Selection in the monadic theory of a countable ordinal

Abstract

A monadic formula ψ(Y) is a selector for a formula φ(Y) in a structure ℳ if there exists a unique subset P of ℳ which satisfies ψ and this P also satisfies φ. We show that for every ordinal α≥ ω^ω there are formulas having no selector in the structure (α, <). For α ≤ ω₁, we decide which formulas have a selector in (α, <), and construct selectors for them. We deduce the impossibility of a full generalization of the Büchi-Landweber solvability theorem from (ω, <) to (ω^ω, <). We state a partial extension of that theorem to all countable ordinals. To each formula we assign a selection degree which measures “how difficult it is to select”. We show that in a countable ordinal all non-selectable formulas share the same degree.