Selection in the monadic theory of a countable ordinal

A monadic formula (Y ) is a selector for a formula '(Y ) in a structure M if there exists a unique subset P of M which satis es and this P also satis es '. We show that for every ordinal ! there are formulas having no selector in the structure ( ;<). For !1, we decide which formulas have a selector in ( ;<), and construct selectors for them. We deduce the impossibility of a full generalization of the B uchi-Landweber solvability theorem from (!;<) to (!; <). We state a partial extension of that theorem to all countable ordinals. To each formula we assign a selection degree which measures \how di cult it is to select". We show that in a countable ordinal all non-selectable formulas share the same degree.