Selection over classes of ordinals expanded by monadic predicates

Abstract

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 that φ is a selector for φ over C . For a monadic formula φ ( X , Y ) and ordinals α ≤ ω 1 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 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 ω such that in the structure ( ω 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.