Locally Finite ω-Languages and Effective Analytic Sets Have the Same Topological Complexity
Résumé
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^CK there exist some Σ¨0_α-complete and some Π ^0_α-complete locally finite ω-languages, and the supremum of the set of Borel ranks of locally finite ω-languages is the ordinal γ^1_2 , which is strictly greater than the first non-recursive ordinal ω_1^ CK. This gives an answer to the question of the topological complexity of locally finite ω-languages, 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 Π^1_2-complete, hence highly undecidable.
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