Borel Hierarchy and Omega Context Free Languages

We give in this paper additional answers to questions of Lescow and Thomas [Logical Specifications of Infinite Computations, In:"A Decade of Concurrency", Springer LNCS 803 (1994), 583-621], proving new topological properties of omega context free languages : there exist some omega-CFL which are non Borel sets. And one cannot decide whether an omega-CFL is a Borel set. We give also an answer to questions of Niwinski and Simonnet about omega powers of finitary languages, giving an example of a finitary context free language L such that L^omega is not a Borel set. Then we prove some recursive analogues to preceding properties: in particular one cannot decide whether an omega-CFL is an arithmetical set.