# When does every definable nonempty set have a definable element?

## TL;DR

This paper explores the conditions under which definable nonempty sets have definable elements, establishing equivalences with the principle V=HOD and examining models with specific definability properties.

## Contribution

It proves the equivalence of the statement that every definable set has a definable element with V=HOD, and constructs models where certain definability conditions hold despite V≠HOD.

## Key findings

- Equivalence of all definable sets having definable elements and V=HOD.
- Existence of forcing extensions where V≠HOD but definability conditions hold.
- Results extend to HOD of reals and other classes.

## 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 $\text{HOD}(X)$.