Some Natural Zero One Laws for Ordinals Below ε 0

We are going to prove that every ordinal α with ε0 > α ≥ ω satisfies a natural zero one law in the following sense. For α < ε0 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 α with ω ≤ α < ε0 and any sentence φ in the language of linear orders the limit δφ(α) = limn→∞ #{β<α:(β,∈)|=φ ∧ Nβ=n} #{β<α:Nβ=n} exists and that δφ(α) ∈ {0, 1}. We further show that for any such sentence φ the limit δφ(ε0) exists although this limit is in general in between 0 and 1. We also investigate corresponding asymptotic densities for ordinals below ω.