Locally finite ω-languages and effective analytic sets have the same topological complexity

Local sentences and the formal languages they define were introduced by Ressayre in [Res88]. We prove that locally finite ω-languages and effective analytic sets have the same topological complexity: the Borel and Wadge hierarchies of the class of locally finite ω-languages are equal to the Borel and Wadge hierarchies of the class of effective analytic sets. In particular, for each non-null recursive ordinal α < ω 1 there exist some Σ 0 α-complete and some Π 0 αcomplete locally finite ω-languages, and the supremum of the set of Borel ranks of locally finite ω-languages is the ordinal γ 2 , which is strictly greater than the first non-recursive ordinal ω 1 . This gives an answer to the question of the topological complexity of locally finite ωlanguages, which was asked by Simonnet [Sim92] and also by Duparc, Finkel, and Ressayre in [DFR01]. Moreover we show that the topological complexity of a locally finite ω-language defined by a local sentence φ may depend on the models of the Zermelo-Fraenkel axiomatic system ZFC. Using similar constructions as in the proof of the above results we also show that the equivalence, the inclusion, and the universality problems for locally finite ω-languages are Π2-complete, hence highly undecidable.