Selection over classes of ordinals expanded by monadic predicates

A monadic formula ψ(Y ) is a selector for a monadic formula φ(Y ) in a structure M if ψ defines inM a unique subset P of the domain and this P also satisfies φ inM. If C is a class of structures and φ is a selector for ψ in everyM ∈ C, we say that 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 classC of ordinals to have the property that everymonadic 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. © 2009 Elsevier B.V. All rights reserved.