When does every definable nonempty set have a definable element?

François G. Dorais; Joel David Hamkins

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arXiv · 1706.07285

When does every definable nonempty set have a definable element?

François G. Dorais·Joel David Hamkins

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Abstract

The assertion that every definable set has a definable element is equivalent over ZF to the principle $V=\text{HOD}$, and indeed, we prove, so is the assertion merely that every $\Pi_2$-definable set has an ordinal-definable element. Meanwhile, every model of ZFC has a forcing extension satisfying $V\neq\text{HOD}$ in which every $\Sigma_2$-definable set has an ordinal-definable element. Similar results hold for $\text{HOD}(\mathbb{R})$ and $\text{HOD}(\text{Ord}^\omega)$ and other natural instances of $\text{HOD}(X)$.

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BibTeX RIS

François G. Dorais, Joel David Hamkins. 2017-06-22. When does every definable nonempty set have a definable element?. https://arxiv.org/abs/1706.07285

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