Selection over classes of ordinals expanded by monadic predicates

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