Thin equivalence relations in scaled pointclasses

We give a new proof via inner model theory that every thin Σα 1 equivalence relation is ∆α 1 , where α begins a Σ1 gap and Σ Jα(R) 1 is closed under number quanti cation, assuming ADJα(R). In the recent past several results were shown with inner model theory which had been previously proved by direct application of the axiom of determinacy. We show a result of Harrington and Sami [1] about thin equivalence relations with inner model theory from an improved determinacy assumption. 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 ADJα(R). Also, setting Γ = Σ Jα(R) 1 , assume Γ to be closed under number quanti cation, i.e., ∀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 Γ̆ 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 which capture more complicated sets of reals than the projective sets. The pointclass Γ = Σ Jα(R) 1 as in the statement of Theorem 0.1 is scaled under ADJα(R) (cf. [7]). The remaining cases for α which we address in this paper are subsumed in