We will discuss the problem of interpreting fragments of set theory into subsystems of second-order arithmetic. We will see how and why $\mathsf{ATR}_0$ is the weakest subsystem of second-order arithmetic that has a robust interpretation of set theory. We will discuss what can be done in weaker subsystems such as $\mathsf{ACA}_0$ and $\mathsf{ACA}_0^+$. Finally, we will discuss potential applications to reverse mathematics.

We will begin with a review of second-order arithmetic, so familiarity with $\mathsf{ACA}_0$, $\mathsf{ACA}_0^+$, $\mathsf{ATR}_0$ and other subsystems of second-order arithmetic is not necessary.