Some Natural Zero One Laws for Ordinals Below ε_0

摘要

We are going to prove that every ordinal α with ε_0 > α ≥ω~ω satisfies a natural zero one law in the following sense. For α < ε_0 let Nα be the number of occurences of ω in the Cantor normal form of α. (Nα is then the number of edges in the unordered tree which can canonically be associated with α.) We prove that for any a with ω~ω≤ α ≤ε_0 and any sentence ψ in the language of linear orders the limit δ_ψ(α) = lim_(n→∞)#{β<α:(β,∈)|=ψ∧Nβ=n}/#{β<α:Nβ=n} exists and that δ_ψ(α) ∈ {0,1}. We further show that for any such sentence ψ the limit δ_ψ(ε_0) exists although this limit is in general in between 0 and 1. We also investigate corresponding asymptotic densities for ordinals below ω~ω.