Thin Equivalence Relations in Scaled Pointclasses

• Main Author: Ralf Schindler
• Other Authors: The Pennsylvania State University CiteSeerX Archives
• Format: Text
• Language: English
• Published: 2011
• Subjects: sami
• Online Access: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.643.2723; http://www.math.uni-bonn.de/people/schlicht/Thin/thin_revised_03.pdf

An inner model-theoretic proof that every thin ΣJα(R)1 equivalence relation is ∆Jα(R)1 in a certain parameter is presented for ordinals α beginning a Σ1 gap in L(R) where ΣJα(R)1 is closed under number quantification. We use the (optimal) hypothesis ADJα(R). Several results in descriptive set theory proved from determinacy have been studied from an inner model-theoretic perspective [3, 4]. In this line of research we present a new proof of a result of Harrington and Sami [1] on thin equivalence relations. The proof allows us to isolate the optimal hypothesis for the following type of equivalence relation. Theorem 0.1. Let α ≥ 2 begin a Σ1 gap in L(R) and assume ADJα(R). Also, setting Γ = ΣJα(R)1, assume Γ to be closed under number quantification, i.e., ∀NΓ ⊂ Γ. Let E be a thin Γ equivalence relation and N an α-suitable mouse with a capturing term for a universal Γ set. Then E is Γ ̆ in any real coding N as a parameter. An equivalence relation E is called thin if there is no perfect set of pairwiseE–inequivalent