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.

Examples

 * ω: 0, 1, 2, ...
 * ω2: 0, ω, ω2, ...
 * ω3: 0, ω2, ω22, ...
 * ωω: 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, $$\omega^{\varepsilon_0+1}$$, $$\omega^{\omega^{\varepsilon_0+1}}$$, ...
 * εω: ε0, ε1, ε2, ...
 * ζ0: 0, ε0, $$\varepsilon_{\varepsilon_0}$$, ...
 * ζ1: ζ0 + 1, $$\varepsilon_{\zeta_0 + 1}$$, $$\varepsilon_{\varepsilon_{\zeta_0 + 1}}$$, ...
 * ζω: 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), ...
 * Γω: 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), $$\psi(\Omega^{\Omega^{2\Omega^2}})$$, ...
 * 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, ...