ordinal (after any countable ordinal)
Definition of Madore's psi ψ(α)
C(α) = the set of all ordinals constructible using 0, 1, ω and Ω and finite applications of addition, multiplication, exponentiation and ψ(κ) (previously defined values), where κ is contained in C(β) and β < α.
ψ(α) is the smallest ordinal not in the set (NOT the ordinal coming after all of them)
ψ(0) = ε_0 (cannot be written as ω^ω^ω^ω^... in finite applications of exponentiation)
ψ(1) = ε_1 (ψ(0) = ε_0 is now in the set and still ε_1 cannot be constructed)
ψ(α) = ε_α (for α =< ζ_0)
ψ(ζ_0) = ζ_0 (because it's the fixed point of α = ε_α, now ε_α for any α is contained in C(α))
ψ(>ζ_0) = ζ_0 (ζ_0 cannot be in C(α) because it cannot be expressed as ε_ε_ε_ε_... in finite applications, so it's stuck at ζ_0 up to Ω)
ψ(Ω) = ζ_0
ψ(Ω+1) = ε_(ζ_0+1) (Ω, and for that matter, ψ(Ω) = ζ_0, are finally in C(α). This is the next number that cannot be constructed)
ψ(Ω+α) = ε_(ζ_0+α) (α =< ζ_1)
ψ(Ω+ζ_1) = ζ_1 (stuck at ζ_1 up to Ω2)
ψ(Ω+>ζ_1) = ζ_1 (The first ordinal not constructible from ε_α is ζ_1, ζ_α isn't contained in the set of functions yet)
ψ(Ω2) = ζ_1
ψ(Ω(1+α)) = ζ_α (α =< φ(3,0)) (now ζ_α is in the set of functions)
ψ(Ω*Ω) = ψ(Ω^2) = η_0 = φ(3,0)
ψ(Ω^α) = φ((α+1),0) (α =< Γ_0) (now φ(α,0) is in the set)
To be continued...