## Abstract

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≤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 (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.

Original language | English |
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Pages (from-to) | 1006-1023 |

Number of pages | 18 |

Journal | Annals of Pure and Applied Logic |

Volume | 161 |

Issue number | 8 |

DOIs | |

State | Published - May 2010 |

## Keywords

- Decidability
- Monadic logic of order
- Selection problem
- Uniformization problem