The envelope of a pointclass under a local determinacy hypothesis.

Given an inductive-like pointclass Γ ˜ and assuming the Axiom of Determinacy, Martin identified and analyzed a pointclass that contains the prewellorderings of the next scale beyond Γ ˜ if such a scale exists. We show that much of Martin's analysis can be carried out assuming only ZF + DC R and Δ ˜ Γ ˜ determinacy by adapting arguments of Kechris and Woodin [10] and Martin [13] . This generalization can be used to show that every set of reals is Suslin in the intersection of two divergent models of A D + , giving a new proof of a theorem of Woodin, as well as to show that every set of reals is Suslin in the derived model at an indestructibly weakly compact limit of Woodin cardinals. [ABSTRACT FROM AUTHOR]