Title: When does every definable nonempty set have a definable element?,
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
$\text{HOD}(X)$.