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When does every definable nonempty set have a definable element? (English) Zbl 07197332
Summary: The assertion that every definable set has a definable element is equivalent over \(\mathsf{ZF}\) to the principle \(\mathbf{V}=\mathbf{HOD}\), and indeed, we prove, so is the assertion merely that every \(\Pi_2\)-definable set has an ordinal-definable element. Meanwhile, every model of \(\mathsf{ZFC}\) has a forcing extension satisfying \(\mathbf{V}\neq\mathbf{HOD}\) in which every \(\Sigma_2\)-definable set has an ordinal-definable element. Similar results hold for \(\mathbf{HOD}(\mathbb{R})\) and \(\mathbf{HOD}(\operatorname{Ord}^\omega)\) and other natural instances of \(\mathbf{HOD}(X)\).
MSC:
03 Mathematical logic and foundations
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