**February 27**

**Corey Switzer**,
CUNY

**Sets at arm's length**

We will present the following striking theorem, due to Towsner: If $(M, \mathcal X) \models \mathsf{RCA}_0 + I\Sigma_n$ is countable, then for *any* (!!!) $W \subseteq M$ there is a countable expansion with the same first order part, $(M, \mathcal Y)$, where $\mathcal X \subseteq \mathcal Y$ so that $(M, \mathcal Y) \models \mathsf{RCA}_0 + I\Sigma_n$ and the set $W$ is $\Delta_{n+1}$ definable. Time permitting, we will then sketch some nice applications of this theorem to the study of conservative extensions of fragments of arithmetic.