Projective Hausdorff gaps |
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Authors: | Yurii Khomskii |
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Institution: | 1. Kurt G?del Research Center for Mathematical Logic (KGRC), University of Vienna, W?hringer Strasse 25, 1090, Vienna, Austria
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Abstract: | Todor?evi? (Fund Math 150(1):55–66, 1996) shows that there is no Hausdorff gap (A, B) if A is analytic. In this note we extend the result by showing that the assertion “there is no Hausdorff gap (A, B) if A is coanalytic” is equivalent to “there is no Hausdorff gap (A, B) if A is ${{\bf \it{\Sigma}}^{1}_{2}}$ ”, and equivalent to ${\forall r (\aleph_1^{Lr]}\,< \aleph_1)}$ . We also consider real-valued games corresponding to Hausdorff gaps, and show that ${\mathsf{AD}_\mathbb{R}}$ for pointclasses Γ implies that there are no Hausdorff gaps (A, B) if ${{\it{A}} \in {\bf \it{\Gamma}}}$ . |
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