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Thin equivalence relations in scaled pointclasses
Author(s) -
Schindler Ralf,
Schlicht Philipp
Publication year - 2011
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.200920050
Subject(s) - equivalence relation , combinatorics , equivalence (formal languages) , mathematics , sigma , physics , discrete mathematics , quantum mechanics
Abstract For ordinals α beginning a Σ 1 gap in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathrm{L}(\mathbb {R})$\end{document} , where \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Sigma _{1}^{\mathrm{J}_{\alpha }(\mathbb {R})}$\end{document} is closed under number quantification, we give an inner model‐theoretic proof that every thin \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Sigma _{1}^{\mathrm{J}_{\alpha }(\mathbb {R})}$\end{document} equivalence relation is \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Delta _{1}^{\mathrm{J}_{\alpha }(\mathbb {R})}$\end{document} in a real parameter from the (optimal) hypothesis \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {AD}^{\mathrm{J}_{\alpha }(\mathbb {R})}$\end{document} .