Topological and Arithmetical Properties of Infinitary Rational Relations

We prove that there exist some infinitary rational relations which are analytic but non Borel sets, giving an answer to a question of Simonnet [Sim92]. Then we show that for every countable ordinal α one cannot decide whether a given infinitary rational relation is in the Borel class Σα ( respectively Π 0 α). Furthermore one cannot decide whether a given infinitary rational relation is a Borel set or a Σ1-complete set. We prove some recursive analogues to these properties. In particular one cannot decide whether an infinitary rational relation is an arithmetical set. We then deduce from the proof of these results some other ones, like: one cannot decide whether the complement of an infinitary rational relation is also an infinitary rational relation