This will be a talk for the CUNY set theory seminar, October 10, 2014, 12pm GC 6417.

**Abstract**. Although the concept of `being definable’ is not generally expressible in the language of set theory, it turns out that the models of ZF in which every definable nonempty set has a definable element are precisely the models of V=HOD. Indeed, V=HOD is equivalent to the assertion merely that every $\Pi_2$-definable set has an ordinal-definable element. Meanwhile, this is not true in the case of $\Sigma_2$-definability, because 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.

This is joint work with François G. Dorais and Emil Jeřábek, growing out of some questions and answers on MathOverflow, namely,

- Definable collections without definable members
- A question asked by Ashutosh five years ago, in which François and I gradually came upon the answer together.
- Is it consistent that every definable set has a definable member?
- A similar question asked last week by (anonymous) user38200
- Can $V\neq\text{HOD}$ if every $\Sigma_2$-definable set has an ordinal-definable member?
- A question I had regarding the limits of an issue in my answer to the previous question.

In this talk, I shall present the answers to all these questions and place the results in the context of classical results on definability, including a review of basic concepts for graduate students.

Isn’t it fun when MathOverflow (or Math.SE) questions and answers turn into talks?

Oh, yes, and in this case, it is also turning into a short paper!