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Preserving levels of projective determinacy by tree forcings. (English) Zbl 1506.03101

Summary: We prove that various classical tree forcings – for instance Sacks forcing, Mathias forcing, Laver forcing, Miller forcing and Silver forcing – preserve the statement that every real has a sharp and hence analytic determinacy. We then lift this result via methods of inner model theory to obtain level-by-level preservation of projective determinacy (PD). Assuming PD, we further prove that projective generic absoluteness holds and no new equivalence classes are added to thin projective transitive relations by these forcings.

MSC:

03E15 Descriptive set theory
03E35 Consistency and independence results
03E45 Inner models, including constructibility, ordinal definability, and core models
03E55 Large cardinals
03E57 Generic absoluteness and forcing axioms
03E60 Determinacy principles
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