User:Binary198/Table of fundamental sequences

A fundamental sequence for a limit ordinal is a sequence of ordinals approaching the limit ordinal from below. This article lists (most) of them.

Table

Table
Ordinal	S[0]	S[n]
ω	0	n
ω²	0	ωn
ω^α+1	0	ω^αn
ω^α for limit α	ω^α[0]	ω^α[n]
ω^α + ω^β + ... + ω^x + ω^m+1 for α ≥ β ≥ ... ≥ x ≥ m + 1	ω^α + ω^β + ... + ω^x	ω^α + ω^β + ... + ω^mn
δ^βγ + δ^εζ ... + δ^xy for limit δ	δ^βγ + δ^εζ ... + δ^xy(y-1) + δ^x-1δ[0]	δ^βγ + δ^εζ ... + δ^xy(y-1) + δ^x-1δ[n]
δ^βγ + δ^εζ ... + δ^xy for limit y	δ^βγ + δ^εζ ... + (δ^xy)[0]	δ^βγ + δ^εζ ... + (δ^xy)[n]
δ^βγ + δ^εζ ... + δ^xy for limit x	δ^βγ + δ^εζ ... + δ^x(y-1) + δ^x[0]	δ^βγ + δ^εζ ... + δ^x(y-1) + δ^x[n]
ψ_ν(β+1) in Buchholz's psi	0	ψ_ν(β)n
ε₀	0	$\omega ^{\varepsilon _{0}[n-1]}$
ε_α+1	ε_α + 1	$\omega ^{\varepsilon _{\alpha +1}[n-1]}$
ε_α for limit α < ε_α	ε_α[0]	ε_α[n]
ψ(α+1) in Madore's psi	1	ψ(α) ↑↑ n in Knuth's up arrow notation
ψ(α) in for limit α (with countable cofinality)	ψ(α[0])	ψ(α[n])
ψ(α) in for limit α (with uncountable cofinality)	See ordinal collapsing function	See ordinal collapsing function
φ(α+1, 0)	0	φ(α, φ(α+1, 0)[n-1])
φ(α+1, β+1)	φ(α+1, β) + 1	φ(α, φ(α+1, β+1)[n-1])
φ(α, β) for limit β < φ(α, β)	φ(α, β[0])	φ(α, β[n])
φ(α, 0) for limit α < φ(α, 0)	φ(α[0], 0)	φ(α[n], 0)
φ(α, β+1) for limit α	φ(α[0], φ(α, β) + 1)	φ(α[n], φ(α, β) + 1)
Γ₀	0	φ(Γ₀[n-1], 0)
Γ_α+1	Γ_α + 1	φ(Γ_α+1[n-1], 0)
Γ_α for limit α	Γ_α[0]	Γ_α[n]

ω: 0, 1, 2, ...
ω²: 0, ω, ω2, ...
ω³: 0, ω², ω²2, ...
ω^ω: 1, ω, ω², ...
ω³ + ω³ + ω² + ω: ω³ + ω² + ω², ω³ + ω² + ω² + 1, ω³ + ω² + ω² + 2, ...
ε₀: 0, 1, ω, ...
ε₁: ε₀ + 1, $\omega ^{\varepsilon _{0}+1}$ , $\omega ^{\omega ^{\varepsilon _{0}+1}}$ , ...
ε_ω: ε₀, ε₁, ε₂, ...
ζ₀: 0, ε₀, $\varepsilon _{\varepsilon _{0}}$ , ...
ζ₁: ζ₀ + 1, $\varepsilon _{\zeta _{0}+1}$ , $\varepsilon _{\varepsilon _{\zeta _{0}+1}}$ , ...
ζ_ω: ζ₀, ζ₁, ζ₂, ...
φ(ω, 0): 1, ε₀, ζ₀, ...
φ(ω, 1): $\omega ^{\varphi (\omega ,0)+1}$ , $\varepsilon _{\varphi (\omega ,0)+1}$ , $\zeta _{\varphi (\omega ,0)+1}$ , ...
Γ₀: 0, 1, ε₀, ...
Γ₁: Γ₀ + 1, φ(Γ₀ + 1, 0), φ(φ(Γ₀ + 1, 0), 0), ...
Γ_ω: Γ₀, Γ₁, Γ₂, ...
Ackermann ordinal: ε₀, φ(ε₀, 0, 0), φ(φ(1, 0, ε₀), 0, 0), ...
Small Veblen ordinal: ε₁, φ(1, 0, 0, 0), $\psi (\Omega ^{\Omega ^{2\Omega ^{2}}})$ $\psi (\Omega ^{\Omega ^{2\Omega ^{2}}})$ , ...
- This is the "canonical sequence". One could create a non-canonical, yet more intuitive sequence: ζ₀, Γ₀, φ(1, 0, 0, 0), ...
Large Veblen ordinal: ε₀, $\psi (\Omega ^{\Omega ^{\varepsilon _{0}}})$ , $\psi (\Omega ^{\Omega ^{\psi (\Omega ^{\varepsilon _{0}})}})$ , ...
Bachmann-Howard ordinal: ζ₀, Γ₀, LVO, ...
Buchholz's ordinal: ε₁, ζ₀, BHO, ...