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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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