Abstract

A monadic formula ψ is a selector for a monadic formula φ in a structure if ψ defines in a unique subset P of the domain and this P also satisfies φ in . If is a class of structures and φ is a selector for ψ in every , we say that φ is a selector for φ over .For a monadic formula φ and ordinals α≤ω1 and δ<ωω, we decide whether there exists a monadic formula ψ such that for every Pα of order-type smaller than δ, ψ selects φ in . If so, we construct such a ψ.We introduce a criterion for a class of ordinals to have the property that every monadic formula φ has a selector over it. We deduce the existence of Sωω such that in the structure every formula has a selector.Given a monadic sentence π and a monadic formula φ, we decide whether φ has a selector over the class of countable ordinals satisfying π, and if so, construct one for it