Thin Equivalence Relations in Scaled

• Main Author: Ralf Schindler
• Other Authors: The Pennsylvania State University CiteSeerX Archives
• Format: Text
• Language: English
• Published: 2010
• Subjects: sami
• Online Access: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.180.1730; http://wwwmath.uni-muenster.de/math/inst/logik/org/staff/rds/Thin.pdf

We give a new proof via inner model theory that every thin Σ Jα(R) 1 equivalence relation is ∆ Jα(R) 1, where α begins a Σ1 gap and Σ Jα(R) 1 is closed under number quanti cation, assuming AD Jα(R). In the recent past several results previously proved by direct applications of the axiom of determinacy were shown via an inner model theoretic approach. Here we give an inner model theoretic proof of a result of Harrington and Sami [1] on thin equivalence relations. The proof makes it possible to isolate optimal hypotheses in the case that Σ Jα(R) 1 is closed under number quanti cation, where α begins a Σ1 gap. Recall that an equivalence relation E is called thin if there is no perfect set of pairwise E inequivalent reals. Theorem 0.1. Let α ≥ 2 begin a Σ1 gap in L(R). Assume AD Jα(R). Also, setting Γ = Σ Jα(R) 1, assume Γ to be closed under number quanti cation, i.e., ∀ ω Γ ⊂ Γ. Let E be a thin Γ equivalence relation. Let N be an α-suitable mouse with a capturing term for the complete Γ set. Then E is ˘ Γ in any real coding N as a parameter. The notion of α suitable mice with capturing terms (which is due to Woodin), is described in our section 1 and in detail in [6]. Such α suitable mice are in a sense analogues of M # n (capturing Σ 1 n+2) which capture more complicated sets of reals. The pointclass Γ = Σ Jα(R) 1 as in the statement of Theorem 0.1 is scaled