Ordinals are a type of number which can describe the sizes of sets.
We start with
0, 1, 2...
And we find the limit, which is ω. And we continue from there.
ω, ω+1, ω+2...
The limit of this sequence is ω2.
We can do the same thing until we reach our limit.
ω^2
We can continue exponentiating ω until we get
ω^ω^ω... or ε_0
Now, we've gone as high as Cantor's normal form allows. What is Cantor's normal form? It is:
α = ω^α_1 + ω^α_2 + … + ω^α_n where where α_1 ≥ α_2 ≥ … ≥ α_n.
Of course, we can't go higher than ζ_0 or ε_ε_ε_ε_ε_ε_ε...
This is where Veblen Phi comes in.
φ(0, β) = ω^β
φ(α+1, β)= the 1+β1+βth ordinal in the set {γ|φ(α, γ) = γ}
If α is a limit ordinal, φ(α, β) = φ(α, β) = the 1+β1+βth ordinal that is in the intersection of the sets {γ|φ(δ, γ) = γ}{γ|φ(δ, γ)= γ} for all δ<α.
Essentially, we are defining fixed points for ordinals.
The Veblen normal form is:
α = φ(β_1, γ_1)+φ(β_2, γ_2)+…+φ(β_n, γ_n) where the terms are decreasing and γ_i<φ(β_i, γ_i) for all i.
Binary Veblen Phi can only go up so far, halting at Γ_0.
TBD