Quasi-bounded trees and analytic inductions

Abstract

Abstract.A tree T on ω is said to be cofinal if for every α ∈ ω ω there is some branch β of T such that α ≤ β, and quasi-bounded otherwise. We prove that the set of quasi-bounded trees is a complete Σ 1 1 -inductive set. In particular, it is neither analytic nor co-analytic.In a recent joint work with G. Debs, we were led to study the complexity of the set of cofinal trees as a subset of the compact set of all trees on ω, in fact to show that this set is not Π 1 1 . The aim of this paper is to compute the exact complexity of this set, which appears to be beyond the σ-algebra generated by the analytic sets. We also prove similar results concerning the set of cofinal or quasi-bounded closed subsets of the Baire space with respect to the Effros Borel structure on the set F (ω ω ) of closed nonempty subsets of ω ω . Most of the definitions and results we recall here can be found in [4], which we refer to for all undefined notions and basic properties of classical descriptive classes. Sequences and trees.For any set E we denote by Seq(E) the set of finite sequences of elements of E. If s = e 0 , e 1 , . . . , e k−1 ∈ Seq(E) we denote by |s| its length k. As usual, for any two s = e 0 , e 1 , . . . , e k−1 and t = a 0 , a 1 , . . . , a l−1 in Seq(E) we say that t extends s or that s is a beginning of t, and write s ≺ t if |s| < |t| and e i = a i for i < |s|. And we write s t iff s ≺ t or s = t. When s ∈ Seq(E) and k ≤ |s|, we denote by s |k the sequence

An upper bound on the complexity of recognizable tree languages

“…In [MN12], D. Niwinski and H. Michalewski are also interested in the problem of finding upper bounds for the class of regular tree languages. Using a method developed by J. Saint Raymond in [SR06], they prove that the game tree language W 1,3 is complete for the class of Σ 1 1 -inductive sets. The Σ 1 1 -inductive sets are known to contain more complex sets than the σ-algebra generated by the analytic sets.…”

The descriptive complexity of the set of all closed zero-dimensional subsets of a Polish space

“…We mention the following two simple examples: -(Louveau [9]): There exists a G δ set G ⊂ 2 ω × ω ω such that if π denotes the projection mapping on the first coordinate and B = {π(K); K ∈ K(G)} then the σ-ideal of K(2 ω ) generated by B is Σ 0 2 -complete. -(Saint Raymond [12]): Let T ⊂ 2 ω <ω denote the set of all trees on ω. Then the set QB = {T ∈ T : ∃α ∈ ω ω , ∀β ∈ ⌈T ⌉ α ≤ β} (where ≤ denotes the product order on ω ω ) is Σ 0 2 -complete.…”