Preserving levels of projective determinacy by tree forcings |
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Authors: | Fabiana Castiblanco Philipp Schlicht |
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Institution: | 1. Institut für Mathematische Logik und Grundlagenforschung, Universität Münster, Einsteinstraße 62, 48149 Münster, Germany;2. School of Mathematics, University of Bristol, Fry Building, Woodland Road, Bristol, BS8 1UG, UK |
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Abstract: | 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. |
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Keywords: | Tree forcing Projective determinacy Thin relations Sharps Mouse operator Uniform indiscernibles |
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