Selection over classes of ordinals expanded by monadic predicates - Alexander Rabinovich and Amit Shomrat.dvi

A monadic formula ψ(Y) is a selector for a monadic formula φ(Y) in a structure M if ψ defines in M a unique subset P of the domain and this P also satisfies φ in M. If C is a class of structures and φ is a selector for ψ in every M ∈ C, we say φ is a selector for φ over C. For a monadic formula φ(X,Y) and ordinals α ≤ ω1 and δ < 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 φ(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 φ has a selector over it. We deduce the existence of S ⊆ ω such that in the structure (ω, <, S ) every formula has a selector. Given a monadic sentence π and a monadic formula φ(Y), we decide whether φ has a selector over the class of countable ordinals satisfying π, and if so, construct one for it.